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| 1 | 115 | storres | r""" |
|---|---|---|---|
| 2 | 115 | storres | Sage core functions needed for the implementation of SLZ. |
| 3 | 90 | storres | |
| 4 | 115 | storres | AUTHORS: |
| 5 | 115 | storres | - S.T. (2013-08): initial version |
| 6 | 90 | storres | |
| 7 | 115 | storres | Examples: |
| 8 | 119 | storres | |
| 9 | 119 | storres | TODO:: |
| 10 | 90 | storres | """ |
| 11 | 246 | storres | sys.stderr.write("sageSLZ loading...\n")
|
| 12 | 115 | storres | # |
| 13 | 225 | storres | import inspect |
| 14 | 225 | storres | # |
| 15 | 165 | storres | def slz_compute_binade(number): |
| 16 | 165 | storres | """" |
| 17 | 165 | storres | For a given number, compute the "binade" that is integer m such that |
| 18 | 165 | storres | 2^m <= number < 2^(m+1). If number == 0 return None. |
| 19 | 165 | storres | """ |
| 20 | 165 | storres | # Checking the parameter. |
| 21 | 172 | storres | # The exception construction is used to detect if number is a RealNumber |
| 22 | 165 | storres | # since not all numbers have |
| 23 | 165 | storres | # the mro() method. sage.rings.real_mpfr.RealNumber do. |
| 24 | 165 | storres | try: |
| 25 | 165 | storres | classTree = [number.__class__] + number.mro() |
| 26 | 172 | storres | # If the number is not a RealNumber (or offspring thereof) try |
| 27 | 165 | storres | # to transform it. |
| 28 | 165 | storres | if not sage.rings.real_mpfr.RealNumber in classTree: |
| 29 | 165 | storres | numberAsRR = RR(number) |
| 30 | 165 | storres | else: |
| 31 | 165 | storres | numberAsRR = number |
| 32 | 165 | storres | except AttributeError: |
| 33 | 165 | storres | return None |
| 34 | 165 | storres | # Zero special case. |
| 35 | 165 | storres | if numberAsRR == 0: |
| 36 | 165 | storres | return RR(-infinity) |
| 37 | 165 | storres | else: |
| 38 | 176 | storres | realField = numberAsRR.parent() |
| 39 | 176 | storres | numberLog2 = numberAsRR.abs().log2() |
| 40 | 176 | storres | floorNumberLog2 = floor(numberLog2) |
| 41 | 176 | storres | ## Do not get caught by rounding of log2() both ways. |
| 42 | 176 | storres | ## When numberLog2 is an integer, compare numberAsRR |
| 43 | 176 | storres | # with 2^numberLog2. |
| 44 | 176 | storres | if floorNumberLog2 == numberLog2: |
| 45 | 176 | storres | if numberAsRR.abs() < realField(2^floorNumberLog2): |
| 46 | 176 | storres | return floorNumberLog2 - 1 |
| 47 | 176 | storres | else: |
| 48 | 176 | storres | return floorNumberLog2 |
| 49 | 176 | storres | else: |
| 50 | 176 | storres | return floorNumberLog2 |
| 51 | 165 | storres | # End slz_compute_binade |
| 52 | 165 | storres | |
| 53 | 115 | storres | # |
| 54 | 121 | storres | def slz_compute_binade_bounds(number, emin, emax=sys.maxint): |
| 55 | 119 | storres | """ |
| 56 | 119 | storres | For given "real number", compute the bounds of the binade it belongs to. |
| 57 | 121 | storres | |
| 58 | 121 | storres | NOTE:: |
| 59 | 121 | storres | When number >= 2^(emax+1), we return the "fake" binade |
| 60 | 121 | storres | [2^(emax+1), +infinity]. Ditto for number <= -2^(emax+1) |
| 61 | 125 | storres | with interval [-infinity, -2^(emax+1)]. We want to distinguish |
| 62 | 125 | storres | this case from that of "really" invalid arguments. |
| 63 | 121 | storres | |
| 64 | 119 | storres | """ |
| 65 | 121 | storres | # Check the parameters. |
| 66 | 125 | storres | # RealNumbers or RealNumber offspring only. |
| 67 | 165 | storres | # The exception construction is necessary since not all objects have |
| 68 | 125 | storres | # the mro() method. sage.rings.real_mpfr.RealNumber do. |
| 69 | 124 | storres | try: |
| 70 | 124 | storres | classTree = [number.__class__] + number.mro() |
| 71 | 124 | storres | if not sage.rings.real_mpfr.RealNumber in classTree: |
| 72 | 124 | storres | return None |
| 73 | 124 | storres | except AttributeError: |
| 74 | 121 | storres | return None |
| 75 | 121 | storres | # Non zero negative integers only for emin. |
| 76 | 121 | storres | if emin >= 0 or int(emin) != emin: |
| 77 | 121 | storres | return None |
| 78 | 121 | storres | # Non zero positive integers only for emax. |
| 79 | 121 | storres | if emax <= 0 or int(emax) != emax: |
| 80 | 121 | storres | return None |
| 81 | 121 | storres | precision = number.precision() |
| 82 | 121 | storres | RF = RealField(precision) |
| 83 | 125 | storres | if number == 0: |
| 84 | 125 | storres | return (RF(0),RF(2^(emin)) - RF(2^(emin-precision))) |
| 85 | 121 | storres | # A more precise RealField is needed to avoid unwanted rounding effects |
| 86 | 121 | storres | # when computing number.log2(). |
| 87 | 121 | storres | RRF = RealField(max(2048, 2 * precision)) |
| 88 | 121 | storres | # number = 0 special case, the binade bounds are |
| 89 | 121 | storres | # [0, 2^emin - 2^(emin-precision)] |
| 90 | 121 | storres | # Begin general case |
| 91 | 119 | storres | l2 = RRF(number).abs().log2() |
| 92 | 121 | storres | # Another special one: beyond largest representable -> "Fake" binade. |
| 93 | 121 | storres | if l2 >= emax + 1: |
| 94 | 121 | storres | if number > 0: |
| 95 | 125 | storres | return (RF(2^(emax+1)), RF(+infinity) ) |
| 96 | 121 | storres | else: |
| 97 | 121 | storres | return (RF(-infinity), -RF(2^(emax+1))) |
| 98 | 165 | storres | # Regular case cont'd. |
| 99 | 119 | storres | offset = int(l2) |
| 100 | 121 | storres | # number.abs() >= 1. |
| 101 | 119 | storres | if l2 >= 0: |
| 102 | 119 | storres | if number >= 0: |
| 103 | 119 | storres | lb = RF(2^offset) |
| 104 | 119 | storres | ub = RF(2^(offset + 1) - 2^(-precision+offset+1)) |
| 105 | 119 | storres | else: #number < 0 |
| 106 | 119 | storres | lb = -RF(2^(offset + 1) - 2^(-precision+offset+1)) |
| 107 | 119 | storres | ub = -RF(2^offset) |
| 108 | 121 | storres | else: # log2 < 0, number.abs() < 1. |
| 109 | 119 | storres | if l2 < emin: # Denormal |
| 110 | 121 | storres | # print "Denormal:", l2 |
| 111 | 119 | storres | if number >= 0: |
| 112 | 119 | storres | lb = RF(0) |
| 113 | 119 | storres | ub = RF(2^(emin)) - RF(2^(emin-precision)) |
| 114 | 119 | storres | else: # number <= 0 |
| 115 | 119 | storres | lb = - RF(2^(emin)) + RF(2^(emin-precision)) |
| 116 | 119 | storres | ub = RF(0) |
| 117 | 119 | storres | elif l2 > emin: # Normal number other than +/-2^emin. |
| 118 | 119 | storres | if number >= 0: |
| 119 | 121 | storres | if int(l2) == l2: |
| 120 | 121 | storres | lb = RF(2^(offset)) |
| 121 | 121 | storres | ub = RF(2^(offset+1)) - RF(2^(-precision+offset+1)) |
| 122 | 121 | storres | else: |
| 123 | 121 | storres | lb = RF(2^(offset-1)) |
| 124 | 121 | storres | ub = RF(2^(offset)) - RF(2^(-precision+offset)) |
| 125 | 119 | storres | else: # number < 0 |
| 126 | 121 | storres | if int(l2) == l2: # Binade limit. |
| 127 | 121 | storres | lb = -RF(2^(offset+1) - 2^(-precision+offset+1)) |
| 128 | 121 | storres | ub = -RF(2^(offset)) |
| 129 | 121 | storres | else: |
| 130 | 121 | storres | lb = -RF(2^(offset) - 2^(-precision+offset)) |
| 131 | 121 | storres | ub = -RF(2^(offset-1)) |
| 132 | 121 | storres | else: # l2== emin, number == +/-2^emin |
| 133 | 119 | storres | if number >= 0: |
| 134 | 119 | storres | lb = RF(2^(offset)) |
| 135 | 119 | storres | ub = RF(2^(offset+1)) - RF(2^(-precision+offset+1)) |
| 136 | 119 | storres | else: # number < 0 |
| 137 | 119 | storres | lb = -RF(2^(offset+1) - 2^(-precision+offset+1)) |
| 138 | 119 | storres | ub = -RF(2^(offset)) |
| 139 | 119 | storres | return (lb, ub) |
| 140 | 119 | storres | # End slz_compute_binade_bounds |
| 141 | 119 | storres | # |
| 142 | 123 | storres | def slz_compute_coppersmith_reduced_polynomials(inputPolynomial, |
| 143 | 123 | storres | alpha, |
| 144 | 123 | storres | N, |
| 145 | 123 | storres | iBound, |
| 146 | 223 | storres | tBound, |
| 147 | 223 | storres | debug = False): |
| 148 | 123 | storres | """ |
| 149 | 123 | storres | For a given set of arguments (see below), compute a list |
| 150 | 123 | storres | of "reduced polynomials" that could be used to compute roots |
| 151 | 123 | storres | of the inputPolynomial. |
| 152 | 124 | storres | INPUT: |
| 153 | 124 | storres | |
| 154 | 124 | storres | - "inputPolynomial" -- (no default) a bivariate integer polynomial; |
| 155 | 124 | storres | - "alpha" -- the alpha parameter of the Coppersmith algorithm; |
| 156 | 124 | storres | - "N" -- the modulus; |
| 157 | 124 | storres | - "iBound" -- the bound on the first variable; |
| 158 | 124 | storres | - "tBound" -- the bound on the second variable. |
| 159 | 124 | storres | |
| 160 | 124 | storres | OUTPUT: |
| 161 | 124 | storres | |
| 162 | 124 | storres | A list of bivariate integer polynomial obtained using the Coppersmith |
| 163 | 124 | storres | algorithm. The polynomials correspond to the rows of the LLL-reduce |
| 164 | 124 | storres | reduced base that comply with the Coppersmith condition. |
| 165 | 123 | storres | """ |
| 166 | 123 | storres | # Arguments check. |
| 167 | 123 | storres | if iBound == 0 or tBound == 0: |
| 168 | 179 | storres | return None |
| 169 | 123 | storres | # End arguments check. |
| 170 | 123 | storres | nAtAlpha = N^alpha |
| 171 | 123 | storres | ## Building polynomials for matrix. |
| 172 | 123 | storres | polyRing = inputPolynomial.parent() |
| 173 | 123 | storres | # Whatever the 2 variables are actually called, we call them |
| 174 | 123 | storres | # 'i' and 't' in all the variable names. |
| 175 | 123 | storres | (iVariable, tVariable) = inputPolynomial.variables()[:2] |
| 176 | 123 | storres | #print polyVars[0], type(polyVars[0]) |
| 177 | 123 | storres | initialPolynomial = inputPolynomial.subs({iVariable:iVariable * iBound,
|
| 178 | 123 | storres | tVariable:tVariable * tBound}) |
| 179 | 223 | storres | if debug: |
| 180 | 223 | storres | polynomialsList = \ |
| 181 | 223 | storres | spo_polynomial_to_polynomials_list_8(initialPolynomial, |
| 182 | 223 | storres | alpha, |
| 183 | 223 | storres | N, |
| 184 | 223 | storres | iBound, |
| 185 | 223 | storres | tBound, |
| 186 | 223 | storres | 20) |
| 187 | 223 | storres | else: |
| 188 | 223 | storres | polynomialsList = \ |
| 189 | 223 | storres | spo_polynomial_to_polynomials_list_8(initialPolynomial, |
| 190 | 223 | storres | alpha, |
| 191 | 223 | storres | N, |
| 192 | 223 | storres | iBound, |
| 193 | 223 | storres | tBound, |
| 194 | 223 | storres | 0) |
| 195 | 123 | storres | #print "Polynomials list:", polynomialsList |
| 196 | 123 | storres | ## Building the proto matrix. |
| 197 | 123 | storres | knownMonomials = [] |
| 198 | 123 | storres | protoMatrix = [] |
| 199 | 223 | storres | if debug: |
| 200 | 223 | storres | for poly in polynomialsList: |
| 201 | 223 | storres | spo_add_polynomial_coeffs_to_matrix_row(poly, |
| 202 | 223 | storres | knownMonomials, |
| 203 | 223 | storres | protoMatrix, |
| 204 | 223 | storres | 20) |
| 205 | 223 | storres | else: |
| 206 | 223 | storres | for poly in polynomialsList: |
| 207 | 223 | storres | spo_add_polynomial_coeffs_to_matrix_row(poly, |
| 208 | 223 | storres | knownMonomials, |
| 209 | 223 | storres | protoMatrix, |
| 210 | 223 | storres | 0) |
| 211 | 123 | storres | matrixToReduce = spo_proto_to_row_matrix(protoMatrix) |
| 212 | 123 | storres | #print matrixToReduce |
| 213 | 123 | storres | ## Reduction and checking. |
| 214 | 163 | storres | ## S.T. changed 'fp' to None as of Sage 6.6 complying to |
| 215 | 163 | storres | # error message issued when previous code was used. |
| 216 | 163 | storres | #reducedMatrix = matrixToReduce.LLL(fp='fp') |
| 217 | 163 | storres | reducedMatrix = matrixToReduce.LLL(fp=None) |
| 218 | 123 | storres | isLLLReduced = reducedMatrix.is_LLL_reduced() |
| 219 | 123 | storres | if not isLLLReduced: |
| 220 | 179 | storres | return None |
| 221 | 123 | storres | monomialsCount = len(knownMonomials) |
| 222 | 123 | storres | monomialsCountSqrt = sqrt(monomialsCount) |
| 223 | 123 | storres | #print "Monomials count:", monomialsCount, monomialsCountSqrt.n() |
| 224 | 123 | storres | #print reducedMatrix |
| 225 | 123 | storres | ## Check the Coppersmith condition for each row and build the reduced |
| 226 | 123 | storres | # polynomials. |
| 227 | 123 | storres | ccReducedPolynomialsList = [] |
| 228 | 123 | storres | for row in reducedMatrix.rows(): |
| 229 | 123 | storres | l2Norm = row.norm(2) |
| 230 | 123 | storres | if (l2Norm * monomialsCountSqrt) < nAtAlpha: |
| 231 | 123 | storres | #print (l2Norm * monomialsCountSqrt).n() |
| 232 | 125 | storres | #print l2Norm.n() |
| 233 | 123 | storres | ccReducedPolynomial = \ |
| 234 | 123 | storres | slz_compute_reduced_polynomial(row, |
| 235 | 123 | storres | knownMonomials, |
| 236 | 123 | storres | iVariable, |
| 237 | 123 | storres | iBound, |
| 238 | 123 | storres | tVariable, |
| 239 | 123 | storres | tBound) |
| 240 | 123 | storres | if not ccReducedPolynomial is None: |
| 241 | 123 | storres | ccReducedPolynomialsList.append(ccReducedPolynomial) |
| 242 | 123 | storres | else: |
| 243 | 125 | storres | #print l2Norm.n() , ">", nAtAlpha |
| 244 | 123 | storres | pass |
| 245 | 123 | storres | if len(ccReducedPolynomialsList) < 2: |
| 246 | 125 | storres | print "Less than 2 Coppersmith condition compliant vectors." |
| 247 | 123 | storres | return () |
| 248 | 125 | storres | #print ccReducedPolynomialsList |
| 249 | 123 | storres | return ccReducedPolynomialsList |
| 250 | 123 | storres | # End slz_compute_coppersmith_reduced_polynomials |
| 251 | 243 | storres | # |
| 252 | 243 | storres | def slz_compute_coppersmith_reduced_polynomials_gram(inputPolynomial, |
| 253 | 243 | storres | alpha, |
| 254 | 243 | storres | N, |
| 255 | 243 | storres | iBound, |
| 256 | 243 | storres | tBound, |
| 257 | 243 | storres | debug = False): |
| 258 | 243 | storres | """ |
| 259 | 243 | storres | For a given set of arguments (see below), compute a list |
| 260 | 243 | storres | of "reduced polynomials" that could be used to compute roots |
| 261 | 243 | storres | of the inputPolynomial. |
| 262 | 243 | storres | INPUT: |
| 263 | 243 | storres | |
| 264 | 243 | storres | - "inputPolynomial" -- (no default) a bivariate integer polynomial; |
| 265 | 243 | storres | - "alpha" -- the alpha parameter of the Coppersmith algorithm; |
| 266 | 243 | storres | - "N" -- the modulus; |
| 267 | 243 | storres | - "iBound" -- the bound on the first variable; |
| 268 | 243 | storres | - "tBound" -- the bound on the second variable. |
| 269 | 243 | storres | |
| 270 | 243 | storres | OUTPUT: |
| 271 | 243 | storres | |
| 272 | 243 | storres | A list of bivariate integer polynomial obtained using the Coppersmith |
| 273 | 243 | storres | algorithm. The polynomials correspond to the rows of the LLL-reduce |
| 274 | 243 | storres | reduced base that comply with the Coppersmith condition. |
| 275 | 243 | storres | """ |
| 276 | 243 | storres | # Arguments check. |
| 277 | 243 | storres | if iBound == 0 or tBound == 0: |
| 278 | 243 | storres | return None |
| 279 | 243 | storres | # End arguments check. |
| 280 | 243 | storres | nAtAlpha = N^alpha |
| 281 | 243 | storres | ## Building polynomials for matrix. |
| 282 | 243 | storres | polyRing = inputPolynomial.parent() |
| 283 | 243 | storres | # Whatever the 2 variables are actually called, we call them |
| 284 | 243 | storres | # 'i' and 't' in all the variable names. |
| 285 | 243 | storres | (iVariable, tVariable) = inputPolynomial.variables()[:2] |
| 286 | 243 | storres | #print polyVars[0], type(polyVars[0]) |
| 287 | 243 | storres | initialPolynomial = inputPolynomial.subs({iVariable:iVariable * iBound,
|
| 288 | 243 | storres | tVariable:tVariable * tBound}) |
| 289 | 243 | storres | if debug: |
| 290 | 243 | storres | polynomialsList = \ |
| 291 | 243 | storres | spo_polynomial_to_polynomials_list_8(initialPolynomial, |
| 292 | 243 | storres | alpha, |
| 293 | 243 | storres | N, |
| 294 | 243 | storres | iBound, |
| 295 | 243 | storres | tBound, |
| 296 | 243 | storres | 20) |
| 297 | 243 | storres | else: |
| 298 | 243 | storres | polynomialsList = \ |
| 299 | 243 | storres | spo_polynomial_to_polynomials_list_8(initialPolynomial, |
| 300 | 243 | storres | alpha, |
| 301 | 243 | storres | N, |
| 302 | 243 | storres | iBound, |
| 303 | 243 | storres | tBound, |
| 304 | 243 | storres | 0) |
| 305 | 243 | storres | #print "Polynomials list:", polynomialsList |
| 306 | 243 | storres | ## Building the proto matrix. |
| 307 | 243 | storres | knownMonomials = [] |
| 308 | 243 | storres | protoMatrix = [] |
| 309 | 243 | storres | if debug: |
| 310 | 243 | storres | for poly in polynomialsList: |
| 311 | 243 | storres | spo_add_polynomial_coeffs_to_matrix_row(poly, |
| 312 | 243 | storres | knownMonomials, |
| 313 | 243 | storres | protoMatrix, |
| 314 | 243 | storres | 20) |
| 315 | 243 | storres | else: |
| 316 | 243 | storres | for poly in polynomialsList: |
| 317 | 243 | storres | spo_add_polynomial_coeffs_to_matrix_row(poly, |
| 318 | 243 | storres | knownMonomials, |
| 319 | 243 | storres | protoMatrix, |
| 320 | 243 | storres | 0) |
| 321 | 243 | storres | matrixToReduce = spo_proto_to_row_matrix(protoMatrix) |
| 322 | 243 | storres | #print matrixToReduce |
| 323 | 243 | storres | ## Reduction and checking. |
| 324 | 243 | storres | ### In this variant we use the Pari LLL_gram reduction function. |
| 325 | 243 | storres | # It works with the Gram matrix instead of the plain matrix. |
| 326 | 243 | storres | matrixToReduceTransposed = matrixToReduce.transpose() |
| 327 | 243 | storres | matrixToReduceGram = matrixToReduce * matrixToReduceTransposed |
| 328 | 243 | storres | ### LLL_gram returns a unimodular transformation matrix such that: |
| 329 | 243 | storres | # umt.transpose() * matrixToReduce * umt is reduced.. |
| 330 | 243 | storres | umt = matrixToReduceGram.LLL_gram() |
| 331 | 243 | storres | #print "Unimodular transformation matrix:" |
| 332 | 243 | storres | #for row in umt.rows(): |
| 333 | 243 | storres | # print row |
| 334 | 243 | storres | ### The computed transformation matrix is transposed and applied to the |
| 335 | 243 | storres | # left side of matrixToReduce. |
| 336 | 243 | storres | reducedMatrix = umt.transpose() * matrixToReduce |
| 337 | 243 | storres | #print "Reduced matrix:" |
| 338 | 243 | storres | #for row in reducedMatrix.rows(): |
| 339 | 243 | storres | # print row |
| 340 | 243 | storres | isLLLReduced = reducedMatrix.is_LLL_reduced() |
| 341 | 243 | storres | #if not isLLLReduced: |
| 342 | 243 | storres | # return None |
| 343 | 243 | storres | monomialsCount = len(knownMonomials) |
| 344 | 243 | storres | monomialsCountSqrt = sqrt(monomialsCount) |
| 345 | 243 | storres | #print "Monomials count:", monomialsCount, monomialsCountSqrt.n() |
| 346 | 243 | storres | #print reducedMatrix |
| 347 | 243 | storres | ## Check the Coppersmith condition for each row and build the reduced |
| 348 | 243 | storres | # polynomials. |
| 349 | 243 | storres | ccReducedPolynomialsList = [] |
| 350 | 243 | storres | for row in reducedMatrix.rows(): |
| 351 | 243 | storres | l2Norm = row.norm(2) |
| 352 | 243 | storres | if (l2Norm * monomialsCountSqrt) < nAtAlpha: |
| 353 | 243 | storres | #print (l2Norm * monomialsCountSqrt).n() |
| 354 | 243 | storres | #print l2Norm.n() |
| 355 | 243 | storres | ccReducedPolynomial = \ |
| 356 | 243 | storres | slz_compute_reduced_polynomial(row, |
| 357 | 243 | storres | knownMonomials, |
| 358 | 243 | storres | iVariable, |
| 359 | 243 | storres | iBound, |
| 360 | 243 | storres | tVariable, |
| 361 | 243 | storres | tBound) |
| 362 | 243 | storres | if not ccReducedPolynomial is None: |
| 363 | 243 | storres | ccReducedPolynomialsList.append(ccReducedPolynomial) |
| 364 | 243 | storres | else: |
| 365 | 243 | storres | #print l2Norm.n() , ">", nAtAlpha |
| 366 | 243 | storres | pass |
| 367 | 243 | storres | if len(ccReducedPolynomialsList) < 2: |
| 368 | 243 | storres | print "Less than 2 Coppersmith condition compliant vectors." |
| 369 | 243 | storres | return () |
| 370 | 243 | storres | #print ccReducedPolynomialsList |
| 371 | 243 | storres | return ccReducedPolynomialsList |
| 372 | 243 | storres | # End slz_compute_coppersmith_reduced_polynomials_gram |
| 373 | 243 | storres | # |
| 374 | 247 | storres | def slz_compute_coppersmith_reduced_polynomials_proj(inputPolynomial, |
| 375 | 247 | storres | alpha, |
| 376 | 247 | storres | N, |
| 377 | 247 | storres | iBound, |
| 378 | 247 | storres | tBound, |
| 379 | 247 | storres | debug = False): |
| 380 | 247 | storres | """ |
| 381 | 247 | storres | For a given set of arguments (see below), compute a list |
| 382 | 247 | storres | of "reduced polynomials" that could be used to compute roots |
| 383 | 247 | storres | of the inputPolynomial. |
| 384 | 247 | storres | INPUT: |
| 385 | 247 | storres | |
| 386 | 247 | storres | - "inputPolynomial" -- (no default) a bivariate integer polynomial; |
| 387 | 247 | storres | - "alpha" -- the alpha parameter of the Coppersmith algorithm; |
| 388 | 247 | storres | - "N" -- the modulus; |
| 389 | 247 | storres | - "iBound" -- the bound on the first variable; |
| 390 | 247 | storres | - "tBound" -- the bound on the second variable. |
| 391 | 247 | storres | |
| 392 | 247 | storres | OUTPUT: |
| 393 | 247 | storres | |
| 394 | 247 | storres | A list of bivariate integer polynomial obtained using the Coppersmith |
| 395 | 247 | storres | algorithm. The polynomials correspond to the rows of the LLL-reduce |
| 396 | 247 | storres | reduced base that comply with the Coppersmith condition. |
| 397 | 247 | storres | """ |
| 398 | 247 | storres | #@par Changes from runSLZ-113.sage |
| 399 | 247 | storres | # LLL reduction is not performed on the matrix itself but rather on the |
| 400 | 247 | storres | # product of the matrix with a uniform random matrix. |
| 401 | 247 | storres | # The reduced matrix obtained is discarded but the transformation matrix |
| 402 | 247 | storres | # obtained is used to multiply the original matrix in order to reduced it. |
| 403 | 247 | storres | # If a sufficient level of reduction is obtained, we stop here. If not |
| 404 | 247 | storres | # the product matrix obtained above is LLL reduced. But as it has been |
| 405 | 247 | storres | # pre-reduced at the above step, reduction is supposed to be much faster. |
| 406 | 247 | storres | # |
| 407 | 247 | storres | # Arguments check. |
| 408 | 247 | storres | if iBound == 0 or tBound == 0: |
| 409 | 247 | storres | return None |
| 410 | 247 | storres | # End arguments check. |
| 411 | 247 | storres | nAtAlpha = N^alpha |
| 412 | 247 | storres | ## Building polynomials for matrix. |
| 413 | 247 | storres | polyRing = inputPolynomial.parent() |
| 414 | 247 | storres | # Whatever the 2 variables are actually called, we call them |
| 415 | 247 | storres | # 'i' and 't' in all the variable names. |
| 416 | 247 | storres | (iVariable, tVariable) = inputPolynomial.variables()[:2] |
| 417 | 247 | storres | #print polyVars[0], type(polyVars[0]) |
| 418 | 247 | storres | initialPolynomial = inputPolynomial.subs({iVariable:iVariable * iBound,
|
| 419 | 247 | storres | tVariable:tVariable * tBound}) |
| 420 | 247 | storres | if debug: |
| 421 | 247 | storres | polynomialsList = \ |
| 422 | 247 | storres | spo_polynomial_to_polynomials_list_8(initialPolynomial, |
| 423 | 247 | storres | alpha, |
| 424 | 247 | storres | N, |
| 425 | 247 | storres | iBound, |
| 426 | 247 | storres | tBound, |
| 427 | 247 | storres | 20) |
| 428 | 247 | storres | else: |
| 429 | 247 | storres | polynomialsList = \ |
| 430 | 247 | storres | spo_polynomial_to_polynomials_list_8(initialPolynomial, |
| 431 | 247 | storres | alpha, |
| 432 | 247 | storres | N, |
| 433 | 247 | storres | iBound, |
| 434 | 247 | storres | tBound, |
| 435 | 247 | storres | 0) |
| 436 | 247 | storres | #print "Polynomials list:", polynomialsList |
| 437 | 247 | storres | ## Building the proto matrix. |
| 438 | 247 | storres | knownMonomials = [] |
| 439 | 247 | storres | protoMatrix = [] |
| 440 | 247 | storres | if debug: |
| 441 | 247 | storres | for poly in polynomialsList: |
| 442 | 247 | storres | spo_add_polynomial_coeffs_to_matrix_row(poly, |
| 443 | 247 | storres | knownMonomials, |
| 444 | 247 | storres | protoMatrix, |
| 445 | 247 | storres | 20) |
| 446 | 247 | storres | else: |
| 447 | 247 | storres | for poly in polynomialsList: |
| 448 | 247 | storres | spo_add_polynomial_coeffs_to_matrix_row(poly, |
| 449 | 247 | storres | knownMonomials, |
| 450 | 247 | storres | protoMatrix, |
| 451 | 247 | storres | 0) |
| 452 | 247 | storres | matrixToReduce = spo_proto_to_row_matrix(protoMatrix) |
| 453 | 247 | storres | #print matrixToReduce |
| 454 | 247 | storres | ## Reduction and checking. |
| 455 | 247 | storres | ### Reduction with projection |
| 456 | 249 | storres | (reducedMatrixStep1, reductionMatrixStep1) = \ |
| 457 | 249 | storres | slz_reduce_lll_proj(matrixToReduce,16) |
| 458 | 247 | storres | #print "Reduced matrix:" |
| 459 | 249 | storres | #print reducedMatrixStep1 |
| 460 | 247 | storres | #for row in reducedMatrix.rows(): |
| 461 | 247 | storres | # print row |
| 462 | 247 | storres | monomialsCount = len(knownMonomials) |
| 463 | 247 | storres | monomialsCountSqrt = sqrt(monomialsCount) |
| 464 | 247 | storres | #print "Monomials count:", monomialsCount, monomialsCountSqrt.n() |
| 465 | 247 | storres | #print reducedMatrix |
| 466 | 247 | storres | ## Check the Coppersmith condition for each row and build the reduced |
| 467 | 247 | storres | # polynomials. |
| 468 | 247 | storres | ccReducedPolynomialsList = [] |
| 469 | 249 | storres | for row in reducedMatrixStep1.rows(): |
| 470 | 249 | storres | l2Norm = row.norm(2) |
| 471 | 249 | storres | if (l2Norm * monomialsCountSqrt) < nAtAlpha: |
| 472 | 249 | storres | #print (l2Norm * monomialsCountSqrt).n() |
| 473 | 249 | storres | #print l2Norm.n() |
| 474 | 249 | storres | ccReducedPolynomial = \ |
| 475 | 249 | storres | slz_compute_reduced_polynomial(row, |
| 476 | 249 | storres | knownMonomials, |
| 477 | 249 | storres | iVariable, |
| 478 | 249 | storres | iBound, |
| 479 | 249 | storres | tVariable, |
| 480 | 249 | storres | tBound) |
| 481 | 249 | storres | if not ccReducedPolynomial is None: |
| 482 | 249 | storres | ccReducedPolynomialsList.append(ccReducedPolynomial) |
| 483 | 249 | storres | else: |
| 484 | 249 | storres | #print l2Norm.n() , ">", nAtAlpha |
| 485 | 249 | storres | pass |
| 486 | 249 | storres | if len(ccReducedPolynomialsList) < 2: # Insufficient reduction. |
| 487 | 249 | storres | print "Less than 2 Coppersmith condition compliant vectors." |
| 488 | 249 | storres | print "Extra reduction starting..." |
| 489 | 249 | storres | reducedMatrix = reducedMatrixStep1.LLL(algorithm='fpLLL:wrapper') |
| 490 | 253 | storres | ### If uncommented, the following statement avoids performing |
| 491 | 253 | storres | # an actual LLL reduction. This allows for demonstrating |
| 492 | 253 | storres | # the behavior of our pseudo-reduction alone. |
| 493 | 253 | storres | #return () |
| 494 | 249 | storres | else: |
| 495 | 253 | storres | print "First step of reduction afforded enough vectors" |
| 496 | 249 | storres | return ccReducedPolynomialsList |
| 497 | 249 | storres | #print ccReducedPolynomialsList |
| 498 | 249 | storres | ## Check again the Coppersmith condition for each row and build the reduced |
| 499 | 249 | storres | # polynomials. |
| 500 | 249 | storres | ccReducedPolynomialsList = [] |
| 501 | 247 | storres | for row in reducedMatrix.rows(): |
| 502 | 247 | storres | l2Norm = row.norm(2) |
| 503 | 247 | storres | if (l2Norm * monomialsCountSqrt) < nAtAlpha: |
| 504 | 247 | storres | #print (l2Norm * monomialsCountSqrt).n() |
| 505 | 247 | storres | #print l2Norm.n() |
| 506 | 247 | storres | ccReducedPolynomial = \ |
| 507 | 247 | storres | slz_compute_reduced_polynomial(row, |
| 508 | 247 | storres | knownMonomials, |
| 509 | 247 | storres | iVariable, |
| 510 | 247 | storres | iBound, |
| 511 | 247 | storres | tVariable, |
| 512 | 247 | storres | tBound) |
| 513 | 247 | storres | if not ccReducedPolynomial is None: |
| 514 | 247 | storres | ccReducedPolynomialsList.append(ccReducedPolynomial) |
| 515 | 247 | storres | else: |
| 516 | 247 | storres | #print l2Norm.n() , ">", nAtAlpha |
| 517 | 247 | storres | pass |
| 518 | 249 | storres | if len(ccReducedPolynomialsList) < 2: # Insufficient reduction. |
| 519 | 253 | storres | print "Less than 2 Coppersmith condition compliant vectors after extra reduction." |
| 520 | 247 | storres | return () |
| 521 | 249 | storres | else: |
| 522 | 249 | storres | return ccReducedPolynomialsList |
| 523 | 247 | storres | # End slz_compute_coppersmith_reduced_polynomials_proj |
| 524 | 277 | storres | # |
| 525 | 277 | storres | def slz_compute_weak_coppersmith_reduced_polynomials_proj_02(inputPolynomial, |
| 526 | 277 | storres | alpha, |
| 527 | 277 | storres | N, |
| 528 | 277 | storres | iBound, |
| 529 | 277 | storres | tBound, |
| 530 | 277 | storres | debug = False): |
| 531 | 277 | storres | """ |
| 532 | 277 | storres | For a given set of arguments (see below), compute a list |
| 533 | 277 | storres | of "reduced polynomials" that could be used to compute roots |
| 534 | 277 | storres | of the inputPolynomial. |
| 535 | 277 | storres | INPUT: |
| 536 | 277 | storres | |
| 537 | 277 | storres | - "inputPolynomial" -- (no default) a bivariate integer polynomial; |
| 538 | 277 | storres | - "alpha" -- the alpha parameter of the Coppersmith algorithm; |
| 539 | 277 | storres | - "N" -- the modulus; |
| 540 | 277 | storres | - "iBound" -- the bound on the first variable; |
| 541 | 277 | storres | - "tBound" -- the bound on the second variable. |
| 542 | 277 | storres | |
| 543 | 277 | storres | OUTPUT: |
| 544 | 277 | storres | |
| 545 | 277 | storres | A list of bivariate integer polynomial obtained using the Coppersmith |
| 546 | 277 | storres | algorithm. The polynomials correspond to the rows of the LLL-reduce |
| 547 | 277 | storres | reduced base that comply with the weak version of Coppersmith condition. |
| 548 | 277 | storres | """ |
| 549 | 277 | storres | #@par Changes from runSLZ-113.sage |
| 550 | 277 | storres | # LLL reduction is not performed on the matrix itself but rather on the |
| 551 | 277 | storres | # product of the matrix with a uniform random matrix. |
| 552 | 277 | storres | # The reduced matrix obtained is discarded but the transformation matrix |
| 553 | 277 | storres | # obtained is used to multiply the original matrix in order to reduced it. |
| 554 | 277 | storres | # If a sufficient level of reduction is obtained, we stop here. If not |
| 555 | 277 | storres | # the product matrix obtained above is LLL reduced. But as it has been |
| 556 | 277 | storres | # pre-reduced at the above step, reduction is supposed to be much faster. |
| 557 | 277 | storres | # |
| 558 | 277 | storres | # Arguments check. |
| 559 | 277 | storres | if iBound == 0 or tBound == 0: |
| 560 | 277 | storres | return None |
| 561 | 277 | storres | # End arguments check. |
| 562 | 277 | storres | nAtAlpha = N^alpha |
| 563 | 277 | storres | ## Building polynomials for matrix. |
| 564 | 277 | storres | polyRing = inputPolynomial.parent() |
| 565 | 277 | storres | # Whatever the 2 variables are actually called, we call them |
| 566 | 277 | storres | # 'i' and 't' in all the variable names. |
| 567 | 277 | storres | (iVariable, tVariable) = inputPolynomial.variables()[:2] |
| 568 | 277 | storres | #print polyVars[0], type(polyVars[0]) |
| 569 | 277 | storres | initialPolynomial = inputPolynomial.subs({iVariable:iVariable * iBound,
|
| 570 | 277 | storres | tVariable:tVariable * tBound}) |
| 571 | 277 | storres | if debug: |
| 572 | 277 | storres | polynomialsList = \ |
| 573 | 277 | storres | spo_polynomial_to_polynomials_list_8(initialPolynomial, |
| 574 | 277 | storres | alpha, |
| 575 | 277 | storres | N, |
| 576 | 277 | storres | iBound, |
| 577 | 277 | storres | tBound, |
| 578 | 277 | storres | 20) |
| 579 | 277 | storres | else: |
| 580 | 277 | storres | polynomialsList = \ |
| 581 | 277 | storres | spo_polynomial_to_polynomials_list_8(initialPolynomial, |
| 582 | 277 | storres | alpha, |
| 583 | 277 | storres | N, |
| 584 | 277 | storres | iBound, |
| 585 | 277 | storres | tBound, |
| 586 | 277 | storres | 0) |
| 587 | 277 | storres | #print "Polynomials list:", polynomialsList |
| 588 | 277 | storres | ## Building the proto matrix. |
| 589 | 277 | storres | knownMonomials = [] |
| 590 | 277 | storres | protoMatrix = [] |
| 591 | 277 | storres | if debug: |
| 592 | 277 | storres | for poly in polynomialsList: |
| 593 | 277 | storres | spo_add_polynomial_coeffs_to_matrix_row(poly, |
| 594 | 277 | storres | knownMonomials, |
| 595 | 277 | storres | protoMatrix, |
| 596 | 277 | storres | 20) |
| 597 | 277 | storres | else: |
| 598 | 277 | storres | for poly in polynomialsList: |
| 599 | 277 | storres | spo_add_polynomial_coeffs_to_matrix_row(poly, |
| 600 | 277 | storres | knownMonomials, |
| 601 | 277 | storres | protoMatrix, |
| 602 | 277 | storres | 0) |
| 603 | 277 | storres | matrixToReduce = spo_proto_to_row_matrix(protoMatrix) |
| 604 | 277 | storres | #print matrixToReduce |
| 605 | 277 | storres | ## Reduction and checking. |
| 606 | 277 | storres | ### Reduction with projection |
| 607 | 277 | storres | (reducedMatrixStep1, reductionMatrixStep1) = \ |
| 608 | 277 | storres | slz_reduce_lll_proj_02(matrixToReduce,16) |
| 609 | 277 | storres | #print "Reduced matrix:" |
| 610 | 277 | storres | #print reducedMatrixStep1 |
| 611 | 277 | storres | #for row in reducedMatrix.rows(): |
| 612 | 277 | storres | # print row |
| 613 | 277 | storres | monomialsCount = len(knownMonomials) |
| 614 | 277 | storres | monomialsCountSqrt = sqrt(monomialsCount) |
| 615 | 277 | storres | #print "Monomials count:", monomialsCount, monomialsCountSqrt.n() |
| 616 | 277 | storres | #print reducedMatrix |
| 617 | 277 | storres | ## Check the Coppersmith condition for each row and build the reduced |
| 618 | 277 | storres | # polynomials. |
| 619 | 277 | storres | ccReducedPolynomialsList = [] |
| 620 | 277 | storres | for row in reducedMatrixStep1.rows(): |
| 621 | 277 | storres | l1Norm = row.norm(1) |
| 622 | 277 | storres | l2Norm = row.norm(2) |
| 623 | 277 | storres | if l2Norm * monomialsCountSqrt < l1Norm: |
| 624 | 277 | storres | print "l1norm is smaller than l2norm*sqrt(w)." |
| 625 | 277 | storres | else: |
| 626 | 277 | storres | print "l1norm is NOT smaller than l2norm*sqrt(w)." |
| 627 | 277 | storres | print (l2Norm * monomialsCountSqrt).n(), 'vs ', l1Norm.n() |
| 628 | 277 | storres | if l1Norm < nAtAlpha: |
| 629 | 277 | storres | #print l2Norm.n() |
| 630 | 277 | storres | ccReducedPolynomial = \ |
| 631 | 277 | storres | slz_compute_reduced_polynomial(row, |
| 632 | 277 | storres | knownMonomials, |
| 633 | 277 | storres | iVariable, |
| 634 | 277 | storres | iBound, |
| 635 | 277 | storres | tVariable, |
| 636 | 277 | storres | tBound) |
| 637 | 277 | storres | if not ccReducedPolynomial is None: |
| 638 | 277 | storres | ccReducedPolynomialsList.append(ccReducedPolynomial) |
| 639 | 277 | storres | else: |
| 640 | 277 | storres | #print l2Norm.n() , ">", nAtAlpha |
| 641 | 277 | storres | pass |
| 642 | 277 | storres | if len(ccReducedPolynomialsList) < 2: # Insufficient reduction. |
| 643 | 277 | storres | print "Less than 2 Coppersmith condition compliant vectors." |
| 644 | 277 | storres | print "Extra reduction starting..." |
| 645 | 277 | storres | reducedMatrix = reducedMatrixStep1.LLL(algorithm='fpLLL:wrapper') |
| 646 | 277 | storres | ### If uncommented, the following statement avoids performing |
| 647 | 277 | storres | # an actual LLL reduction. This allows for demonstrating |
| 648 | 277 | storres | # the behavior of our pseudo-reduction alone. |
| 649 | 277 | storres | #return () |
| 650 | 277 | storres | else: |
| 651 | 277 | storres | print "First step of reduction afforded enough vectors" |
| 652 | 277 | storres | return ccReducedPolynomialsList |
| 653 | 277 | storres | #print ccReducedPolynomialsList |
| 654 | 277 | storres | ## Check again the Coppersmith condition for each row and build the reduced |
| 655 | 277 | storres | # polynomials. |
| 656 | 277 | storres | ccReducedPolynomialsList = [] |
| 657 | 277 | storres | for row in reducedMatrix.rows(): |
| 658 | 277 | storres | l2Norm = row.norm(2) |
| 659 | 277 | storres | if (l2Norm * monomialsCountSqrt) < nAtAlpha: |
| 660 | 277 | storres | #print (l2Norm * monomialsCountSqrt).n() |
| 661 | 277 | storres | #print l2Norm.n() |
| 662 | 277 | storres | ccReducedPolynomial = \ |
| 663 | 277 | storres | slz_compute_reduced_polynomial(row, |
| 664 | 277 | storres | knownMonomials, |
| 665 | 277 | storres | iVariable, |
| 666 | 277 | storres | iBound, |
| 667 | 277 | storres | tVariable, |
| 668 | 277 | storres | tBound) |
| 669 | 277 | storres | if not ccReducedPolynomial is None: |
| 670 | 277 | storres | ccReducedPolynomialsList.append(ccReducedPolynomial) |
| 671 | 277 | storres | else: |
| 672 | 277 | storres | #print l2Norm.n() , ">", nAtAlpha |
| 673 | 277 | storres | pass |
| 674 | 277 | storres | if len(ccReducedPolynomialsList) < 2: # Insufficient reduction. |
| 675 | 277 | storres | print "Less than 2 Coppersmith condition compliant vectors after extra reduction." |
| 676 | 277 | storres | return () |
| 677 | 277 | storres | else: |
| 678 | 277 | storres | return ccReducedPolynomialsList |
| 679 | 277 | storres | # End slz_compute_coppersmith_reduced_polynomials_proj_02 |
| 680 | 277 | storres | # |
| 681 | 277 | storres | def slz_compute_coppersmith_reduced_polynomials_proj(inputPolynomial, |
| 682 | 277 | storres | alpha, |
| 683 | 277 | storres | N, |
| 684 | 277 | storres | iBound, |
| 685 | 277 | storres | tBound, |
| 686 | 277 | storres | debug = False): |
| 687 | 277 | storres | """ |
| 688 | 277 | storres | For a given set of arguments (see below), compute a list |
| 689 | 277 | storres | of "reduced polynomials" that could be used to compute roots |
| 690 | 277 | storres | of the inputPolynomial. |
| 691 | 277 | storres | INPUT: |
| 692 | 277 | storres | |
| 693 | 277 | storres | - "inputPolynomial" -- (no default) a bivariate integer polynomial; |
| 694 | 277 | storres | - "alpha" -- the alpha parameter of the Coppersmith algorithm; |
| 695 | 277 | storres | - "N" -- the modulus; |
| 696 | 277 | storres | - "iBound" -- the bound on the first variable; |
| 697 | 277 | storres | - "tBound" -- the bound on the second variable. |
| 698 | 277 | storres | |
| 699 | 277 | storres | OUTPUT: |
| 700 | 277 | storres | |
| 701 | 277 | storres | A list of bivariate integer polynomial obtained using the Coppersmith |
| 702 | 277 | storres | algorithm. The polynomials correspond to the rows of the LLL-reduce |
| 703 | 277 | storres | reduced base that comply with the Coppersmith condition. |
| 704 | 277 | storres | """ |
| 705 | 277 | storres | #@par Changes from runSLZ-113.sage |
| 706 | 277 | storres | # LLL reduction is not performed on the matrix itself but rather on the |
| 707 | 277 | storres | # product of the matrix with a uniform random matrix. |
| 708 | 277 | storres | # The reduced matrix obtained is discarded but the transformation matrix |
| 709 | 277 | storres | # obtained is used to multiply the original matrix in order to reduced it. |
| 710 | 277 | storres | # If a sufficient level of reduction is obtained, we stop here. If not |
| 711 | 277 | storres | # the product matrix obtained above is LLL reduced. But as it has been |
| 712 | 277 | storres | # pre-reduced at the above step, reduction is supposed to be much faster. |
| 713 | 277 | storres | # |
| 714 | 277 | storres | # Arguments check. |
| 715 | 277 | storres | if iBound == 0 or tBound == 0: |
| 716 | 277 | storres | return None |
| 717 | 277 | storres | # End arguments check. |
| 718 | 277 | storres | nAtAlpha = N^alpha |
| 719 | 277 | storres | ## Building polynomials for matrix. |
| 720 | 277 | storres | polyRing = inputPolynomial.parent() |
| 721 | 277 | storres | # Whatever the 2 variables are actually called, we call them |
| 722 | 277 | storres | # 'i' and 't' in all the variable names. |
| 723 | 277 | storres | (iVariable, tVariable) = inputPolynomial.variables()[:2] |
| 724 | 277 | storres | #print polyVars[0], type(polyVars[0]) |
| 725 | 277 | storres | initialPolynomial = inputPolynomial.subs({iVariable:iVariable * iBound,
|
| 726 | 277 | storres | tVariable:tVariable * tBound}) |
| 727 | 277 | storres | if debug: |
| 728 | 277 | storres | polynomialsList = \ |
| 729 | 277 | storres | spo_polynomial_to_polynomials_list_8(initialPolynomial, |
| 730 | 277 | storres | alpha, |
| 731 | 277 | storres | N, |
| 732 | 277 | storres | iBound, |
| 733 | 277 | storres | tBound, |
| 734 | 277 | storres | 20) |
| 735 | 277 | storres | else: |
| 736 | 277 | storres | polynomialsList = \ |
| 737 | 277 | storres | spo_polynomial_to_polynomials_list_8(initialPolynomial, |
| 738 | 277 | storres | alpha, |
| 739 | 277 | storres | N, |
| 740 | 277 | storres | iBound, |
| 741 | 277 | storres | tBound, |
| 742 | 277 | storres | 0) |
| 743 | 277 | storres | #print "Polynomials list:", polynomialsList |
| 744 | 277 | storres | ## Building the proto matrix. |
| 745 | 277 | storres | knownMonomials = [] |
| 746 | 277 | storres | protoMatrix = [] |
| 747 | 277 | storres | if debug: |
| 748 | 277 | storres | for poly in polynomialsList: |
| 749 | 277 | storres | spo_add_polynomial_coeffs_to_matrix_row(poly, |
| 750 | 277 | storres | knownMonomials, |
| 751 | 277 | storres | protoMatrix, |
| 752 | 277 | storres | 20) |
| 753 | 277 | storres | else: |
| 754 | 277 | storres | for poly in polynomialsList: |
| 755 | 277 | storres | spo_add_polynomial_coeffs_to_matrix_row(poly, |
| 756 | 277 | storres | knownMonomials, |
| 757 | 277 | storres | protoMatrix, |
| 758 | 277 | storres | 0) |
| 759 | 277 | storres | matrixToReduce = spo_proto_to_row_matrix(protoMatrix) |
| 760 | 277 | storres | #print matrixToReduce |
| 761 | 277 | storres | ## Reduction and checking. |
| 762 | 277 | storres | ### Reduction with projection |
| 763 | 277 | storres | (reducedMatrixStep1, reductionMatrixStep1) = \ |
| 764 | 277 | storres | slz_reduce_lll_proj(matrixToReduce,16) |
| 765 | 277 | storres | #print "Reduced matrix:" |
| 766 | 277 | storres | #print reducedMatrixStep1 |
| 767 | 277 | storres | #for row in reducedMatrix.rows(): |
| 768 | 277 | storres | # print row |
| 769 | 277 | storres | monomialsCount = len(knownMonomials) |
| 770 | 277 | storres | monomialsCountSqrt = sqrt(monomialsCount) |
| 771 | 277 | storres | #print "Monomials count:", monomialsCount, monomialsCountSqrt.n() |
| 772 | 277 | storres | #print reducedMatrix |
| 773 | 277 | storres | ## Check the Coppersmith condition for each row and build the reduced |
| 774 | 277 | storres | # polynomials. |
| 775 | 277 | storres | ccReducedPolynomialsList = [] |
| 776 | 277 | storres | for row in reducedMatrixStep1.rows(): |
| 777 | 277 | storres | l2Norm = row.norm(2) |
| 778 | 277 | storres | if (l2Norm * monomialsCountSqrt) < nAtAlpha: |
| 779 | 277 | storres | #print (l2Norm * monomialsCountSqrt).n() |
| 780 | 277 | storres | #print l2Norm.n() |
| 781 | 277 | storres | ccReducedPolynomial = \ |
| 782 | 277 | storres | slz_compute_reduced_polynomial(row, |
| 783 | 277 | storres | knownMonomials, |
| 784 | 277 | storres | iVariable, |
| 785 | 277 | storres | iBound, |
| 786 | 277 | storres | tVariable, |
| 787 | 277 | storres | tBound) |
| 788 | 277 | storres | if not ccReducedPolynomial is None: |
| 789 | 277 | storres | ccReducedPolynomialsList.append(ccReducedPolynomial) |
| 790 | 277 | storres | else: |
| 791 | 277 | storres | #print l2Norm.n() , ">", nAtAlpha |
| 792 | 277 | storres | pass |
| 793 | 277 | storres | if len(ccReducedPolynomialsList) < 2: # Insufficient reduction. |
| 794 | 277 | storres | print "Less than 2 Coppersmith condition compliant vectors." |
| 795 | 277 | storres | print "Extra reduction starting..." |
| 796 | 277 | storres | reducedMatrix = reducedMatrixStep1.LLL(algorithm='fpLLL:wrapper') |
| 797 | 277 | storres | ### If uncommented, the following statement avoids performing |
| 798 | 277 | storres | # an actual LLL reduction. This allows for demonstrating |
| 799 | 277 | storres | # the behavior of our pseudo-reduction alone. |
| 800 | 277 | storres | #return () |
| 801 | 277 | storres | else: |
| 802 | 277 | storres | print "First step of reduction afforded enough vectors" |
| 803 | 277 | storres | return ccReducedPolynomialsList |
| 804 | 277 | storres | #print ccReducedPolynomialsList |
| 805 | 277 | storres | ## Check again the Coppersmith condition for each row and build the reduced |
| 806 | 277 | storres | # polynomials. |
| 807 | 277 | storres | ccReducedPolynomialsList = [] |
| 808 | 277 | storres | for row in reducedMatrix.rows(): |
| 809 | 277 | storres | l2Norm = row.norm(2) |
| 810 | 277 | storres | if (l2Norm * monomialsCountSqrt) < nAtAlpha: |
| 811 | 277 | storres | #print (l2Norm * monomialsCountSqrt).n() |
| 812 | 277 | storres | #print l2Norm.n() |
| 813 | 277 | storres | ccReducedPolynomial = \ |
| 814 | 277 | storres | slz_compute_reduced_polynomial(row, |
| 815 | 277 | storres | knownMonomials, |
| 816 | 277 | storres | iVariable, |
| 817 | 277 | storres | iBound, |
| 818 | 277 | storres | tVariable, |
| 819 | 277 | storres | tBound) |
| 820 | 277 | storres | if not ccReducedPolynomial is None: |
| 821 | 277 | storres | ccReducedPolynomialsList.append(ccReducedPolynomial) |
| 822 | 277 | storres | else: |
| 823 | 277 | storres | #print l2Norm.n() , ">", nAtAlpha |
| 824 | 277 | storres | pass |
| 825 | 277 | storres | if len(ccReducedPolynomialsList) < 2: # Insufficient reduction. |
| 826 | 277 | storres | print "Less than 2 Coppersmith condition compliant vectors after extra reduction." |
| 827 | 277 | storres | return () |
| 828 | 277 | storres | else: |
| 829 | 277 | storres | return ccReducedPolynomialsList |
| 830 | 277 | storres | # End slz_compute_coppersmith_reduced_polynomials_proj |
| 831 | 274 | storres | def slz_compute_weak_coppersmith_reduced_polynomials_proj(inputPolynomial, |
| 832 | 274 | storres | alpha, |
| 833 | 274 | storres | N, |
| 834 | 274 | storres | iBound, |
| 835 | 274 | storres | tBound, |
| 836 | 274 | storres | debug = False): |
| 837 | 274 | storres | """ |
| 838 | 274 | storres | For a given set of arguments (see below), compute a list |
| 839 | 274 | storres | of "reduced polynomials" that could be used to compute roots |
| 840 | 274 | storres | of the inputPolynomial. |
| 841 | 274 | storres | INPUT: |
| 842 | 274 | storres | |
| 843 | 274 | storres | - "inputPolynomial" -- (no default) a bivariate integer polynomial; |
| 844 | 274 | storres | - "alpha" -- the alpha parameter of the Coppersmith algorithm; |
| 845 | 274 | storres | - "N" -- the modulus; |
| 846 | 274 | storres | - "iBound" -- the bound on the first variable; |
| 847 | 274 | storres | - "tBound" -- the bound on the second variable. |
| 848 | 274 | storres | |
| 849 | 274 | storres | OUTPUT: |
| 850 | 274 | storres | |
| 851 | 274 | storres | A list of bivariate integer polynomial obtained using the Coppersmith |
| 852 | 274 | storres | algorithm. The polynomials correspond to the rows of the LLL-reduce |
| 853 | 274 | storres | reduced base that comply with the weak version of Coppersmith condition. |
| 854 | 274 | storres | """ |
| 855 | 274 | storres | #@par Changes from runSLZ-113.sage |
| 856 | 274 | storres | # LLL reduction is not performed on the matrix itself but rather on the |
| 857 | 274 | storres | # product of the matrix with a uniform random matrix. |
| 858 | 274 | storres | # The reduced matrix obtained is discarded but the transformation matrix |
| 859 | 274 | storres | # obtained is used to multiply the original matrix in order to reduced it. |
| 860 | 274 | storres | # If a sufficient level of reduction is obtained, we stop here. If not |
| 861 | 274 | storres | # the product matrix obtained above is LLL reduced. But as it has been |
| 862 | 274 | storres | # pre-reduced at the above step, reduction is supposed to be much faster. |
| 863 | 274 | storres | # |
| 864 | 274 | storres | # Arguments check. |
| 865 | 274 | storres | if iBound == 0 or tBound == 0: |
| 866 | 274 | storres | return None |
| 867 | 274 | storres | # End arguments check. |
| 868 | 274 | storres | nAtAlpha = N^alpha |
| 869 | 274 | storres | ## Building polynomials for matrix. |
| 870 | 274 | storres | polyRing = inputPolynomial.parent() |
| 871 | 274 | storres | # Whatever the 2 variables are actually called, we call them |
| 872 | 274 | storres | # 'i' and 't' in all the variable names. |
| 873 | 274 | storres | (iVariable, tVariable) = inputPolynomial.variables()[:2] |
| 874 | 274 | storres | #print polyVars[0], type(polyVars[0]) |
| 875 | 274 | storres | initialPolynomial = inputPolynomial.subs({iVariable:iVariable * iBound,
|
| 876 | 274 | storres | tVariable:tVariable * tBound}) |
| 877 | 274 | storres | if debug: |
| 878 | 274 | storres | polynomialsList = \ |
| 879 | 274 | storres | spo_polynomial_to_polynomials_list_8(initialPolynomial, |
| 880 | 274 | storres | alpha, |
| 881 | 274 | storres | N, |
| 882 | 274 | storres | iBound, |
| 883 | 274 | storres | tBound, |
| 884 | 274 | storres | 20) |
| 885 | 274 | storres | else: |
| 886 | 274 | storres | polynomialsList = \ |
| 887 | 274 | storres | spo_polynomial_to_polynomials_list_8(initialPolynomial, |
| 888 | 274 | storres | alpha, |
| 889 | 274 | storres | N, |
| 890 | 274 | storres | iBound, |
| 891 | 274 | storres | tBound, |
| 892 | 274 | storres | 0) |
| 893 | 274 | storres | #print "Polynomials list:", polynomialsList |
| 894 | 274 | storres | ## Building the proto matrix. |
| 895 | 274 | storres | knownMonomials = [] |
| 896 | 274 | storres | protoMatrix = [] |
| 897 | 274 | storres | if debug: |
| 898 | 274 | storres | for poly in polynomialsList: |
| 899 | 274 | storres | spo_add_polynomial_coeffs_to_matrix_row(poly, |
| 900 | 274 | storres | knownMonomials, |
| 901 | 274 | storres | protoMatrix, |
| 902 | 274 | storres | 20) |
| 903 | 274 | storres | else: |
| 904 | 274 | storres | for poly in polynomialsList: |
| 905 | 274 | storres | spo_add_polynomial_coeffs_to_matrix_row(poly, |
| 906 | 274 | storres | knownMonomials, |
| 907 | 274 | storres | protoMatrix, |
| 908 | 274 | storres | 0) |
| 909 | 274 | storres | matrixToReduce = spo_proto_to_row_matrix(protoMatrix) |
| 910 | 274 | storres | #print matrixToReduce |
| 911 | 274 | storres | ## Reduction and checking. |
| 912 | 274 | storres | ### Reduction with projection |
| 913 | 274 | storres | (reducedMatrixStep1, reductionMatrixStep1) = \ |
| 914 | 274 | storres | slz_reduce_lll_proj(matrixToReduce,16) |
| 915 | 274 | storres | #print "Reduced matrix:" |
| 916 | 274 | storres | #print reducedMatrixStep1 |
| 917 | 274 | storres | #for row in reducedMatrix.rows(): |
| 918 | 274 | storres | # print row |
| 919 | 274 | storres | monomialsCount = len(knownMonomials) |
| 920 | 274 | storres | monomialsCountSqrt = sqrt(monomialsCount) |
| 921 | 274 | storres | #print "Monomials count:", monomialsCount, monomialsCountSqrt.n() |
| 922 | 274 | storres | #print reducedMatrix |
| 923 | 274 | storres | ## Check the Coppersmith condition for each row and build the reduced |
| 924 | 274 | storres | # polynomials. |
| 925 | 274 | storres | ccReducedPolynomialsList = [] |
| 926 | 274 | storres | for row in reducedMatrixStep1.rows(): |
| 927 | 274 | storres | l1Norm = row.norm(1) |
| 928 | 274 | storres | l2Norm = row.norm(2) |
| 929 | 274 | storres | if l2Norm * monomialsCountSqrt < l1Norm: |
| 930 | 274 | storres | print "l1norm is smaller than l2norm*sqrt(w)." |
| 931 | 274 | storres | else: |
| 932 | 274 | storres | print "l1norm is NOT smaller than l2norm*sqrt(w)." |
| 933 | 274 | storres | print (l2Norm * monomialsCountSqrt).n(), 'vs ', l1Norm.n() |
| 934 | 274 | storres | if l1Norm < nAtAlpha: |
| 935 | 274 | storres | #print l2Norm.n() |
| 936 | 274 | storres | ccReducedPolynomial = \ |
| 937 | 274 | storres | slz_compute_reduced_polynomial(row, |
| 938 | 274 | storres | knownMonomials, |
| 939 | 274 | storres | iVariable, |
| 940 | 274 | storres | iBound, |
| 941 | 274 | storres | tVariable, |
| 942 | 274 | storres | tBound) |
| 943 | 274 | storres | if not ccReducedPolynomial is None: |
| 944 | 274 | storres | ccReducedPolynomialsList.append(ccReducedPolynomial) |
| 945 | 274 | storres | else: |
| 946 | 274 | storres | #print l2Norm.n() , ">", nAtAlpha |
| 947 | 274 | storres | pass |
| 948 | 274 | storres | if len(ccReducedPolynomialsList) < 2: # Insufficient reduction. |
| 949 | 274 | storres | print "Less than 2 Coppersmith condition compliant vectors." |
| 950 | 274 | storres | print "Extra reduction starting..." |
| 951 | 274 | storres | reducedMatrix = reducedMatrixStep1.LLL(algorithm='fpLLL:wrapper') |
| 952 | 274 | storres | ### If uncommented, the following statement avoids performing |
| 953 | 274 | storres | # an actual LLL reduction. This allows for demonstrating |
| 954 | 274 | storres | # the behavior of our pseudo-reduction alone. |
| 955 | 274 | storres | #return () |
| 956 | 274 | storres | else: |
| 957 | 274 | storres | print "First step of reduction afforded enough vectors" |
| 958 | 274 | storres | return ccReducedPolynomialsList |
| 959 | 274 | storres | #print ccReducedPolynomialsList |
| 960 | 274 | storres | ## Check again the Coppersmith condition for each row and build the reduced |
| 961 | 274 | storres | # polynomials. |
| 962 | 274 | storres | ccReducedPolynomialsList = [] |
| 963 | 274 | storres | for row in reducedMatrix.rows(): |
| 964 | 274 | storres | l2Norm = row.norm(2) |
| 965 | 274 | storres | if (l2Norm * monomialsCountSqrt) < nAtAlpha: |
| 966 | 274 | storres | #print (l2Norm * monomialsCountSqrt).n() |
| 967 | 274 | storres | #print l2Norm.n() |
| 968 | 274 | storres | ccReducedPolynomial = \ |
| 969 | 274 | storres | slz_compute_reduced_polynomial(row, |
| 970 | 274 | storres | knownMonomials, |
| 971 | 274 | storres | iVariable, |
| 972 | 274 | storres | iBound, |
| 973 | 274 | storres | tVariable, |
| 974 | 274 | storres | tBound) |
| 975 | 274 | storres | if not ccReducedPolynomial is None: |
| 976 | 274 | storres | ccReducedPolynomialsList.append(ccReducedPolynomial) |
| 977 | 274 | storres | else: |
| 978 | 274 | storres | #print l2Norm.n() , ">", nAtAlpha |
| 979 | 274 | storres | pass |
| 980 | 274 | storres | if len(ccReducedPolynomialsList) < 2: # Insufficient reduction. |
| 981 | 274 | storres | print "Less than 2 Coppersmith condition compliant vectors after extra reduction." |
| 982 | 274 | storres | return () |
| 983 | 274 | storres | else: |
| 984 | 274 | storres | return ccReducedPolynomialsList |
| 985 | 277 | storres | # End slz_compute_coppersmith_reduced_polynomials_proj2 |
| 986 | 247 | storres | # |
| 987 | 212 | storres | def slz_compute_coppersmith_reduced_polynomials_with_lattice_volume(inputPolynomial, |
| 988 | 212 | storres | alpha, |
| 989 | 212 | storres | N, |
| 990 | 212 | storres | iBound, |
| 991 | 229 | storres | tBound, |
| 992 | 229 | storres | debug = False): |
| 993 | 212 | storres | """ |
| 994 | 212 | storres | For a given set of arguments (see below), compute a list |
| 995 | 212 | storres | of "reduced polynomials" that could be used to compute roots |
| 996 | 212 | storres | of the inputPolynomial. |
| 997 | 212 | storres | Print the volume of the initial basis as well. |
| 998 | 212 | storres | INPUT: |
| 999 | 212 | storres | |
| 1000 | 212 | storres | - "inputPolynomial" -- (no default) a bivariate integer polynomial; |
| 1001 | 212 | storres | - "alpha" -- the alpha parameter of the Coppersmith algorithm; |
| 1002 | 212 | storres | - "N" -- the modulus; |
| 1003 | 212 | storres | - "iBound" -- the bound on the first variable; |
| 1004 | 212 | storres | - "tBound" -- the bound on the second variable. |
| 1005 | 212 | storres | |
| 1006 | 212 | storres | OUTPUT: |
| 1007 | 212 | storres | |
| 1008 | 212 | storres | A list of bivariate integer polynomial obtained using the Coppersmith |
| 1009 | 212 | storres | algorithm. The polynomials correspond to the rows of the LLL-reduce |
| 1010 | 212 | storres | reduced base that comply with the Coppersmith condition. |
| 1011 | 212 | storres | """ |
| 1012 | 212 | storres | # Arguments check. |
| 1013 | 212 | storres | if iBound == 0 or tBound == 0: |
| 1014 | 212 | storres | return None |
| 1015 | 212 | storres | # End arguments check. |
| 1016 | 212 | storres | nAtAlpha = N^alpha |
| 1017 | 229 | storres | if debug: |
| 1018 | 229 | storres | print "N at alpha:", nAtAlpha |
| 1019 | 212 | storres | ## Building polynomials for matrix. |
| 1020 | 212 | storres | polyRing = inputPolynomial.parent() |
| 1021 | 212 | storres | # Whatever the 2 variables are actually called, we call them |
| 1022 | 212 | storres | # 'i' and 't' in all the variable names. |
| 1023 | 212 | storres | (iVariable, tVariable) = inputPolynomial.variables()[:2] |
| 1024 | 212 | storres | #print polyVars[0], type(polyVars[0]) |
| 1025 | 212 | storres | initialPolynomial = inputPolynomial.subs({iVariable:iVariable * iBound,
|
| 1026 | 212 | storres | tVariable:tVariable * tBound}) |
| 1027 | 212 | storres | ## polynomialsList = \ |
| 1028 | 212 | storres | ## spo_polynomial_to_polynomials_list_8(initialPolynomial, |
| 1029 | 212 | storres | ## spo_polynomial_to_polynomials_list_5(initialPolynomial, |
| 1030 | 212 | storres | polynomialsList = \ |
| 1031 | 212 | storres | spo_polynomial_to_polynomials_list_5(initialPolynomial, |
| 1032 | 212 | storres | alpha, |
| 1033 | 212 | storres | N, |
| 1034 | 212 | storres | iBound, |
| 1035 | 212 | storres | tBound, |
| 1036 | 212 | storres | 0) |
| 1037 | 212 | storres | #print "Polynomials list:", polynomialsList |
| 1038 | 212 | storres | ## Building the proto matrix. |
| 1039 | 212 | storres | knownMonomials = [] |
| 1040 | 212 | storres | protoMatrix = [] |
| 1041 | 212 | storres | for poly in polynomialsList: |
| 1042 | 212 | storres | spo_add_polynomial_coeffs_to_matrix_row(poly, |
| 1043 | 212 | storres | knownMonomials, |
| 1044 | 212 | storres | protoMatrix, |
| 1045 | 212 | storres | 0) |
| 1046 | 212 | storres | matrixToReduce = spo_proto_to_row_matrix(protoMatrix) |
| 1047 | 212 | storres | matrixToReduceTranspose = matrixToReduce.transpose() |
| 1048 | 212 | storres | squareMatrix = matrixToReduce * matrixToReduceTranspose |
| 1049 | 212 | storres | squareMatDet = det(squareMatrix) |
| 1050 | 212 | storres | latticeVolume = sqrt(squareMatDet) |
| 1051 | 212 | storres | print "Lattice volume:", latticeVolume.n() |
| 1052 | 212 | storres | print "Lattice volume / N:", (latticeVolume/N).n() |
| 1053 | 212 | storres | #print matrixToReduce |
| 1054 | 212 | storres | ## Reduction and checking. |
| 1055 | 212 | storres | ## S.T. changed 'fp' to None as of Sage 6.6 complying to |
| 1056 | 212 | storres | # error message issued when previous code was used. |
| 1057 | 212 | storres | #reducedMatrix = matrixToReduce.LLL(fp='fp') |
| 1058 | 212 | storres | reductionTimeStart = cputime() |
| 1059 | 212 | storres | reducedMatrix = matrixToReduce.LLL(fp=None) |
| 1060 | 212 | storres | reductionTime = cputime(reductionTimeStart) |
| 1061 | 212 | storres | print "Reduction time:", reductionTime |
| 1062 | 212 | storres | isLLLReduced = reducedMatrix.is_LLL_reduced() |
| 1063 | 212 | storres | if not isLLLReduced: |
| 1064 | 229 | storres | return None |
| 1065 | 229 | storres | # |
| 1066 | 229 | storres | if debug: |
| 1067 | 229 | storres | matrixFile = file('/tmp/reducedMatrix.txt', 'w')
|
| 1068 | 229 | storres | for row in reducedMatrix.rows(): |
| 1069 | 229 | storres | matrixFile.write(str(row) + "\n") |
| 1070 | 229 | storres | matrixFile.close() |
| 1071 | 229 | storres | # |
| 1072 | 212 | storres | monomialsCount = len(knownMonomials) |
| 1073 | 212 | storres | monomialsCountSqrt = sqrt(monomialsCount) |
| 1074 | 212 | storres | #print "Monomials count:", monomialsCount, monomialsCountSqrt.n() |
| 1075 | 212 | storres | #print reducedMatrix |
| 1076 | 212 | storres | ## Check the Coppersmith condition for each row and build the reduced |
| 1077 | 212 | storres | # polynomials. |
| 1078 | 229 | storres | ccVectorsCount = 0 |
| 1079 | 212 | storres | ccReducedPolynomialsList = [] |
| 1080 | 212 | storres | for row in reducedMatrix.rows(): |
| 1081 | 212 | storres | l2Norm = row.norm(2) |
| 1082 | 212 | storres | if (l2Norm * monomialsCountSqrt) < nAtAlpha: |
| 1083 | 212 | storres | #print (l2Norm * monomialsCountSqrt).n() |
| 1084 | 212 | storres | #print l2Norm.n() |
| 1085 | 229 | storres | ccVectorsCount +=1 |
| 1086 | 212 | storres | ccReducedPolynomial = \ |
| 1087 | 212 | storres | slz_compute_reduced_polynomial(row, |
| 1088 | 212 | storres | knownMonomials, |
| 1089 | 212 | storres | iVariable, |
| 1090 | 212 | storres | iBound, |
| 1091 | 212 | storres | tVariable, |
| 1092 | 212 | storres | tBound) |
| 1093 | 212 | storres | if not ccReducedPolynomial is None: |
| 1094 | 212 | storres | ccReducedPolynomialsList.append(ccReducedPolynomial) |
| 1095 | 212 | storres | else: |
| 1096 | 212 | storres | #print l2Norm.n() , ">", nAtAlpha |
| 1097 | 212 | storres | pass |
| 1098 | 229 | storres | if debug: |
| 1099 | 229 | storres | print ccVectorsCount, "out of ", len(ccReducedPolynomialsList), |
| 1100 | 229 | storres | print "took Coppersmith text." |
| 1101 | 212 | storres | if len(ccReducedPolynomialsList) < 2: |
| 1102 | 212 | storres | print "Less than 2 Coppersmith condition compliant vectors." |
| 1103 | 212 | storres | return () |
| 1104 | 229 | storres | if debug: |
| 1105 | 229 | storres | print "Reduced and Coppersmith compliant polynomials list", ccReducedPolynomialsList |
| 1106 | 212 | storres | return ccReducedPolynomialsList |
| 1107 | 212 | storres | # End slz_compute_coppersmith_reduced_polynomials_with_lattice volume |
| 1108 | 212 | storres | |
| 1109 | 219 | storres | def slz_compute_initial_lattice_matrix(inputPolynomial, |
| 1110 | 219 | storres | alpha, |
| 1111 | 219 | storres | N, |
| 1112 | 219 | storres | iBound, |
| 1113 | 228 | storres | tBound, |
| 1114 | 228 | storres | debug = False): |
| 1115 | 219 | storres | """ |
| 1116 | 219 | storres | For a given set of arguments (see below), compute the initial lattice |
| 1117 | 219 | storres | that could be reduced. |
| 1118 | 219 | storres | INPUT: |
| 1119 | 219 | storres | |
| 1120 | 219 | storres | - "inputPolynomial" -- (no default) a bivariate integer polynomial; |
| 1121 | 219 | storres | - "alpha" -- the alpha parameter of the Coppersmith algorithm; |
| 1122 | 219 | storres | - "N" -- the modulus; |
| 1123 | 219 | storres | - "iBound" -- the bound on the first variable; |
| 1124 | 219 | storres | - "tBound" -- the bound on the second variable. |
| 1125 | 219 | storres | |
| 1126 | 219 | storres | OUTPUT: |
| 1127 | 219 | storres | |
| 1128 | 219 | storres | A list of bivariate integer polynomial obtained using the Coppersmith |
| 1129 | 219 | storres | algorithm. The polynomials correspond to the rows of the LLL-reduce |
| 1130 | 219 | storres | reduced base that comply with the Coppersmith condition. |
| 1131 | 219 | storres | """ |
| 1132 | 219 | storres | # Arguments check. |
| 1133 | 219 | storres | if iBound == 0 or tBound == 0: |
| 1134 | 219 | storres | return None |
| 1135 | 219 | storres | # End arguments check. |
| 1136 | 219 | storres | nAtAlpha = N^alpha |
| 1137 | 219 | storres | ## Building polynomials for matrix. |
| 1138 | 219 | storres | polyRing = inputPolynomial.parent() |
| 1139 | 219 | storres | # Whatever the 2 variables are actually called, we call them |
| 1140 | 219 | storres | # 'i' and 't' in all the variable names. |
| 1141 | 219 | storres | (iVariable, tVariable) = inputPolynomial.variables()[:2] |
| 1142 | 219 | storres | #print polyVars[0], type(polyVars[0]) |
| 1143 | 219 | storres | initialPolynomial = inputPolynomial.subs({iVariable:iVariable * iBound,
|
| 1144 | 219 | storres | tVariable:tVariable * tBound}) |
| 1145 | 219 | storres | polynomialsList = \ |
| 1146 | 219 | storres | spo_polynomial_to_polynomials_list_8(initialPolynomial, |
| 1147 | 219 | storres | alpha, |
| 1148 | 219 | storres | N, |
| 1149 | 219 | storres | iBound, |
| 1150 | 219 | storres | tBound, |
| 1151 | 219 | storres | 0) |
| 1152 | 219 | storres | #print "Polynomials list:", polynomialsList |
| 1153 | 219 | storres | ## Building the proto matrix. |
| 1154 | 219 | storres | knownMonomials = [] |
| 1155 | 219 | storres | protoMatrix = [] |
| 1156 | 219 | storres | for poly in polynomialsList: |
| 1157 | 219 | storres | spo_add_polynomial_coeffs_to_matrix_row(poly, |
| 1158 | 219 | storres | knownMonomials, |
| 1159 | 219 | storres | protoMatrix, |
| 1160 | 219 | storres | 0) |
| 1161 | 219 | storres | matrixToReduce = spo_proto_to_row_matrix(protoMatrix) |
| 1162 | 228 | storres | if debug: |
| 1163 | 228 | storres | print "Initial basis polynomials" |
| 1164 | 228 | storres | for poly in polynomialsList: |
| 1165 | 228 | storres | print poly |
| 1166 | 219 | storres | return matrixToReduce |
| 1167 | 219 | storres | # End slz_compute_initial_lattice_matrix. |
| 1168 | 219 | storres | |
| 1169 | 122 | storres | def slz_compute_integer_polynomial_modular_roots(inputPolynomial, |
| 1170 | 122 | storres | alpha, |
| 1171 | 122 | storres | N, |
| 1172 | 122 | storres | iBound, |
| 1173 | 122 | storres | tBound): |
| 1174 | 122 | storres | """ |
| 1175 | 123 | storres | For a given set of arguments (see below), compute the polynomial modular |
| 1176 | 122 | storres | roots, if any. |
| 1177 | 124 | storres | |
| 1178 | 122 | storres | """ |
| 1179 | 123 | storres | # Arguments check. |
| 1180 | 123 | storres | if iBound == 0 or tBound == 0: |
| 1181 | 123 | storres | return set() |
| 1182 | 123 | storres | # End arguments check. |
| 1183 | 122 | storres | nAtAlpha = N^alpha |
| 1184 | 122 | storres | ## Building polynomials for matrix. |
| 1185 | 122 | storres | polyRing = inputPolynomial.parent() |
| 1186 | 122 | storres | # Whatever the 2 variables are actually called, we call them |
| 1187 | 122 | storres | # 'i' and 't' in all the variable names. |
| 1188 | 122 | storres | (iVariable, tVariable) = inputPolynomial.variables()[:2] |
| 1189 | 125 | storres | ccReducedPolynomialsList = \ |
| 1190 | 125 | storres | slz_compute_coppersmith_reduced_polynomials (inputPolynomial, |
| 1191 | 125 | storres | alpha, |
| 1192 | 125 | storres | N, |
| 1193 | 125 | storres | iBound, |
| 1194 | 125 | storres | tBound) |
| 1195 | 125 | storres | if len(ccReducedPolynomialsList) == 0: |
| 1196 | 125 | storres | return set() |
| 1197 | 122 | storres | ## Create the valid (poly1 and poly2 are algebraically independent) |
| 1198 | 122 | storres | # resultant tuples (poly1, poly2, resultant(poly1, poly2)). |
| 1199 | 122 | storres | # Try to mix and match all the polynomial pairs built from the |
| 1200 | 122 | storres | # ccReducedPolynomialsList to obtain non zero resultants. |
| 1201 | 122 | storres | resultantsInITuplesList = [] |
| 1202 | 122 | storres | for polyOuterIndex in xrange(0, len(ccReducedPolynomialsList)-1): |
| 1203 | 122 | storres | for polyInnerIndex in xrange(polyOuterIndex+1, |
| 1204 | 122 | storres | len(ccReducedPolynomialsList)): |
| 1205 | 122 | storres | # Compute the resultant in resultants in the |
| 1206 | 122 | storres | # first variable (is it the optimal choice?). |
| 1207 | 122 | storres | resultantInI = \ |
| 1208 | 122 | storres | ccReducedPolynomialsList[polyOuterIndex].resultant(ccReducedPolynomialsList[polyInnerIndex], |
| 1209 | 122 | storres | ccReducedPolynomialsList[0].parent(str(iVariable))) |
| 1210 | 122 | storres | #print "Resultant", resultantInI |
| 1211 | 122 | storres | # Test algebraic independence. |
| 1212 | 122 | storres | if not resultantInI.is_zero(): |
| 1213 | 122 | storres | resultantsInITuplesList.append((ccReducedPolynomialsList[polyOuterIndex], |
| 1214 | 122 | storres | ccReducedPolynomialsList[polyInnerIndex], |
| 1215 | 122 | storres | resultantInI)) |
| 1216 | 122 | storres | # If no non zero resultant was found: we can't get no algebraically |
| 1217 | 122 | storres | # independent polynomials pair. Give up! |
| 1218 | 122 | storres | if len(resultantsInITuplesList) == 0: |
| 1219 | 123 | storres | return set() |
| 1220 | 123 | storres | #print resultantsInITuplesList |
| 1221 | 122 | storres | # Compute the roots. |
| 1222 | 122 | storres | Zi = ZZ[str(iVariable)] |
| 1223 | 122 | storres | Zt = ZZ[str(tVariable)] |
| 1224 | 122 | storres | polynomialRootsSet = set() |
| 1225 | 122 | storres | # First, solve in the second variable since resultants are in the first |
| 1226 | 122 | storres | # variable. |
| 1227 | 122 | storres | for resultantInITuple in resultantsInITuplesList: |
| 1228 | 122 | storres | tRootsList = Zt(resultantInITuple[2]).roots() |
| 1229 | 122 | storres | # For each tRoot, compute the corresponding iRoots and check |
| 1230 | 123 | storres | # them in the input polynomial. |
| 1231 | 122 | storres | for tRoot in tRootsList: |
| 1232 | 123 | storres | #print "tRoot:", tRoot |
| 1233 | 122 | storres | # Roots returned by root() are (value, multiplicity) tuples. |
| 1234 | 122 | storres | iRootsList = \ |
| 1235 | 122 | storres | Zi(resultantInITuple[0].subs({resultantInITuple[0].variables()[1]:tRoot[0]})).roots()
|
| 1236 | 123 | storres | print iRootsList |
| 1237 | 122 | storres | # The iRootsList can be empty, hence the test. |
| 1238 | 122 | storres | if len(iRootsList) != 0: |
| 1239 | 122 | storres | for iRoot in iRootsList: |
| 1240 | 122 | storres | polyEvalModN = inputPolynomial(iRoot[0], tRoot[0]) / N |
| 1241 | 122 | storres | # polyEvalModN must be an integer. |
| 1242 | 122 | storres | if polyEvalModN == int(polyEvalModN): |
| 1243 | 122 | storres | polynomialRootsSet.add((iRoot[0],tRoot[0])) |
| 1244 | 122 | storres | return polynomialRootsSet |
| 1245 | 122 | storres | # End slz_compute_integer_polynomial_modular_roots. |
| 1246 | 122 | storres | # |
| 1247 | 125 | storres | def slz_compute_integer_polynomial_modular_roots_2(inputPolynomial, |
| 1248 | 125 | storres | alpha, |
| 1249 | 125 | storres | N, |
| 1250 | 125 | storres | iBound, |
| 1251 | 125 | storres | tBound): |
| 1252 | 125 | storres | """ |
| 1253 | 125 | storres | For a given set of arguments (see below), compute the polynomial modular |
| 1254 | 125 | storres | roots, if any. |
| 1255 | 125 | storres | This version differs in the way resultants are computed. |
| 1256 | 125 | storres | """ |
| 1257 | 125 | storres | # Arguments check. |
| 1258 | 125 | storres | if iBound == 0 or tBound == 0: |
| 1259 | 125 | storres | return set() |
| 1260 | 125 | storres | # End arguments check. |
| 1261 | 125 | storres | nAtAlpha = N^alpha |
| 1262 | 125 | storres | ## Building polynomials for matrix. |
| 1263 | 125 | storres | polyRing = inputPolynomial.parent() |
| 1264 | 125 | storres | # Whatever the 2 variables are actually called, we call them |
| 1265 | 125 | storres | # 'i' and 't' in all the variable names. |
| 1266 | 125 | storres | (iVariable, tVariable) = inputPolynomial.variables()[:2] |
| 1267 | 125 | storres | #print polyVars[0], type(polyVars[0]) |
| 1268 | 125 | storres | ccReducedPolynomialsList = \ |
| 1269 | 125 | storres | slz_compute_coppersmith_reduced_polynomials (inputPolynomial, |
| 1270 | 125 | storres | alpha, |
| 1271 | 125 | storres | N, |
| 1272 | 125 | storres | iBound, |
| 1273 | 125 | storres | tBound) |
| 1274 | 125 | storres | if len(ccReducedPolynomialsList) == 0: |
| 1275 | 125 | storres | return set() |
| 1276 | 125 | storres | ## Create the valid (poly1 and poly2 are algebraically independent) |
| 1277 | 125 | storres | # resultant tuples (poly1, poly2, resultant(poly1, poly2)). |
| 1278 | 125 | storres | # Try to mix and match all the polynomial pairs built from the |
| 1279 | 125 | storres | # ccReducedPolynomialsList to obtain non zero resultants. |
| 1280 | 125 | storres | resultantsInTTuplesList = [] |
| 1281 | 125 | storres | for polyOuterIndex in xrange(0, len(ccReducedPolynomialsList)-1): |
| 1282 | 125 | storres | for polyInnerIndex in xrange(polyOuterIndex+1, |
| 1283 | 125 | storres | len(ccReducedPolynomialsList)): |
| 1284 | 125 | storres | # Compute the resultant in resultants in the |
| 1285 | 125 | storres | # first variable (is it the optimal choice?). |
| 1286 | 125 | storres | resultantInT = \ |
| 1287 | 125 | storres | ccReducedPolynomialsList[polyOuterIndex].resultant(ccReducedPolynomialsList[polyInnerIndex], |
| 1288 | 125 | storres | ccReducedPolynomialsList[0].parent(str(tVariable))) |
| 1289 | 125 | storres | #print "Resultant", resultantInT |
| 1290 | 125 | storres | # Test algebraic independence. |
| 1291 | 125 | storres | if not resultantInT.is_zero(): |
| 1292 | 125 | storres | resultantsInTTuplesList.append((ccReducedPolynomialsList[polyOuterIndex], |
| 1293 | 125 | storres | ccReducedPolynomialsList[polyInnerIndex], |
| 1294 | 125 | storres | resultantInT)) |
| 1295 | 125 | storres | # If no non zero resultant was found: we can't get no algebraically |
| 1296 | 125 | storres | # independent polynomials pair. Give up! |
| 1297 | 125 | storres | if len(resultantsInTTuplesList) == 0: |
| 1298 | 125 | storres | return set() |
| 1299 | 125 | storres | #print resultantsInITuplesList |
| 1300 | 125 | storres | # Compute the roots. |
| 1301 | 125 | storres | Zi = ZZ[str(iVariable)] |
| 1302 | 125 | storres | Zt = ZZ[str(tVariable)] |
| 1303 | 125 | storres | polynomialRootsSet = set() |
| 1304 | 125 | storres | # First, solve in the second variable since resultants are in the first |
| 1305 | 125 | storres | # variable. |
| 1306 | 125 | storres | for resultantInTTuple in resultantsInTTuplesList: |
| 1307 | 125 | storres | iRootsList = Zi(resultantInTTuple[2]).roots() |
| 1308 | 125 | storres | # For each iRoot, compute the corresponding tRoots and check |
| 1309 | 125 | storres | # them in the input polynomial. |
| 1310 | 125 | storres | for iRoot in iRootsList: |
| 1311 | 125 | storres | #print "iRoot:", iRoot |
| 1312 | 125 | storres | # Roots returned by root() are (value, multiplicity) tuples. |
| 1313 | 125 | storres | tRootsList = \ |
| 1314 | 125 | storres | Zt(resultantInTTuple[0].subs({resultantInTTuple[0].variables()[0]:iRoot[0]})).roots()
|
| 1315 | 125 | storres | print tRootsList |
| 1316 | 125 | storres | # The tRootsList can be empty, hence the test. |
| 1317 | 125 | storres | if len(tRootsList) != 0: |
| 1318 | 125 | storres | for tRoot in tRootsList: |
| 1319 | 125 | storres | polyEvalModN = inputPolynomial(iRoot[0],tRoot[0]) / N |
| 1320 | 125 | storres | # polyEvalModN must be an integer. |
| 1321 | 125 | storres | if polyEvalModN == int(polyEvalModN): |
| 1322 | 125 | storres | polynomialRootsSet.add((iRoot[0],tRoot[0])) |
| 1323 | 125 | storres | return polynomialRootsSet |
| 1324 | 125 | storres | # End slz_compute_integer_polynomial_modular_roots_2. |
| 1325 | 125 | storres | # |
| 1326 | 61 | storres | def slz_compute_polynomial_and_interval(functionSo, degreeSo, lowerBoundSa, |
| 1327 | 218 | storres | upperBoundSa, approxAccurSa, |
| 1328 | 272 | storres | precSa=None, debug=False): |
| 1329 | 61 | storres | """ |
| 1330 | 61 | storres | Under the assumptions listed for slz_get_intervals_and_polynomials, compute |
| 1331 | 61 | storres | a polynomial that approximates the function on a an interval starting |
| 1332 | 61 | storres | at lowerBoundSa and finishing at a value that guarantees that the polynomial |
| 1333 | 61 | storres | approximates with the expected precision. |
| 1334 | 61 | storres | The interval upper bound is lowered until the expected approximation |
| 1335 | 61 | storres | precision is reached. |
| 1336 | 61 | storres | The polynomial, the bounds, the center of the interval and the error |
| 1337 | 61 | storres | are returned. |
| 1338 | 272 | storres | Argument debug is not used. |
| 1339 | 156 | storres | OUTPUT: |
| 1340 | 124 | storres | A tuple made of 4 Sollya objects: |
| 1341 | 124 | storres | - a polynomial; |
| 1342 | 124 | storres | - an range (an interval, not in the sense of number given as an interval); |
| 1343 | 124 | storres | - the center of the interval; |
| 1344 | 124 | storres | - the maximum error in the approximation of the input functionSo by the |
| 1345 | 218 | storres | output polynomial ; this error <= approxAccurSaS. |
| 1346 | 124 | storres | |
| 1347 | 61 | storres | """ |
| 1348 | 218 | storres | #print"In slz_compute_polynomial_and_interval..." |
| 1349 | 166 | storres | ## Superficial argument check. |
| 1350 | 166 | storres | if lowerBoundSa > upperBoundSa: |
| 1351 | 166 | storres | return None |
| 1352 | 218 | storres | ## Change Sollya precision, if requested. |
| 1353 | 218 | storres | if precSa is None: |
| 1354 | 218 | storres | precSa = ceil((RR('1.5') * abs(RR(approxAccurSa).log2())) / 64) * 64
|
| 1355 | 218 | storres | #print "Computed internal precision:", precSa |
| 1356 | 218 | storres | if precSa < 192: |
| 1357 | 218 | storres | precSa = 192 |
| 1358 | 226 | storres | sollyaPrecChanged = False |
| 1359 | 226 | storres | (initialSollyaPrecSo, initialSollyaPrecSa) = pobyso_get_prec_so_so_sa() |
| 1360 | 226 | storres | if precSa > initialSollyaPrecSa: |
| 1361 | 226 | storres | if precSa <= 2: |
| 1362 | 226 | storres | print inspect.stack()[0][3], ": precision change <=2 requested." |
| 1363 | 226 | storres | pobyso_set_prec_sa_so(precSa) |
| 1364 | 218 | storres | sollyaPrecChanged = True |
| 1365 | 61 | storres | RRR = lowerBoundSa.parent() |
| 1366 | 176 | storres | intervalShrinkConstFactorSa = RRR('0.9')
|
| 1367 | 61 | storres | absoluteErrorTypeSo = pobyso_absolute_so_so() |
| 1368 | 61 | storres | currentRangeSo = pobyso_bounds_to_range_sa_so(lowerBoundSa, upperBoundSa) |
| 1369 | 61 | storres | currentUpperBoundSa = upperBoundSa |
| 1370 | 61 | storres | currentLowerBoundSa = lowerBoundSa |
| 1371 | 61 | storres | # What we want here is the polynomial without the variable change, |
| 1372 | 61 | storres | # since our actual variable will be x-intervalCenter defined over the |
| 1373 | 61 | storres | # domain [lowerBound-intervalCenter , upperBound-intervalCenter]. |
| 1374 | 61 | storres | (polySo, intervalCenterSo, maxErrorSo) = \ |
| 1375 | 61 | storres | pobyso_taylor_expansion_no_change_var_so_so(functionSo, degreeSo, |
| 1376 | 61 | storres | currentRangeSo, |
| 1377 | 61 | storres | absoluteErrorTypeSo) |
| 1378 | 61 | storres | maxErrorSa = pobyso_get_constant_as_rn_with_rf_so_sa(maxErrorSo) |
| 1379 | 218 | storres | while maxErrorSa > approxAccurSa: |
| 1380 | 181 | storres | print "++Approximation error:", maxErrorSa.n() |
| 1381 | 81 | storres | sollya_lib_clear_obj(polySo) |
| 1382 | 81 | storres | sollya_lib_clear_obj(intervalCenterSo) |
| 1383 | 120 | storres | sollya_lib_clear_obj(maxErrorSo) |
| 1384 | 181 | storres | # Very empirical shrinking factor. |
| 1385 | 218 | storres | shrinkFactorSa = 1 / (maxErrorSa/approxAccurSa).log2().abs() |
| 1386 | 181 | storres | print "Shrink factor:", \ |
| 1387 | 181 | storres | shrinkFactorSa.n(), \ |
| 1388 | 181 | storres | intervalShrinkConstFactorSa |
| 1389 | 182 | storres | |
| 1390 | 218 | storres | #errorRatioSa = approxAccurSa/maxErrorSa |
| 1391 | 61 | storres | #print "Error ratio: ", errorRatioSa |
| 1392 | 181 | storres | # Make sure interval shrinks. |
| 1393 | 81 | storres | if shrinkFactorSa > intervalShrinkConstFactorSa: |
| 1394 | 81 | storres | actualShrinkFactorSa = intervalShrinkConstFactorSa |
| 1395 | 81 | storres | #print "Fixed" |
| 1396 | 61 | storres | else: |
| 1397 | 81 | storres | actualShrinkFactorSa = shrinkFactorSa |
| 1398 | 81 | storres | #print "Computed",shrinkFactorSa,maxErrorSa |
| 1399 | 81 | storres | #print shrinkFactorSa, maxErrorSa |
| 1400 | 101 | storres | #print "Shrink factor", actualShrinkFactorSa |
| 1401 | 81 | storres | currentUpperBoundSa = currentLowerBoundSa + \ |
| 1402 | 181 | storres | (currentUpperBoundSa - currentLowerBoundSa) * \ |
| 1403 | 181 | storres | actualShrinkFactorSa |
| 1404 | 71 | storres | #print "Current upper bound:", currentUpperBoundSa |
| 1405 | 61 | storres | sollya_lib_clear_obj(currentRangeSo) |
| 1406 | 181 | storres | # Check what is left with the bounds. |
| 1407 | 101 | storres | if currentUpperBoundSa <= currentLowerBoundSa or \ |
| 1408 | 101 | storres | currentUpperBoundSa == currentLowerBoundSa.nextabove(): |
| 1409 | 86 | storres | sollya_lib_clear_obj(absoluteErrorTypeSo) |
| 1410 | 86 | storres | print "Can't find an interval." |
| 1411 | 86 | storres | print "Use either or both a higher polynomial degree or a higher", |
| 1412 | 86 | storres | print "internal precision." |
| 1413 | 86 | storres | print "Aborting!" |
| 1414 | 218 | storres | if sollyaPrecChanged: |
| 1415 | 226 | storres | pobyso_set_prec_so_so(initialSollyaPrecSo) |
| 1416 | 226 | storres | sollya_lib_clear_obj(initialSollyaPrecSo) |
| 1417 | 218 | storres | return None |
| 1418 | 61 | storres | currentRangeSo = pobyso_bounds_to_range_sa_so(currentLowerBoundSa, |
| 1419 | 61 | storres | currentUpperBoundSa) |
| 1420 | 86 | storres | # print "New interval:", |
| 1421 | 86 | storres | # pobyso_autoprint(currentRangeSo) |
| 1422 | 120 | storres | #print "Second Taylor expansion call." |
| 1423 | 61 | storres | (polySo, intervalCenterSo, maxErrorSo) = \ |
| 1424 | 61 | storres | pobyso_taylor_expansion_no_change_var_so_so(functionSo, degreeSo, |
| 1425 | 61 | storres | currentRangeSo, |
| 1426 | 61 | storres | absoluteErrorTypeSo) |
| 1427 | 61 | storres | #maxErrorSa = pobyso_get_constant_as_rn_with_rf_so_sa(maxErrorSo, RRR) |
| 1428 | 85 | storres | #print "Max errorSo:", |
| 1429 | 85 | storres | #pobyso_autoprint(maxErrorSo) |
| 1430 | 61 | storres | maxErrorSa = pobyso_get_constant_as_rn_with_rf_so_sa(maxErrorSo) |
| 1431 | 85 | storres | #print "Max errorSa:", maxErrorSa |
| 1432 | 85 | storres | #print "Sollya prec:", |
| 1433 | 85 | storres | #pobyso_autoprint(sollya_lib_get_prec(None)) |
| 1434 | 218 | storres | # End while |
| 1435 | 61 | storres | sollya_lib_clear_obj(absoluteErrorTypeSo) |
| 1436 | 218 | storres | if sollyaPrecChanged: |
| 1437 | 226 | storres | pobyso_set_prec_so_so(initialSollyaPrecSo) |
| 1438 | 226 | storres | sollya_lib_clear_obj(initialSollyaPrecSo) |
| 1439 | 176 | storres | return (polySo, currentRangeSo, intervalCenterSo, maxErrorSo) |
| 1440 | 81 | storres | # End slz_compute_polynomial_and_interval |
| 1441 | 218 | storres | |
| 1442 | 218 | storres | def slz_compute_polynomial_and_interval_01(functionSo, degreeSo, lowerBoundSa, |
| 1443 | 218 | storres | upperBoundSa, approxAccurSa, |
| 1444 | 219 | storres | sollyaPrecSa=None, debug=False): |
| 1445 | 219 | storres | """ |
| 1446 | 219 | storres | Under the assumptions listed for slz_get_intervals_and_polynomials, compute |
| 1447 | 219 | storres | a polynomial that approximates the function on a an interval starting |
| 1448 | 219 | storres | at lowerBoundSa and finishing at a value that guarantees that the polynomial |
| 1449 | 219 | storres | approximates with the expected precision. |
| 1450 | 219 | storres | The interval upper bound is lowered until the expected approximation |
| 1451 | 219 | storres | precision is reached. |
| 1452 | 219 | storres | The polynomial, the bounds, the center of the interval and the error |
| 1453 | 219 | storres | are returned. |
| 1454 | 219 | storres | OUTPUT: |
| 1455 | 219 | storres | A tuple made of 4 Sollya objects: |
| 1456 | 219 | storres | - a polynomial; |
| 1457 | 219 | storres | - an range (an interval, not in the sense of number given as an interval); |
| 1458 | 219 | storres | - the center of the interval; |
| 1459 | 219 | storres | - the maximum error in the approximation of the input functionSo by the |
| 1460 | 219 | storres | output polynomial ; this error <= approxAccurSaS. |
| 1461 | 272 | storres | |
| 1462 | 272 | storres | Changes from version 0 (no number): |
| 1463 | 272 | storres | - make use of debug argument; |
| 1464 | 272 | storres | - polynomial coefficients are "shaved". |
| 1465 | 219 | storres | |
| 1466 | 219 | storres | """ |
| 1467 | 219 | storres | #print"In slz_compute_polynomial_and_interval..." |
| 1468 | 219 | storres | ## Superficial argument check. |
| 1469 | 219 | storres | if lowerBoundSa > upperBoundSa: |
| 1470 | 225 | storres | print inspect.stack()[0][3], ": lower bound is larger than upper bound. " |
| 1471 | 219 | storres | return None |
| 1472 | 219 | storres | ## Change Sollya precision, if requested. |
| 1473 | 226 | storres | (initialSollyaPrecSo, initialSollyaPrecSa) = pobyso_get_prec_so_so_sa() |
| 1474 | 225 | storres | sollyaPrecChangedSa = False |
| 1475 | 225 | storres | if sollyaPrecSa is None: |
| 1476 | 226 | storres | sollyaPrecSa = initialSollyaPrecSa |
| 1477 | 219 | storres | else: |
| 1478 | 226 | storres | if sollyaPrecSa > initialSollyaPrecSa: |
| 1479 | 226 | storres | if sollyaPrecSa <= 2: |
| 1480 | 226 | storres | print inspect.stack()[0][3], ": precision change <= 2 requested." |
| 1481 | 225 | storres | pobyso_set_prec_sa_so(sollyaPrecSa) |
| 1482 | 225 | storres | sollyaPrecChangedSa = True |
| 1483 | 225 | storres | ## Other initializations and data recovery. |
| 1484 | 219 | storres | RRR = lowerBoundSa.parent() |
| 1485 | 219 | storres | intervalShrinkConstFactorSa = RRR('0.9')
|
| 1486 | 219 | storres | absoluteErrorTypeSo = pobyso_absolute_so_so() |
| 1487 | 219 | storres | currentRangeSo = pobyso_bounds_to_range_sa_so(lowerBoundSa, upperBoundSa) |
| 1488 | 219 | storres | currentUpperBoundSa = upperBoundSa |
| 1489 | 219 | storres | currentLowerBoundSa = lowerBoundSa |
| 1490 | 219 | storres | # What we want here is the polynomial without the variable change, |
| 1491 | 219 | storres | # since our actual variable will be x-intervalCenter defined over the |
| 1492 | 219 | storres | # domain [lowerBound-intervalCenter , upperBound-intervalCenter]. |
| 1493 | 219 | storres | (polySo, intervalCenterSo, maxErrorSo) = \ |
| 1494 | 219 | storres | pobyso_taylor_expansion_no_change_var_so_so(functionSo, degreeSo, |
| 1495 | 219 | storres | currentRangeSo, |
| 1496 | 219 | storres | absoluteErrorTypeSo) |
| 1497 | 219 | storres | maxErrorSa = pobyso_get_constant_as_rn_with_rf_so_sa(maxErrorSo) |
| 1498 | 278 | storres | ## Expression "approxAccurSa/2" (instead of just approxAccurSa) is added |
| 1499 | 278 | storres | # since we use polynomial shaving. In order for it to work, we must start |
| 1500 | 278 | storres | # with a better accuracy. |
| 1501 | 278 | storres | while maxErrorSa > approxAccurSa/2: |
| 1502 | 219 | storres | print "++Approximation error:", maxErrorSa.n() |
| 1503 | 219 | storres | sollya_lib_clear_obj(polySo) |
| 1504 | 219 | storres | sollya_lib_clear_obj(intervalCenterSo) |
| 1505 | 219 | storres | sollya_lib_clear_obj(maxErrorSo) |
| 1506 | 219 | storres | # Very empirical shrinking factor. |
| 1507 | 219 | storres | shrinkFactorSa = 1 / (maxErrorSa/approxAccurSa).log2().abs() |
| 1508 | 219 | storres | print "Shrink factor:", \ |
| 1509 | 219 | storres | shrinkFactorSa.n(), \ |
| 1510 | 219 | storres | intervalShrinkConstFactorSa |
| 1511 | 219 | storres | |
| 1512 | 219 | storres | #errorRatioSa = approxAccurSa/maxErrorSa |
| 1513 | 219 | storres | #print "Error ratio: ", errorRatioSa |
| 1514 | 219 | storres | # Make sure interval shrinks. |
| 1515 | 219 | storres | if shrinkFactorSa > intervalShrinkConstFactorSa: |
| 1516 | 219 | storres | actualShrinkFactorSa = intervalShrinkConstFactorSa |
| 1517 | 219 | storres | #print "Fixed" |
| 1518 | 219 | storres | else: |
| 1519 | 219 | storres | actualShrinkFactorSa = shrinkFactorSa |
| 1520 | 219 | storres | #print "Computed",shrinkFactorSa,maxErrorSa |
| 1521 | 219 | storres | #print shrinkFactorSa, maxErrorSa |
| 1522 | 219 | storres | #print "Shrink factor", actualShrinkFactorSa |
| 1523 | 219 | storres | currentUpperBoundSa = currentLowerBoundSa + \ |
| 1524 | 219 | storres | (currentUpperBoundSa - currentLowerBoundSa) * \ |
| 1525 | 219 | storres | actualShrinkFactorSa |
| 1526 | 219 | storres | #print "Current upper bound:", currentUpperBoundSa |
| 1527 | 219 | storres | sollya_lib_clear_obj(currentRangeSo) |
| 1528 | 219 | storres | # Check what is left with the bounds. |
| 1529 | 219 | storres | if currentUpperBoundSa <= currentLowerBoundSa or \ |
| 1530 | 219 | storres | currentUpperBoundSa == currentLowerBoundSa.nextabove(): |
| 1531 | 219 | storres | sollya_lib_clear_obj(absoluteErrorTypeSo) |
| 1532 | 219 | storres | print "Can't find an interval." |
| 1533 | 219 | storres | print "Use either or both a higher polynomial degree or a higher", |
| 1534 | 219 | storres | print "internal precision." |
| 1535 | 219 | storres | print "Aborting!" |
| 1536 | 225 | storres | if sollyaPrecChangedSa: |
| 1537 | 225 | storres | pobyso_set_prec_so_so(initialSollyaPrecSo) |
| 1538 | 226 | storres | sollya_lib_clear_obj(initialSollyaPrecSo) |
| 1539 | 219 | storres | return None |
| 1540 | 219 | storres | currentRangeSo = pobyso_bounds_to_range_sa_so(currentLowerBoundSa, |
| 1541 | 219 | storres | currentUpperBoundSa) |
| 1542 | 219 | storres | # print "New interval:", |
| 1543 | 219 | storres | # pobyso_autoprint(currentRangeSo) |
| 1544 | 219 | storres | #print "Second Taylor expansion call." |
| 1545 | 219 | storres | (polySo, intervalCenterSo, maxErrorSo) = \ |
| 1546 | 219 | storres | pobyso_taylor_expansion_no_change_var_so_so(functionSo, degreeSo, |
| 1547 | 219 | storres | currentRangeSo, |
| 1548 | 219 | storres | absoluteErrorTypeSo) |
| 1549 | 219 | storres | #maxErrorSa = pobyso_get_constant_as_rn_with_rf_so_sa(maxErrorSo, RRR) |
| 1550 | 219 | storres | #print "Max errorSo:", |
| 1551 | 219 | storres | #pobyso_autoprint(maxErrorSo) |
| 1552 | 219 | storres | maxErrorSa = pobyso_get_constant_as_rn_with_rf_so_sa(maxErrorSo) |
| 1553 | 219 | storres | #print "Max errorSa:", maxErrorSa |
| 1554 | 219 | storres | #print "Sollya prec:", |
| 1555 | 219 | storres | #pobyso_autoprint(sollya_lib_get_prec(None)) |
| 1556 | 219 | storres | # End while |
| 1557 | 219 | storres | sollya_lib_clear_obj(absoluteErrorTypeSo) |
| 1558 | 219 | storres | itpSo = pobyso_constant_from_int_sa_so(floor(sollyaPrecSa/3)) |
| 1559 | 219 | storres | ftpSo = pobyso_constant_from_int_sa_so(floor(2*sollyaPrecSa/3)) |
| 1560 | 219 | storres | maxPrecSo = pobyso_constant_from_int_sa_so(sollyaPrecSa) |
| 1561 | 219 | storres | approxAccurSo = pobyso_constant_sa_so(RR(approxAccurSa)) |
| 1562 | 219 | storres | if debug: |
| 1563 | 229 | storres | print inspect.stack()[0][3], "SollyaPrecSa:", sollyaPrecSa |
| 1564 | 219 | storres | print "About to call polynomial rounding with:" |
| 1565 | 219 | storres | print "polySo: ", ; pobyso_autoprint(polySo) |
| 1566 | 219 | storres | print "functionSo: ", ; pobyso_autoprint(functionSo) |
| 1567 | 219 | storres | print "intervalCenterSo: ", ; pobyso_autoprint(intervalCenterSo) |
| 1568 | 219 | storres | print "currentRangeSo: ", ; pobyso_autoprint(currentRangeSo) |
| 1569 | 219 | storres | print "itpSo: ", ; pobyso_autoprint(itpSo) |
| 1570 | 219 | storres | print "ftpSo: ", ; pobyso_autoprint(ftpSo) |
| 1571 | 219 | storres | print "maxPrecSo: ", ; pobyso_autoprint(maxPrecSo) |
| 1572 | 219 | storres | print "approxAccurSo: ", ; pobyso_autoprint(approxAccurSo) |
| 1573 | 224 | storres | """ |
| 1574 | 224 | storres | # Naive rounding. |
| 1575 | 219 | storres | (roundedPolySo, roundedPolyMaxErrSo) = \ |
| 1576 | 219 | storres | pobyso_polynomial_coefficients_progressive_round_so_so(polySo, |
| 1577 | 219 | storres | functionSo, |
| 1578 | 219 | storres | intervalCenterSo, |
| 1579 | 219 | storres | currentRangeSo, |
| 1580 | 219 | storres | itpSo, |
| 1581 | 219 | storres | ftpSo, |
| 1582 | 219 | storres | maxPrecSo, |
| 1583 | 219 | storres | approxAccurSo) |
| 1584 | 224 | storres | """ |
| 1585 | 224 | storres | # Proved rounding. |
| 1586 | 224 | storres | (roundedPolySo, roundedPolyMaxErrSo) = \ |
| 1587 | 224 | storres | pobyso_round_coefficients_progressive_so_so(polySo, |
| 1588 | 224 | storres | functionSo, |
| 1589 | 224 | storres | maxPrecSo, |
| 1590 | 224 | storres | currentRangeSo, |
| 1591 | 224 | storres | intervalCenterSo, |
| 1592 | 224 | storres | maxErrorSo, |
| 1593 | 224 | storres | approxAccurSo, |
| 1594 | 224 | storres | debug=False) |
| 1595 | 224 | storres | #### Comment out the two next lines when polynomial rounding is activated. |
| 1596 | 224 | storres | #roundedPolySo = sollya_lib_copy_obj(polySo) |
| 1597 | 224 | storres | #roundedPolyMaxErrSo = sollya_lib_copy_obj(maxErrorSo) |
| 1598 | 219 | storres | sollya_lib_clear_obj(polySo) |
| 1599 | 219 | storres | sollya_lib_clear_obj(maxErrorSo) |
| 1600 | 219 | storres | sollya_lib_clear_obj(itpSo) |
| 1601 | 219 | storres | sollya_lib_clear_obj(ftpSo) |
| 1602 | 219 | storres | sollya_lib_clear_obj(approxAccurSo) |
| 1603 | 225 | storres | if sollyaPrecChangedSa: |
| 1604 | 230 | storres | pobyso_set_prec_so_so(initialSollyaPrecSo) |
| 1605 | 230 | storres | sollya_lib_clear_obj(initialSollyaPrecSo) |
| 1606 | 219 | storres | if debug: |
| 1607 | 219 | storres | print "1: ", ; pobyso_autoprint(roundedPolySo) |
| 1608 | 219 | storres | print "2: ", ; pobyso_autoprint(currentRangeSo) |
| 1609 | 219 | storres | print "3: ", ; pobyso_autoprint(intervalCenterSo) |
| 1610 | 219 | storres | print "4: ", ; pobyso_autoprint(roundedPolyMaxErrSo) |
| 1611 | 219 | storres | return (roundedPolySo, currentRangeSo, intervalCenterSo, roundedPolyMaxErrSo) |
| 1612 | 219 | storres | # End slz_compute_polynomial_and_interval_01 |
| 1613 | 219 | storres | |
| 1614 | 219 | storres | def slz_compute_polynomial_and_interval_02(functionSo, degreeSo, lowerBoundSa, |
| 1615 | 219 | storres | upperBoundSa, approxAccurSa, |
| 1616 | 227 | storres | sollyaPrecSa=None, debug=True ): |
| 1617 | 218 | storres | """ |
| 1618 | 218 | storres | Under the assumptions listed for slz_get_intervals_and_polynomials, compute |
| 1619 | 218 | storres | a polynomial that approximates the function on a an interval starting |
| 1620 | 218 | storres | at lowerBoundSa and finishing at a value that guarantees that the polynomial |
| 1621 | 218 | storres | approximates with the expected precision. |
| 1622 | 218 | storres | The interval upper bound is lowered until the expected approximation |
| 1623 | 218 | storres | precision is reached. |
| 1624 | 218 | storres | The polynomial, the bounds, the center of the interval and the error |
| 1625 | 218 | storres | are returned. |
| 1626 | 218 | storres | OUTPUT: |
| 1627 | 218 | storres | A tuple made of 4 Sollya objects: |
| 1628 | 218 | storres | - a polynomial; |
| 1629 | 218 | storres | - an range (an interval, not in the sense of number given as an interval); |
| 1630 | 218 | storres | - the center of the interval; |
| 1631 | 218 | storres | - the maximum error in the approximation of the input functionSo by the |
| 1632 | 218 | storres | output polynomial ; this error <= approxAccurSaS. |
| 1633 | 278 | storres | |
| 1634 | 228 | storres | Changes fom v 01: |
| 1635 | 227 | storres | extra verbose. |
| 1636 | 218 | storres | """ |
| 1637 | 218 | storres | print"In slz_compute_polynomial_and_interval..." |
| 1638 | 218 | storres | ## Superficial argument check. |
| 1639 | 218 | storres | if lowerBoundSa > upperBoundSa: |
| 1640 | 218 | storres | return None |
| 1641 | 218 | storres | ## Change Sollya precision, if requested. |
| 1642 | 226 | storres | sollyaPrecChanged = False |
| 1643 | 226 | storres | (initialSollyaPrecSo, initialSollyaPrecSa) = pobyso_get_prec_so_so_sa() |
| 1644 | 227 | storres | #print "Initial Sollya prec:", initialSollyaPrecSa, type(initialSollyaPrecSa) |
| 1645 | 227 | storres | if sollyaPrecSa is None: |
| 1646 | 226 | storres | sollyaPrecSa = initialSollyaPrecSa |
| 1647 | 226 | storres | else: |
| 1648 | 226 | storres | if sollyaPrecSa <= 2: |
| 1649 | 226 | storres | print inspect.stack()[0][3], ": precision change <=2 requested." |
| 1650 | 218 | storres | pobyso_set_prec_sa_so(sollyaPrecSa) |
| 1651 | 226 | storres | sollyaPrecChanged = True |
| 1652 | 218 | storres | RRR = lowerBoundSa.parent() |
| 1653 | 218 | storres | intervalShrinkConstFactorSa = RRR('0.9')
|
| 1654 | 218 | storres | absoluteErrorTypeSo = pobyso_absolute_so_so() |
| 1655 | 218 | storres | currentRangeSo = pobyso_bounds_to_range_sa_so(lowerBoundSa, upperBoundSa) |
| 1656 | 218 | storres | currentUpperBoundSa = upperBoundSa |
| 1657 | 218 | storres | currentLowerBoundSa = lowerBoundSa |
| 1658 | 227 | storres | #pobyso_autoprint(functionSo) |
| 1659 | 227 | storres | #pobyso_autoprint(degreeSo) |
| 1660 | 227 | storres | #pobyso_autoprint(currentRangeSo) |
| 1661 | 227 | storres | #pobyso_autoprint(absoluteErrorTypeSo) |
| 1662 | 227 | storres | ## What we want here is the polynomial without the variable change, |
| 1663 | 227 | storres | # since our actual variable will be x-intervalCenter defined over the |
| 1664 | 227 | storres | # domain [lowerBound-intervalCenter , upperBound-intervalCenter]. |
| 1665 | 218 | storres | (polySo, intervalCenterSo, maxErrorSo) = \ |
| 1666 | 218 | storres | pobyso_taylor_expansion_no_change_var_so_so(functionSo, degreeSo, |
| 1667 | 218 | storres | currentRangeSo, |
| 1668 | 218 | storres | absoluteErrorTypeSo) |
| 1669 | 218 | storres | maxErrorSa = pobyso_get_constant_as_rn_with_rf_so_sa(maxErrorSo) |
| 1670 | 227 | storres | print "...after Taylor expansion." |
| 1671 | 278 | storres | ## Expression "approxAccurSa/2" (instead of just approxAccurSa) is added |
| 1672 | 278 | storres | # since we use polynomial shaving. In order for it to work, we must start |
| 1673 | 278 | storres | # with a better accuracy. |
| 1674 | 278 | storres | while maxErrorSa > approxAccurSa/2: |
| 1675 | 218 | storres | print "++Approximation error:", maxErrorSa.n() |
| 1676 | 218 | storres | sollya_lib_clear_obj(polySo) |
| 1677 | 218 | storres | sollya_lib_clear_obj(intervalCenterSo) |
| 1678 | 218 | storres | sollya_lib_clear_obj(maxErrorSo) |
| 1679 | 218 | storres | # Very empirical shrinking factor. |
| 1680 | 218 | storres | shrinkFactorSa = 1 / (maxErrorSa/approxAccurSa).log2().abs() |
| 1681 | 218 | storres | print "Shrink factor:", \ |
| 1682 | 218 | storres | shrinkFactorSa.n(), \ |
| 1683 | 218 | storres | intervalShrinkConstFactorSa |
| 1684 | 218 | storres | |
| 1685 | 218 | storres | #errorRatioSa = approxAccurSa/maxErrorSa |
| 1686 | 218 | storres | #print "Error ratio: ", errorRatioSa |
| 1687 | 218 | storres | # Make sure interval shrinks. |
| 1688 | 218 | storres | if shrinkFactorSa > intervalShrinkConstFactorSa: |
| 1689 | 218 | storres | actualShrinkFactorSa = intervalShrinkConstFactorSa |
| 1690 | 218 | storres | #print "Fixed" |
| 1691 | 218 | storres | else: |
| 1692 | 218 | storres | actualShrinkFactorSa = shrinkFactorSa |
| 1693 | 218 | storres | #print "Computed",shrinkFactorSa,maxErrorSa |
| 1694 | 218 | storres | #print shrinkFactorSa, maxErrorSa |
| 1695 | 218 | storres | #print "Shrink factor", actualShrinkFactorSa |
| 1696 | 218 | storres | currentUpperBoundSa = currentLowerBoundSa + \ |
| 1697 | 218 | storres | (currentUpperBoundSa - currentLowerBoundSa) * \ |
| 1698 | 218 | storres | actualShrinkFactorSa |
| 1699 | 218 | storres | #print "Current upper bound:", currentUpperBoundSa |
| 1700 | 218 | storres | sollya_lib_clear_obj(currentRangeSo) |
| 1701 | 218 | storres | # Check what is left with the bounds. |
| 1702 | 218 | storres | if currentUpperBoundSa <= currentLowerBoundSa or \ |
| 1703 | 218 | storres | currentUpperBoundSa == currentLowerBoundSa.nextabove(): |
| 1704 | 218 | storres | sollya_lib_clear_obj(absoluteErrorTypeSo) |
| 1705 | 218 | storres | print "Can't find an interval." |
| 1706 | 218 | storres | print "Use either or both a higher polynomial degree or a higher", |
| 1707 | 218 | storres | print "internal precision." |
| 1708 | 218 | storres | print "Aborting!" |
| 1709 | 226 | storres | if sollyaPrecChanged: |
| 1710 | 226 | storres | pobyso_set_prec_so_so(initialSollyaPrecSo) |
| 1711 | 226 | storres | sollya_lib_clear_obj(initialSollyaPrecSo) |
| 1712 | 218 | storres | return None |
| 1713 | 218 | storres | currentRangeSo = pobyso_bounds_to_range_sa_so(currentLowerBoundSa, |
| 1714 | 218 | storres | currentUpperBoundSa) |
| 1715 | 218 | storres | # print "New interval:", |
| 1716 | 218 | storres | # pobyso_autoprint(currentRangeSo) |
| 1717 | 218 | storres | #print "Second Taylor expansion call." |
| 1718 | 218 | storres | (polySo, intervalCenterSo, maxErrorSo) = \ |
| 1719 | 218 | storres | pobyso_taylor_expansion_no_change_var_so_so(functionSo, degreeSo, |
| 1720 | 218 | storres | currentRangeSo, |
| 1721 | 218 | storres | absoluteErrorTypeSo) |
| 1722 | 218 | storres | #maxErrorSa = pobyso_get_constant_as_rn_with_rf_so_sa(maxErrorSo, RRR) |
| 1723 | 218 | storres | #print "Max errorSo:", |
| 1724 | 218 | storres | #pobyso_autoprint(maxErrorSo) |
| 1725 | 218 | storres | maxErrorSa = pobyso_get_constant_as_rn_with_rf_so_sa(maxErrorSo) |
| 1726 | 218 | storres | #print "Max errorSa:", maxErrorSa |
| 1727 | 218 | storres | #print "Sollya prec:", |
| 1728 | 218 | storres | #pobyso_autoprint(sollya_lib_get_prec(None)) |
| 1729 | 218 | storres | # End while |
| 1730 | 218 | storres | sollya_lib_clear_obj(absoluteErrorTypeSo) |
| 1731 | 218 | storres | itpSo = pobyso_constant_from_int_sa_so(floor(sollyaPrecSa/3)) |
| 1732 | 218 | storres | ftpSo = pobyso_constant_from_int_sa_so(floor(2*sollyaPrecSa/3)) |
| 1733 | 218 | storres | maxPrecSo = pobyso_constant_from_int_sa_so(sollyaPrecSa) |
| 1734 | 218 | storres | approxAccurSo = pobyso_constant_sa_so(RR(approxAccurSa)) |
| 1735 | 218 | storres | print "About to call polynomial rounding with:" |
| 1736 | 218 | storres | print "polySo: ", ; pobyso_autoprint(polySo) |
| 1737 | 218 | storres | print "functionSo: ", ; pobyso_autoprint(functionSo) |
| 1738 | 218 | storres | print "intervalCenterSo: ", ; pobyso_autoprint(intervalCenterSo) |
| 1739 | 218 | storres | print "currentRangeSo: ", ; pobyso_autoprint(currentRangeSo) |
| 1740 | 218 | storres | print "itpSo: ", ; pobyso_autoprint(itpSo) |
| 1741 | 218 | storres | print "ftpSo: ", ; pobyso_autoprint(ftpSo) |
| 1742 | 218 | storres | print "maxPrecSo: ", ; pobyso_autoprint(maxPrecSo) |
| 1743 | 218 | storres | print "approxAccurSo: ", ; pobyso_autoprint(approxAccurSo) |
| 1744 | 218 | storres | (roundedPolySo, roundedPolyMaxErrSo) = \ |
| 1745 | 228 | storres | pobyso_round_coefficients_progressive_so_so(polySo, |
| 1746 | 228 | storres | functionSo, |
| 1747 | 228 | storres | maxPrecSo, |
| 1748 | 228 | storres | currentRangeSo, |
| 1749 | 228 | storres | intervalCenterSo, |
| 1750 | 228 | storres | maxErrorSo, |
| 1751 | 228 | storres | approxAccurSo, |
| 1752 | 228 | storres | debug = True) |
| 1753 | 218 | storres | |
| 1754 | 218 | storres | sollya_lib_clear_obj(polySo) |
| 1755 | 218 | storres | sollya_lib_clear_obj(maxErrorSo) |
| 1756 | 218 | storres | sollya_lib_clear_obj(itpSo) |
| 1757 | 218 | storres | sollya_lib_clear_obj(ftpSo) |
| 1758 | 218 | storres | sollya_lib_clear_obj(approxAccurSo) |
| 1759 | 226 | storres | if sollyaPrecChanged: |
| 1760 | 226 | storres | pobyso_set_prec_so_so(initialSollyaPrecSo) |
| 1761 | 226 | storres | sollya_lib_clear_obj(initialSollyaPrecSo) |
| 1762 | 218 | storres | print "1: ", ; pobyso_autoprint(roundedPolySo) |
| 1763 | 218 | storres | print "2: ", ; pobyso_autoprint(currentRangeSo) |
| 1764 | 218 | storres | print "3: ", ; pobyso_autoprint(intervalCenterSo) |
| 1765 | 218 | storres | print "4: ", ; pobyso_autoprint(roundedPolyMaxErrSo) |
| 1766 | 218 | storres | return (roundedPolySo, currentRangeSo, intervalCenterSo, roundedPolyMaxErrSo) |
| 1767 | 219 | storres | # End slz_compute_polynomial_and_interval_02 |
| 1768 | 218 | storres | |
| 1769 | 122 | storres | def slz_compute_reduced_polynomial(matrixRow, |
| 1770 | 98 | storres | knownMonomials, |
| 1771 | 106 | storres | var1, |
| 1772 | 98 | storres | var1Bound, |
| 1773 | 106 | storres | var2, |
| 1774 | 99 | storres | var2Bound): |
| 1775 | 98 | storres | """ |
| 1776 | 125 | storres | Compute a polynomial from a single reduced matrix row. |
| 1777 | 122 | storres | This function was introduced in order to avoid the computation of the |
| 1778 | 125 | storres | all the polynomials from the full matrix (even those built from rows |
| 1779 | 125 | storres | that do no verify the Coppersmith condition) as this may involves |
| 1780 | 152 | storres | expensive operations over (large) integers. |
| 1781 | 122 | storres | """ |
| 1782 | 122 | storres | ## Check arguments. |
| 1783 | 122 | storres | if len(knownMonomials) == 0: |
| 1784 | 122 | storres | return None |
| 1785 | 122 | storres | # varNounds can be zero since 0^0 returns 1. |
| 1786 | 122 | storres | if (var1Bound < 0) or (var2Bound < 0): |
| 1787 | 122 | storres | return None |
| 1788 | 122 | storres | ## Initialisations. |
| 1789 | 122 | storres | polynomialRing = knownMonomials[0].parent() |
| 1790 | 122 | storres | currentPolynomial = polynomialRing(0) |
| 1791 | 123 | storres | # TODO: use zip instead of indices. |
| 1792 | 122 | storres | for colIndex in xrange(0, len(knownMonomials)): |
| 1793 | 122 | storres | currentCoefficient = matrixRow[colIndex] |
| 1794 | 122 | storres | if currentCoefficient != 0: |
| 1795 | 122 | storres | #print "Current coefficient:", currentCoefficient |
| 1796 | 122 | storres | currentMonomial = knownMonomials[colIndex] |
| 1797 | 122 | storres | #print "Monomial as multivariate polynomial:", \ |
| 1798 | 122 | storres | #currentMonomial, type(currentMonomial) |
| 1799 | 122 | storres | degreeInVar1 = currentMonomial.degree(var1) |
| 1800 | 123 | storres | #print "Degree in var1", var1, ":", degreeInVar1 |
| 1801 | 122 | storres | degreeInVar2 = currentMonomial.degree(var2) |
| 1802 | 123 | storres | #print "Degree in var2", var2, ":", degreeInVar2 |
| 1803 | 122 | storres | if degreeInVar1 > 0: |
| 1804 | 122 | storres | currentCoefficient = \ |
| 1805 | 123 | storres | currentCoefficient / (var1Bound^degreeInVar1) |
| 1806 | 122 | storres | #print "varBound1 in degree:", var1Bound^degreeInVar1 |
| 1807 | 122 | storres | #print "Current coefficient(1)", currentCoefficient |
| 1808 | 122 | storres | if degreeInVar2 > 0: |
| 1809 | 122 | storres | currentCoefficient = \ |
| 1810 | 123 | storres | currentCoefficient / (var2Bound^degreeInVar2) |
| 1811 | 122 | storres | #print "Current coefficient(2)", currentCoefficient |
| 1812 | 122 | storres | #print "Current reduced monomial:", (currentCoefficient * \ |
| 1813 | 122 | storres | # currentMonomial) |
| 1814 | 122 | storres | currentPolynomial += (currentCoefficient * currentMonomial) |
| 1815 | 122 | storres | #print "Current polynomial:", currentPolynomial |
| 1816 | 122 | storres | # End if |
| 1817 | 122 | storres | # End for colIndex. |
| 1818 | 122 | storres | #print "Type of the current polynomial:", type(currentPolynomial) |
| 1819 | 122 | storres | return(currentPolynomial) |
| 1820 | 122 | storres | # End slz_compute_reduced_polynomial |
| 1821 | 122 | storres | # |
| 1822 | 122 | storres | def slz_compute_reduced_polynomials(reducedMatrix, |
| 1823 | 122 | storres | knownMonomials, |
| 1824 | 122 | storres | var1, |
| 1825 | 122 | storres | var1Bound, |
| 1826 | 122 | storres | var2, |
| 1827 | 122 | storres | var2Bound): |
| 1828 | 122 | storres | """ |
| 1829 | 122 | storres | Legacy function, use slz_compute_reduced_polynomials_list |
| 1830 | 122 | storres | """ |
| 1831 | 122 | storres | return(slz_compute_reduced_polynomials_list(reducedMatrix, |
| 1832 | 122 | storres | knownMonomials, |
| 1833 | 122 | storres | var1, |
| 1834 | 122 | storres | var1Bound, |
| 1835 | 122 | storres | var2, |
| 1836 | 122 | storres | var2Bound) |
| 1837 | 122 | storres | ) |
| 1838 | 177 | storres | # |
| 1839 | 122 | storres | def slz_compute_reduced_polynomials_list(reducedMatrix, |
| 1840 | 152 | storres | knownMonomials, |
| 1841 | 152 | storres | var1, |
| 1842 | 152 | storres | var1Bound, |
| 1843 | 152 | storres | var2, |
| 1844 | 152 | storres | var2Bound): |
| 1845 | 122 | storres | """ |
| 1846 | 98 | storres | From a reduced matrix, holding the coefficients, from a monomials list, |
| 1847 | 98 | storres | from the bounds of each variable, compute the corresponding polynomials |
| 1848 | 98 | storres | scaled back by dividing by the "right" powers of the variables bounds. |
| 1849 | 99 | storres | |
| 1850 | 99 | storres | The elements in knownMonomials must be of the "right" polynomial type. |
| 1851 | 172 | storres | They set the polynomial type of the output polynomials in list. |
| 1852 | 152 | storres | @param reducedMatrix: the reduced matrix as output from LLL; |
| 1853 | 152 | storres | @param kwnonMonomials: the ordered list of the monomials used to |
| 1854 | 152 | storres | build the polynomials; |
| 1855 | 152 | storres | @param var1: the first variable (of the "right" type); |
| 1856 | 152 | storres | @param var1Bound: the first variable bound; |
| 1857 | 152 | storres | @param var2: the second variable (of the "right" type); |
| 1858 | 152 | storres | @param var2Bound: the second variable bound. |
| 1859 | 152 | storres | @return: a list of polynomials obtained with the reduced coefficients |
| 1860 | 152 | storres | and scaled down with the bounds |
| 1861 | 98 | storres | """ |
| 1862 | 99 | storres | |
| 1863 | 98 | storres | # TODO: check input arguments. |
| 1864 | 98 | storres | reducedPolynomials = [] |
| 1865 | 106 | storres | #print "type var1:", type(var1), " - type var2:", type(var2) |
| 1866 | 98 | storres | for matrixRow in reducedMatrix.rows(): |
| 1867 | 102 | storres | currentPolynomial = 0 |
| 1868 | 98 | storres | for colIndex in xrange(0, len(knownMonomials)): |
| 1869 | 98 | storres | currentCoefficient = matrixRow[colIndex] |
| 1870 | 106 | storres | if currentCoefficient != 0: |
| 1871 | 106 | storres | #print "Current coefficient:", currentCoefficient |
| 1872 | 106 | storres | currentMonomial = knownMonomials[colIndex] |
| 1873 | 106 | storres | parentRing = currentMonomial.parent() |
| 1874 | 106 | storres | #print "Monomial as multivariate polynomial:", \ |
| 1875 | 106 | storres | #currentMonomial, type(currentMonomial) |
| 1876 | 106 | storres | degreeInVar1 = currentMonomial.degree(parentRing(var1)) |
| 1877 | 106 | storres | #print "Degree in var", var1, ":", degreeInVar1 |
| 1878 | 106 | storres | degreeInVar2 = currentMonomial.degree(parentRing(var2)) |
| 1879 | 106 | storres | #print "Degree in var", var2, ":", degreeInVar2 |
| 1880 | 106 | storres | if degreeInVar1 > 0: |
| 1881 | 167 | storres | currentCoefficient /= var1Bound^degreeInVar1 |
| 1882 | 106 | storres | #print "varBound1 in degree:", var1Bound^degreeInVar1 |
| 1883 | 106 | storres | #print "Current coefficient(1)", currentCoefficient |
| 1884 | 106 | storres | if degreeInVar2 > 0: |
| 1885 | 167 | storres | currentCoefficient /= var2Bound^degreeInVar2 |
| 1886 | 106 | storres | #print "Current coefficient(2)", currentCoefficient |
| 1887 | 106 | storres | #print "Current reduced monomial:", (currentCoefficient * \ |
| 1888 | 106 | storres | # currentMonomial) |
| 1889 | 106 | storres | currentPolynomial += (currentCoefficient * currentMonomial) |
| 1890 | 168 | storres | #if degreeInVar1 == 0 and degreeInVar2 == 0: |
| 1891 | 168 | storres | #print "!!!! constant term !!!!" |
| 1892 | 106 | storres | #print "Current polynomial:", currentPolynomial |
| 1893 | 106 | storres | # End if |
| 1894 | 106 | storres | # End for colIndex. |
| 1895 | 99 | storres | #print "Type of the current polynomial:", type(currentPolynomial) |
| 1896 | 99 | storres | reducedPolynomials.append(currentPolynomial) |
| 1897 | 98 | storres | return reducedPolynomials |
| 1898 | 177 | storres | # End slz_compute_reduced_polynomials_list. |
| 1899 | 98 | storres | |
| 1900 | 177 | storres | def slz_compute_reduced_polynomials_list_from_rows(rowsList, |
| 1901 | 177 | storres | knownMonomials, |
| 1902 | 177 | storres | var1, |
| 1903 | 177 | storres | var1Bound, |
| 1904 | 177 | storres | var2, |
| 1905 | 177 | storres | var2Bound): |
| 1906 | 177 | storres | """ |
| 1907 | 177 | storres | From a list of rows, holding the coefficients, from a monomials list, |
| 1908 | 177 | storres | from the bounds of each variable, compute the corresponding polynomials |
| 1909 | 177 | storres | scaled back by dividing by the "right" powers of the variables bounds. |
| 1910 | 177 | storres | |
| 1911 | 177 | storres | The elements in knownMonomials must be of the "right" polynomial type. |
| 1912 | 177 | storres | They set the polynomial type of the output polynomials in list. |
| 1913 | 177 | storres | @param rowsList: the reduced matrix as output from LLL; |
| 1914 | 177 | storres | @param kwnonMonomials: the ordered list of the monomials used to |
| 1915 | 177 | storres | build the polynomials; |
| 1916 | 177 | storres | @param var1: the first variable (of the "right" type); |
| 1917 | 177 | storres | @param var1Bound: the first variable bound; |
| 1918 | 177 | storres | @param var2: the second variable (of the "right" type); |
| 1919 | 177 | storres | @param var2Bound: the second variable bound. |
| 1920 | 177 | storres | @return: a list of polynomials obtained with the reduced coefficients |
| 1921 | 177 | storres | and scaled down with the bounds |
| 1922 | 177 | storres | """ |
| 1923 | 177 | storres | |
| 1924 | 177 | storres | # TODO: check input arguments. |
| 1925 | 177 | storres | reducedPolynomials = [] |
| 1926 | 177 | storres | #print "type var1:", type(var1), " - type var2:", type(var2) |
| 1927 | 177 | storres | for matrixRow in rowsList: |
| 1928 | 177 | storres | currentPolynomial = 0 |
| 1929 | 177 | storres | for colIndex in xrange(0, len(knownMonomials)): |
| 1930 | 177 | storres | currentCoefficient = matrixRow[colIndex] |
| 1931 | 177 | storres | if currentCoefficient != 0: |
| 1932 | 177 | storres | #print "Current coefficient:", currentCoefficient |
| 1933 | 177 | storres | currentMonomial = knownMonomials[colIndex] |
| 1934 | 177 | storres | parentRing = currentMonomial.parent() |
| 1935 | 177 | storres | #print "Monomial as multivariate polynomial:", \ |
| 1936 | 177 | storres | #currentMonomial, type(currentMonomial) |
| 1937 | 177 | storres | degreeInVar1 = currentMonomial.degree(parentRing(var1)) |
| 1938 | 177 | storres | #print "Degree in var", var1, ":", degreeInVar1 |
| 1939 | 177 | storres | degreeInVar2 = currentMonomial.degree(parentRing(var2)) |
| 1940 | 177 | storres | #print "Degree in var", var2, ":", degreeInVar2 |
| 1941 | 177 | storres | if degreeInVar1 > 0: |
| 1942 | 177 | storres | currentCoefficient /= var1Bound^degreeInVar1 |
| 1943 | 177 | storres | #print "varBound1 in degree:", var1Bound^degreeInVar1 |
| 1944 | 177 | storres | #print "Current coefficient(1)", currentCoefficient |
| 1945 | 177 | storres | if degreeInVar2 > 0: |
| 1946 | 177 | storres | currentCoefficient /= var2Bound^degreeInVar2 |
| 1947 | 177 | storres | #print "Current coefficient(2)", currentCoefficient |
| 1948 | 177 | storres | #print "Current reduced monomial:", (currentCoefficient * \ |
| 1949 | 177 | storres | # currentMonomial) |
| 1950 | 177 | storres | currentPolynomial += (currentCoefficient * currentMonomial) |
| 1951 | 177 | storres | #if degreeInVar1 == 0 and degreeInVar2 == 0: |
| 1952 | 177 | storres | #print "!!!! constant term !!!!" |
| 1953 | 177 | storres | #print "Current polynomial:", currentPolynomial |
| 1954 | 177 | storres | # End if |
| 1955 | 177 | storres | # End for colIndex. |
| 1956 | 177 | storres | #print "Type of the current polynomial:", type(currentPolynomial) |
| 1957 | 177 | storres | reducedPolynomials.append(currentPolynomial) |
| 1958 | 177 | storres | return reducedPolynomials |
| 1959 | 177 | storres | # End slz_compute_reduced_polynomials_list_from_rows. |
| 1960 | 177 | storres | # |
| 1961 | 114 | storres | def slz_compute_scaled_function(functionSa, |
| 1962 | 114 | storres | lowerBoundSa, |
| 1963 | 114 | storres | upperBoundSa, |
| 1964 | 156 | storres | floatingPointPrecSa, |
| 1965 | 156 | storres | debug=False): |
| 1966 | 72 | storres | """ |
| 1967 | 72 | storres | From a function, compute the scaled function whose domain |
| 1968 | 72 | storres | is included in [1, 2) and whose image is also included in [1,2). |
| 1969 | 72 | storres | Return a tuple: |
| 1970 | 72 | storres | [0]: the scaled function |
| 1971 | 72 | storres | [1]: the scaled domain lower bound |
| 1972 | 72 | storres | [2]: the scaled domain upper bound |
| 1973 | 72 | storres | [3]: the scaled image lower bound |
| 1974 | 72 | storres | [4]: the scaled image upper bound |
| 1975 | 72 | storres | """ |
| 1976 | 177 | storres | ## The variable can be called anything. |
| 1977 | 80 | storres | x = functionSa.variables()[0] |
| 1978 | 72 | storres | # Scalling the domain -> [1,2[. |
| 1979 | 72 | storres | boundsIntervalRifSa = RealIntervalField(floatingPointPrecSa) |
| 1980 | 72 | storres | domainBoundsIntervalSa = boundsIntervalRifSa(lowerBoundSa, upperBoundSa) |
| 1981 | 166 | storres | (invDomainScalingExpressionSa, domainScalingExpressionSa) = \ |
| 1982 | 80 | storres | slz_interval_scaling_expression(domainBoundsIntervalSa, x) |
| 1983 | 156 | storres | if debug: |
| 1984 | 156 | storres | print "domainScalingExpression for argument :", \ |
| 1985 | 156 | storres | invDomainScalingExpressionSa |
| 1986 | 190 | storres | print "function: ", functionSa |
| 1987 | 190 | storres | ff = functionSa.subs({x : domainScalingExpressionSa})
|
| 1988 | 190 | storres | ## Bless expression as a function. |
| 1989 | 190 | storres | ff = ff.function(x) |
| 1990 | 72 | storres | #ff = f.subs_expr(x==domainScalingExpressionSa) |
| 1991 | 177 | storres | #domainScalingFunction(x) = invDomainScalingExpressionSa |
| 1992 | 177 | storres | domainScalingFunction = invDomainScalingExpressionSa.function(x) |
| 1993 | 151 | storres | scaledLowerBoundSa = \ |
| 1994 | 151 | storres | domainScalingFunction(lowerBoundSa).n(prec=floatingPointPrecSa) |
| 1995 | 151 | storres | scaledUpperBoundSa = \ |
| 1996 | 151 | storres | domainScalingFunction(upperBoundSa).n(prec=floatingPointPrecSa) |
| 1997 | 156 | storres | if debug: |
| 1998 | 156 | storres | print 'ff:', ff, "- Domain:", scaledLowerBoundSa, \ |
| 1999 | 156 | storres | scaledUpperBoundSa |
| 2000 | 254 | storres | ## If both bounds are negative, swap them. |
| 2001 | 254 | storres | if lowerBoundSa < 0 and upperBoundSa < 0: |
| 2002 | 254 | storres | scaledLowerBoundSa, scaledUpperBoundSa = \ |
| 2003 | 254 | storres | scaledUpperBoundSa,scaledLowerBoundSa |
| 2004 | 72 | storres | # |
| 2005 | 72 | storres | # Scalling the image -> [1,2[. |
| 2006 | 151 | storres | flbSa = ff(scaledLowerBoundSa).n(prec=floatingPointPrecSa) |
| 2007 | 151 | storres | fubSa = ff(scaledUpperBoundSa).n(prec=floatingPointPrecSa) |
| 2008 | 72 | storres | if flbSa <= fubSa: # Increasing |
| 2009 | 72 | storres | imageBinadeBottomSa = floor(flbSa.log2()) |
| 2010 | 72 | storres | else: # Decreasing |
| 2011 | 72 | storres | imageBinadeBottomSa = floor(fubSa.log2()) |
| 2012 | 156 | storres | if debug: |
| 2013 | 156 | storres | print 'ff:', ff, '- Image:', flbSa, fubSa, imageBinadeBottomSa |
| 2014 | 72 | storres | imageBoundsIntervalSa = boundsIntervalRifSa(flbSa, fubSa) |
| 2015 | 166 | storres | (invImageScalingExpressionSa,imageScalingExpressionSa) = \ |
| 2016 | 80 | storres | slz_interval_scaling_expression(imageBoundsIntervalSa, x) |
| 2017 | 156 | storres | if debug: |
| 2018 | 156 | storres | print "imageScalingExpression for argument :", \ |
| 2019 | 156 | storres | invImageScalingExpressionSa |
| 2020 | 72 | storres | iis = invImageScalingExpressionSa.function(x) |
| 2021 | 72 | storres | fff = iis.subs({x:ff})
|
| 2022 | 156 | storres | if debug: |
| 2023 | 156 | storres | print "fff:", fff, |
| 2024 | 156 | storres | print " - Image:", fff(scaledLowerBoundSa), fff(scaledUpperBoundSa) |
| 2025 | 72 | storres | return([fff, scaledLowerBoundSa, scaledUpperBoundSa, \ |
| 2026 | 72 | storres | fff(scaledLowerBoundSa), fff(scaledUpperBoundSa)]) |
| 2027 | 151 | storres | # End slz_compute_scaled_function |
| 2028 | 72 | storres | |
| 2029 | 179 | storres | def slz_fix_bounds_for_binades(lowerBound, |
| 2030 | 179 | storres | upperBound, |
| 2031 | 190 | storres | func = None, |
| 2032 | 179 | storres | domainDirection = -1, |
| 2033 | 179 | storres | imageDirection = -1): |
| 2034 | 179 | storres | """ |
| 2035 | 179 | storres | Assuming the function is increasing or decreasing over the |
| 2036 | 179 | storres | [lowerBound, upperBound] interval, return a lower bound lb and |
| 2037 | 179 | storres | an upper bound ub such that: |
| 2038 | 179 | storres | - lb and ub belong to the same binade; |
| 2039 | 179 | storres | - func(lb) and func(ub) belong to the same binade. |
| 2040 | 179 | storres | domainDirection indicate how bounds move to fit into the same binade: |
| 2041 | 179 | storres | - a negative value move the upper bound down; |
| 2042 | 179 | storres | - a positive value move the lower bound up. |
| 2043 | 179 | storres | imageDirection indicate how bounds move in order to have their image |
| 2044 | 179 | storres | fit into the same binade, variation of the function is also condidered. |
| 2045 | 179 | storres | For an increasing function: |
| 2046 | 179 | storres | - negative value moves the upper bound down (and its image value as well); |
| 2047 | 179 | storres | - a positive values moves the lower bound up (and its image value as well); |
| 2048 | 179 | storres | For a decreasing function it is the other way round. |
| 2049 | 179 | storres | """ |
| 2050 | 179 | storres | ## Arguments check |
| 2051 | 179 | storres | if lowerBound > upperBound: |
| 2052 | 179 | storres | return None |
| 2053 | 190 | storres | if lowerBound == upperBound: |
| 2054 | 190 | storres | return (lowerBound, upperBound) |
| 2055 | 179 | storres | if func is None: |
| 2056 | 179 | storres | return None |
| 2057 | 179 | storres | # |
| 2058 | 179 | storres | #varFunc = func.variables()[0] |
| 2059 | 179 | storres | lb = lowerBound |
| 2060 | 179 | storres | ub = upperBound |
| 2061 | 179 | storres | lbBinade = slz_compute_binade(lb) |
| 2062 | 179 | storres | ubBinade = slz_compute_binade(ub) |
| 2063 | 179 | storres | ## Domain binade. |
| 2064 | 179 | storres | while lbBinade != ubBinade: |
| 2065 | 179 | storres | newIntervalWidth = (ub - lb) / 2 |
| 2066 | 179 | storres | if domainDirection < 0: |
| 2067 | 179 | storres | ub = lb + newIntervalWidth |
| 2068 | 179 | storres | ubBinade = slz_compute_binade(ub) |
| 2069 | 179 | storres | else: |
| 2070 | 179 | storres | lb = lb + newIntervalWidth |
| 2071 | 179 | storres | lbBinade = slz_compute_binade(lb) |
| 2072 | 254 | storres | #print "sfbfb: lower bound:", lb.str(truncate=False) |
| 2073 | 254 | storres | #print "sfbfb: upper bound:", ub.str(truncate=False) |
| 2074 | 254 | storres | ## At this point, both bounds belond to the same binade. |
| 2075 | 179 | storres | ## Image binade. |
| 2076 | 179 | storres | if lb == ub: |
| 2077 | 179 | storres | return (lb, ub) |
| 2078 | 179 | storres | lbImg = func(lb) |
| 2079 | 179 | storres | ubImg = func(ub) |
| 2080 | 254 | storres | funcIsInc = ((ubImg - lbImg) >= 0) |
| 2081 | 179 | storres | lbImgBinade = slz_compute_binade(lbImg) |
| 2082 | 179 | storres | ubImgBinade = slz_compute_binade(ubImg) |
| 2083 | 179 | storres | while lbImgBinade != ubImgBinade: |
| 2084 | 179 | storres | newIntervalWidth = (ub - lb) / 2 |
| 2085 | 179 | storres | if imageDirection < 0: |
| 2086 | 179 | storres | if funcIsInc: |
| 2087 | 179 | storres | ub = lb + newIntervalWidth |
| 2088 | 179 | storres | ubImgBinade = slz_compute_binade(func(ub)) |
| 2089 | 179 | storres | #print ubImgBinade |
| 2090 | 179 | storres | else: |
| 2091 | 179 | storres | lb = lb + newIntervalWidth |
| 2092 | 179 | storres | lbImgBinade = slz_compute_binade(func(lb)) |
| 2093 | 179 | storres | #print lbImgBinade |
| 2094 | 179 | storres | else: |
| 2095 | 179 | storres | if funcIsInc: |
| 2096 | 179 | storres | lb = lb + newIntervalWidth |
| 2097 | 179 | storres | lbImgBinade = slz_compute_binade(func(lb)) |
| 2098 | 179 | storres | #print lbImgBinade |
| 2099 | 179 | storres | else: |
| 2100 | 179 | storres | ub = lb + newIntervalWidth |
| 2101 | 179 | storres | ubImgBinade = slz_compute_binade(func(ub)) |
| 2102 | 179 | storres | #print ubImgBinade |
| 2103 | 179 | storres | # End while lbImgBinade != ubImgBinade: |
| 2104 | 179 | storres | return (lb, ub) |
| 2105 | 179 | storres | # End slz_fix_bounds_for_binades. |
| 2106 | 179 | storres | |
| 2107 | 79 | storres | def slz_float_poly_of_float_to_rat_poly_of_rat(polyOfFloat): |
| 2108 | 79 | storres | # Create a polynomial over the rationals. |
| 2109 | 179 | storres | ratPolynomialRing = QQ[str(polyOfFloat.variables()[0])] |
| 2110 | 179 | storres | return(ratPolynomialRing(polyOfFloat)) |
| 2111 | 86 | storres | # End slz_float_poly_of_float_to_rat_poly_of_rat. |
| 2112 | 81 | storres | |
| 2113 | 188 | storres | def slz_float_poly_of_float_to_rat_poly_of_rat_pow_two(polyOfFloat): |
| 2114 | 188 | storres | """ |
| 2115 | 188 | storres | Create a polynomial over the rationals where all denominators are |
| 2116 | 188 | storres | powers of two. |
| 2117 | 188 | storres | """ |
| 2118 | 188 | storres | polyVariable = polyOfFloat.variables()[0] |
| 2119 | 188 | storres | RPR = QQ[str(polyVariable)] |
| 2120 | 188 | storres | polyCoeffs = polyOfFloat.coefficients() |
| 2121 | 188 | storres | #print polyCoeffs |
| 2122 | 188 | storres | polyExponents = polyOfFloat.exponents() |
| 2123 | 188 | storres | #print polyExponents |
| 2124 | 188 | storres | polyDenomPtwoCoeffs = [] |
| 2125 | 188 | storres | for coeff in polyCoeffs: |
| 2126 | 188 | storres | polyDenomPtwoCoeffs.append(sno_float_to_rat_pow_of_two_denom(coeff)) |
| 2127 | 188 | storres | #print "Converted coefficient:", sno_float_to_rat_pow_of_two_denom(coeff), |
| 2128 | 188 | storres | #print type(sno_float_to_rat_pow_of_two_denom(coeff)) |
| 2129 | 188 | storres | ratPoly = RPR(0) |
| 2130 | 188 | storres | #print type(ratPoly) |
| 2131 | 188 | storres | ## !!! CAUTION !!! Do not use the RPR(coeff * polyVariagle^exponent) |
| 2132 | 188 | storres | # The coefficient becomes plainly wrong when exponent == 0. |
| 2133 | 188 | storres | # No clue as to why. |
| 2134 | 188 | storres | for coeff, exponent in zip(polyDenomPtwoCoeffs, polyExponents): |
| 2135 | 188 | storres | ratPoly += coeff * RPR(polyVariable^exponent) |
| 2136 | 188 | storres | return ratPoly |
| 2137 | 188 | storres | # End slz_float_poly_of_float_to_rat_poly_of_rat. |
| 2138 | 188 | storres | |
| 2139 | 80 | storres | def slz_get_intervals_and_polynomials(functionSa, degreeSa, |
| 2140 | 63 | storres | lowerBoundSa, |
| 2141 | 269 | storres | upperBoundSa, |
| 2142 | 269 | storres | floatingPointPrecSa, |
| 2143 | 269 | storres | internalSollyaPrecSa, |
| 2144 | 269 | storres | approxAccurSa): |
| 2145 | 60 | storres | """ |
| 2146 | 60 | storres | Under the assumption that: |
| 2147 | 60 | storres | - functionSa is monotonic on the [lowerBoundSa, upperBoundSa] interval; |
| 2148 | 60 | storres | - lowerBound and upperBound belong to the same binade. |
| 2149 | 60 | storres | from a: |
| 2150 | 60 | storres | - function; |
| 2151 | 60 | storres | - a degree |
| 2152 | 60 | storres | - a pair of bounds; |
| 2153 | 60 | storres | - the floating-point precision we work on; |
| 2154 | 60 | storres | - the internal Sollya precision; |
| 2155 | 64 | storres | - the requested approximation error |
| 2156 | 61 | storres | The initial interval is, possibly, splitted into smaller intervals. |
| 2157 | 61 | storres | It return a list of tuples, each made of: |
| 2158 | 72 | storres | - a first polynomial (without the changed variable f(x) = p(x-x0)); |
| 2159 | 79 | storres | - a second polynomial (with a changed variable f(x) = q(x)) |
| 2160 | 61 | storres | - the approximation interval; |
| 2161 | 72 | storres | - the center, x0, of the interval; |
| 2162 | 61 | storres | - the corresponding approximation error. |
| 2163 | 100 | storres | TODO: fix endless looping for some parameters sets. |
| 2164 | 60 | storres | """ |
| 2165 | 120 | storres | resultArray = [] |
| 2166 | 101 | storres | # Set Sollya to the necessary internal precision. |
| 2167 | 226 | storres | sollyaPrecChangedSa = False |
| 2168 | 226 | storres | (initialSollyaPrecSo, initialSollyaPrecSa) = pobyso_get_prec_so_so_sa() |
| 2169 | 269 | storres | if internalSollyaPrecSa > initialSollyaPrecSa: |
| 2170 | 226 | storres | if internalSollyaPrecSa <= 2: |
| 2171 | 226 | storres | print inspect.stack()[0][3], ": precision change <=2 requested." |
| 2172 | 85 | storres | pobyso_set_prec_sa_so(internalSollyaPrecSa) |
| 2173 | 226 | storres | sollyaPrecChangedSa = True |
| 2174 | 101 | storres | # |
| 2175 | 80 | storres | x = functionSa.variables()[0] # Actual variable name can be anything. |
| 2176 | 101 | storres | # Scaled function: [1=,2] -> [1,2]. |
| 2177 | 115 | storres | (fff, scaledLowerBoundSa, scaledUpperBoundSa, \ |
| 2178 | 115 | storres | scaledLowerBoundImageSa, scaledUpperBoundImageSa) = \ |
| 2179 | 115 | storres | slz_compute_scaled_function(functionSa, \ |
| 2180 | 115 | storres | lowerBoundSa, \ |
| 2181 | 115 | storres | upperBoundSa, \ |
| 2182 | 80 | storres | floatingPointPrecSa) |
| 2183 | 166 | storres | # In case bounds were in the negative real one may need to |
| 2184 | 166 | storres | # switch scaled bounds. |
| 2185 | 166 | storres | if scaledLowerBoundSa > scaledUpperBoundSa: |
| 2186 | 166 | storres | scaledLowerBoundSa, scaledUpperBoundSa = \ |
| 2187 | 166 | storres | scaledUpperBoundSa, scaledLowerBoundSa |
| 2188 | 166 | storres | #print "Switching!" |
| 2189 | 218 | storres | print "Approximation accuracy: ", RR(approxAccurSa) |
| 2190 | 61 | storres | # Prepare the arguments for the Taylor expansion computation with Sollya. |
| 2191 | 159 | storres | functionSo = \ |
| 2192 | 159 | storres | pobyso_parse_string_sa_so(fff._assume_str().replace('_SAGE_VAR_', ''))
|
| 2193 | 60 | storres | degreeSo = pobyso_constant_from_int_sa_so(degreeSa) |
| 2194 | 61 | storres | scaledBoundsSo = pobyso_bounds_to_range_sa_so(scaledLowerBoundSa, |
| 2195 | 61 | storres | scaledUpperBoundSa) |
| 2196 | 176 | storres | |
| 2197 | 60 | storres | realIntervalField = RealIntervalField(max(lowerBoundSa.parent().precision(), |
| 2198 | 60 | storres | upperBoundSa.parent().precision())) |
| 2199 | 176 | storres | currentScaledLowerBoundSa = scaledLowerBoundSa |
| 2200 | 176 | storres | currentScaledUpperBoundSa = scaledUpperBoundSa |
| 2201 | 176 | storres | while True: |
| 2202 | 176 | storres | ## Compute the first Taylor expansion. |
| 2203 | 176 | storres | print "Computing a Taylor expansion..." |
| 2204 | 176 | storres | (polySo, boundsSo, intervalCenterSo, maxErrorSo) = \ |
| 2205 | 176 | storres | slz_compute_polynomial_and_interval(functionSo, degreeSo, |
| 2206 | 176 | storres | currentScaledLowerBoundSa, |
| 2207 | 176 | storres | currentScaledUpperBoundSa, |
| 2208 | 218 | storres | approxAccurSa, internalSollyaPrecSa) |
| 2209 | 176 | storres | print "...done." |
| 2210 | 176 | storres | ## If slz_compute_polynomial_and_interval fails, it returns None. |
| 2211 | 176 | storres | # This value goes to the first variable: polySo. Other variables are |
| 2212 | 176 | storres | # not assigned and should not be tested. |
| 2213 | 176 | storres | if polySo is None: |
| 2214 | 176 | storres | print "slz_get_intervals_and_polynomials: Aborting and returning None!" |
| 2215 | 176 | storres | if precChangedSa: |
| 2216 | 226 | storres | pobyso_set_prec_so_so(initialSollyaPrecSo) |
| 2217 | 226 | storres | sollya_lib_clear_obj(initialSollyaPrecSo) |
| 2218 | 176 | storres | sollya_lib_clear_obj(functionSo) |
| 2219 | 176 | storres | sollya_lib_clear_obj(degreeSo) |
| 2220 | 176 | storres | sollya_lib_clear_obj(scaledBoundsSo) |
| 2221 | 176 | storres | return None |
| 2222 | 176 | storres | ## Add to the result array. |
| 2223 | 176 | storres | ### Change variable stuff in Sollya x -> x0-x. |
| 2224 | 176 | storres | changeVarExpressionSo = \ |
| 2225 | 176 | storres | sollya_lib_build_function_sub( \ |
| 2226 | 176 | storres | sollya_lib_build_function_free_variable(), |
| 2227 | 101 | storres | sollya_lib_copy_obj(intervalCenterSo)) |
| 2228 | 176 | storres | polyVarChangedSo = \ |
| 2229 | 176 | storres | sollya_lib_evaluate(polySo, changeVarExpressionSo) |
| 2230 | 176 | storres | #### Get rid of the variable change Sollya stuff. |
| 2231 | 115 | storres | sollya_lib_clear_obj(changeVarExpressionSo) |
| 2232 | 176 | storres | resultArray.append((polySo, polyVarChangedSo, boundsSo, |
| 2233 | 101 | storres | intervalCenterSo, maxErrorSo)) |
| 2234 | 176 | storres | boundsSa = pobyso_range_to_interval_so_sa(boundsSo, realIntervalField) |
| 2235 | 101 | storres | errorSa = pobyso_get_constant_as_rn_with_rf_so_sa(maxErrorSo) |
| 2236 | 176 | storres | print "Computed approximation error:", errorSa.n(digits=10) |
| 2237 | 176 | storres | # If the error and interval are OK a the first try, just return. |
| 2238 | 176 | storres | if (boundsSa.endpoints()[1] >= scaledUpperBoundSa) and \ |
| 2239 | 218 | storres | (errorSa <= approxAccurSa): |
| 2240 | 269 | storres | if sollyaPrecChangedSa: |
| 2241 | 226 | storres | pobyso_set_prec_so_so(initialSollyaPrecSo) |
| 2242 | 226 | storres | sollya_lib_clear_obj(initialSollyaPrecSo) |
| 2243 | 176 | storres | sollya_lib_clear_obj(functionSo) |
| 2244 | 176 | storres | sollya_lib_clear_obj(degreeSo) |
| 2245 | 176 | storres | sollya_lib_clear_obj(scaledBoundsSo) |
| 2246 | 101 | storres | #print "Approximation error:", errorSa |
| 2247 | 176 | storres | return resultArray |
| 2248 | 176 | storres | ## The returned interval upper bound does not reach the requested upper |
| 2249 | 176 | storres | # upper bound: compute the next upper bound. |
| 2250 | 176 | storres | ## The following ratio is always >= 1. If errorSa, we may want to |
| 2251 | 176 | storres | # enlarge the interval |
| 2252 | 218 | storres | currentErrorRatio = approxAccurSa / errorSa |
| 2253 | 176 | storres | ## --|--------------------------------------------------------------|-- |
| 2254 | 176 | storres | # --|--------------------|-------------------------------------------- |
| 2255 | 176 | storres | # --|----------------------------|------------------------------------ |
| 2256 | 176 | storres | ## Starting point for the next upper bound. |
| 2257 | 101 | storres | boundsWidthSa = boundsSa.endpoints()[1] - boundsSa.endpoints()[0] |
| 2258 | 101 | storres | # Compute the increment. |
| 2259 | 176 | storres | newBoundsWidthSa = \ |
| 2260 | 176 | storres | ((currentErrorRatio.log() / 10) + 1) * boundsWidthSa |
| 2261 | 176 | storres | currentScaledLowerBoundSa = boundsSa.endpoints()[1] |
| 2262 | 176 | storres | currentScaledUpperBoundSa = boundsSa.endpoints()[1] + newBoundsWidthSa |
| 2263 | 176 | storres | # Take into account the original interval upper bound. |
| 2264 | 176 | storres | if currentScaledUpperBoundSa > scaledUpperBoundSa: |
| 2265 | 176 | storres | currentScaledUpperBoundSa = scaledUpperBoundSa |
| 2266 | 176 | storres | if currentScaledUpperBoundSa == currentScaledLowerBoundSa: |
| 2267 | 85 | storres | print "Can't shrink the interval anymore!" |
| 2268 | 85 | storres | print "You should consider increasing the Sollya internal precision" |
| 2269 | 85 | storres | print "or the polynomial degree." |
| 2270 | 85 | storres | print "Giving up!" |
| 2271 | 176 | storres | if precChangedSa: |
| 2272 | 226 | storres | pobyso_set_prec_so_so(initialSollyaPrecSo) |
| 2273 | 226 | storres | sollya_lib_clear_obj(initialSollyaPrecSo) |
| 2274 | 85 | storres | sollya_lib_clear_obj(functionSo) |
| 2275 | 85 | storres | sollya_lib_clear_obj(degreeSo) |
| 2276 | 85 | storres | sollya_lib_clear_obj(scaledBoundsSo) |
| 2277 | 85 | storres | return None |
| 2278 | 176 | storres | # Compute the other expansions. |
| 2279 | 176 | storres | # Test for insufficient precision. |
| 2280 | 81 | storres | # End slz_get_intervals_and_polynomials |
| 2281 | 60 | storres | |
| 2282 | 80 | storres | def slz_interval_scaling_expression(boundsInterval, expVar): |
| 2283 | 61 | storres | """ |
| 2284 | 151 | storres | Compute the scaling expression to map an interval that spans at most |
| 2285 | 166 | storres | a single binade into [1, 2) and the inverse expression as well. |
| 2286 | 165 | storres | If the interval spans more than one binade, result is wrong since |
| 2287 | 165 | storres | scaling is only based on the lower bound. |
| 2288 | 62 | storres | Not very sure that the transformation makes sense for negative numbers. |
| 2289 | 61 | storres | """ |
| 2290 | 165 | storres | # The "one of the bounds is 0" special case. Here we consider |
| 2291 | 165 | storres | # the interval as the subnormals binade. |
| 2292 | 165 | storres | if boundsInterval.endpoints()[0] == 0 or boundsInterval.endpoints()[1] == 0: |
| 2293 | 165 | storres | # The upper bound is (or should be) positive. |
| 2294 | 165 | storres | if boundsInterval.endpoints()[0] == 0: |
| 2295 | 165 | storres | if boundsInterval.endpoints()[1] == 0: |
| 2296 | 165 | storres | return None |
| 2297 | 165 | storres | binade = slz_compute_binade(boundsInterval.endpoints()[1]) |
| 2298 | 165 | storres | l2 = boundsInterval.endpoints()[1].abs().log2() |
| 2299 | 165 | storres | # The upper bound is a power of two |
| 2300 | 165 | storres | if binade == l2: |
| 2301 | 165 | storres | scalingCoeff = 2^(-binade) |
| 2302 | 165 | storres | invScalingCoeff = 2^(binade) |
| 2303 | 165 | storres | scalingOffset = 1 |
| 2304 | 179 | storres | return \ |
| 2305 | 179 | storres | ((scalingCoeff * expVar + scalingOffset).function(expVar), |
| 2306 | 179 | storres | ((expVar - scalingOffset) * invScalingCoeff).function(expVar)) |
| 2307 | 165 | storres | else: |
| 2308 | 165 | storres | scalingCoeff = 2^(-binade-1) |
| 2309 | 165 | storres | invScalingCoeff = 2^(binade+1) |
| 2310 | 165 | storres | scalingOffset = 1 |
| 2311 | 165 | storres | return((scalingCoeff * expVar + scalingOffset),\ |
| 2312 | 165 | storres | ((expVar - scalingOffset) * invScalingCoeff)) |
| 2313 | 165 | storres | # The lower bound is (or should be) negative. |
| 2314 | 165 | storres | if boundsInterval.endpoints()[1] == 0: |
| 2315 | 165 | storres | if boundsInterval.endpoints()[0] == 0: |
| 2316 | 165 | storres | return None |
| 2317 | 165 | storres | binade = slz_compute_binade(boundsInterval.endpoints()[0]) |
| 2318 | 165 | storres | l2 = boundsInterval.endpoints()[0].abs().log2() |
| 2319 | 165 | storres | # The upper bound is a power of two |
| 2320 | 165 | storres | if binade == l2: |
| 2321 | 165 | storres | scalingCoeff = -2^(-binade) |
| 2322 | 165 | storres | invScalingCoeff = -2^(binade) |
| 2323 | 165 | storres | scalingOffset = 1 |
| 2324 | 165 | storres | return((scalingCoeff * expVar + scalingOffset),\ |
| 2325 | 165 | storres | ((expVar - scalingOffset) * invScalingCoeff)) |
| 2326 | 165 | storres | else: |
| 2327 | 165 | storres | scalingCoeff = -2^(-binade-1) |
| 2328 | 165 | storres | invScalingCoeff = -2^(binade+1) |
| 2329 | 165 | storres | scalingOffset = 1 |
| 2330 | 165 | storres | return((scalingCoeff * expVar + scalingOffset),\ |
| 2331 | 165 | storres | ((expVar - scalingOffset) * invScalingCoeff)) |
| 2332 | 165 | storres | # General case |
| 2333 | 165 | storres | lbBinade = slz_compute_binade(boundsInterval.endpoints()[0]) |
| 2334 | 165 | storres | ubBinade = slz_compute_binade(boundsInterval.endpoints()[1]) |
| 2335 | 165 | storres | # We allow for a single exception in binade spanning is when the |
| 2336 | 165 | storres | # "outward bound" is a power of 2 and its binade is that of the |
| 2337 | 165 | storres | # "inner bound" + 1. |
| 2338 | 165 | storres | if boundsInterval.endpoints()[0] > 0: |
| 2339 | 165 | storres | ubL2 = boundsInterval.endpoints()[1].abs().log2() |
| 2340 | 165 | storres | if lbBinade != ubBinade: |
| 2341 | 165 | storres | print "Different binades." |
| 2342 | 165 | storres | if ubL2 != ubBinade: |
| 2343 | 165 | storres | print "Not a power of 2." |
| 2344 | 165 | storres | return None |
| 2345 | 165 | storres | elif abs(ubBinade - lbBinade) > 1: |
| 2346 | 165 | storres | print "Too large span:", abs(ubBinade - lbBinade) |
| 2347 | 165 | storres | return None |
| 2348 | 165 | storres | else: |
| 2349 | 165 | storres | lbL2 = boundsInterval.endpoints()[0].abs().log2() |
| 2350 | 165 | storres | if lbBinade != ubBinade: |
| 2351 | 165 | storres | print "Different binades." |
| 2352 | 165 | storres | if lbL2 != lbBinade: |
| 2353 | 165 | storres | print "Not a power of 2." |
| 2354 | 165 | storres | return None |
| 2355 | 165 | storres | elif abs(ubBinade - lbBinade) > 1: |
| 2356 | 165 | storres | print "Too large span:", abs(ubBinade - lbBinade) |
| 2357 | 165 | storres | return None |
| 2358 | 165 | storres | #print "Lower bound binade:", binade |
| 2359 | 165 | storres | if boundsInterval.endpoints()[0] > 0: |
| 2360 | 179 | storres | return \ |
| 2361 | 179 | storres | ((2^(-lbBinade) * expVar).function(expVar), |
| 2362 | 179 | storres | (2^(lbBinade) * expVar).function(expVar)) |
| 2363 | 165 | storres | else: |
| 2364 | 179 | storres | return \ |
| 2365 | 179 | storres | ((-(2^(-ubBinade)) * expVar).function(expVar), |
| 2366 | 179 | storres | (-(2^(ubBinade)) * expVar).function(expVar)) |
| 2367 | 165 | storres | """ |
| 2368 | 165 | storres | # Code sent to attic. Should be the base for a |
| 2369 | 165 | storres | # "slz_interval_translate_expression" rather than scale. |
| 2370 | 165 | storres | # Extra control and special cases code added in |
| 2371 | 165 | storres | # slz_interval_scaling_expression could (should ?) be added to |
| 2372 | 165 | storres | # this new function. |
| 2373 | 62 | storres | # The scaling offset is only used for negative numbers. |
| 2374 | 151 | storres | # When the absolute value of the lower bound is < 0. |
| 2375 | 61 | storres | if abs(boundsInterval.endpoints()[0]) < 1: |
| 2376 | 61 | storres | if boundsInterval.endpoints()[0] >= 0: |
| 2377 | 62 | storres | scalingCoeff = 2^floor(boundsInterval.endpoints()[0].log2()) |
| 2378 | 62 | storres | invScalingCoeff = 1/scalingCoeff |
| 2379 | 80 | storres | return((scalingCoeff * expVar, |
| 2380 | 80 | storres | invScalingCoeff * expVar)) |
| 2381 | 60 | storres | else: |
| 2382 | 62 | storres | scalingCoeff = \ |
| 2383 | 62 | storres | 2^(floor((-boundsInterval.endpoints()[0]).log2()) - 1) |
| 2384 | 62 | storres | scalingOffset = -3 * scalingCoeff |
| 2385 | 80 | storres | return((scalingCoeff * expVar + scalingOffset, |
| 2386 | 80 | storres | 1/scalingCoeff * expVar + 3)) |
| 2387 | 61 | storres | else: |
| 2388 | 61 | storres | if boundsInterval.endpoints()[0] >= 0: |
| 2389 | 62 | storres | scalingCoeff = 2^floor(boundsInterval.endpoints()[0].log2()) |
| 2390 | 61 | storres | scalingOffset = 0 |
| 2391 | 80 | storres | return((scalingCoeff * expVar, |
| 2392 | 80 | storres | 1/scalingCoeff * expVar)) |
| 2393 | 61 | storres | else: |
| 2394 | 62 | storres | scalingCoeff = \ |
| 2395 | 62 | storres | 2^(floor((-boundsInterval.endpoints()[1]).log2())) |
| 2396 | 62 | storres | scalingOffset = -3 * scalingCoeff |
| 2397 | 62 | storres | #scalingOffset = 0 |
| 2398 | 80 | storres | return((scalingCoeff * expVar + scalingOffset, |
| 2399 | 80 | storres | 1/scalingCoeff * expVar + 3)) |
| 2400 | 165 | storres | """ |
| 2401 | 151 | storres | # End slz_interval_scaling_expression |
| 2402 | 61 | storres | |
| 2403 | 83 | storres | def slz_interval_and_polynomial_to_sage(polyRangeCenterErrorSo): |
| 2404 | 72 | storres | """ |
| 2405 | 72 | storres | Compute the Sage version of the Taylor polynomial and it's |
| 2406 | 72 | storres | companion data (interval, center...) |
| 2407 | 72 | storres | The input parameter is a five elements tuple: |
| 2408 | 79 | storres | - [0]: the polyomial (without variable change), as polynomial over a |
| 2409 | 79 | storres | real ring; |
| 2410 | 79 | storres | - [1]: the polyomial (with variable change done in Sollya), as polynomial |
| 2411 | 79 | storres | over a real ring; |
| 2412 | 72 | storres | - [2]: the interval (as Sollya range); |
| 2413 | 72 | storres | - [3]: the interval center; |
| 2414 | 72 | storres | - [4]: the approximation error. |
| 2415 | 72 | storres | |
| 2416 | 218 | storres | The function returns a 5 elements tuple: formed with all the |
| 2417 | 72 | storres | input elements converted into their Sollya counterpart. |
| 2418 | 72 | storres | """ |
| 2419 | 233 | storres | #print "Sollya polynomial to convert:", |
| 2420 | 233 | storres | #pobyso_autoprint(polyRangeCenterErrorSo) |
| 2421 | 218 | storres | polynomialSa = pobyso_float_poly_so_sa(polyRangeCenterErrorSo[0]) |
| 2422 | 218 | storres | #print "Polynomial after first conversion: ", pobyso_autoprint(polyRangeCenterErrorSo[1]) |
| 2423 | 218 | storres | polynomialChangedVarSa = pobyso_float_poly_so_sa(polyRangeCenterErrorSo[1]) |
| 2424 | 60 | storres | intervalSa = \ |
| 2425 | 64 | storres | pobyso_get_interval_from_range_so_sa(polyRangeCenterErrorSo[2]) |
| 2426 | 60 | storres | centerSa = \ |
| 2427 | 64 | storres | pobyso_get_constant_as_rn_with_rf_so_sa(polyRangeCenterErrorSo[3]) |
| 2428 | 60 | storres | errorSa = \ |
| 2429 | 64 | storres | pobyso_get_constant_as_rn_with_rf_so_sa(polyRangeCenterErrorSo[4]) |
| 2430 | 64 | storres | return((polynomialSa, polynomialChangedVarSa, intervalSa, centerSa, errorSa)) |
| 2431 | 83 | storres | # End slz_interval_and_polynomial_to_sage |
| 2432 | 62 | storres | |
| 2433 | 269 | storres | def slz_is_htrn(argument, |
| 2434 | 269 | storres | function, |
| 2435 | 269 | storres | targetAccuracy, |
| 2436 | 269 | storres | targetRF = None, |
| 2437 | 269 | storres | targetPlusOnePrecRF = None, |
| 2438 | 269 | storres | quasiExactRF = None): |
| 2439 | 172 | storres | """ |
| 2440 | 172 | storres | Check if an element (argument) of the domain of function (function) |
| 2441 | 172 | storres | yields a HTRN case (rounding to next) for the target precision |
| 2442 | 183 | storres | (as impersonated by targetRF) for a given accuracy (targetAccuracy). |
| 2443 | 205 | storres | |
| 2444 | 205 | storres | The strategy is: |
| 2445 | 205 | storres | - compute the image at high (quasi-exact) precision; |
| 2446 | 205 | storres | - round it to nearest to precision; |
| 2447 | 205 | storres | - round it to exactly to (precision+1), the computed number has two |
| 2448 | 205 | storres | midpoint neighbors; |
| 2449 | 205 | storres | - check the distance between these neighbors and the quasi-exact value; |
| 2450 | 205 | storres | - if none of them is closer than the targetAccuracy, return False, |
| 2451 | 205 | storres | - else return true. |
| 2452 | 205 | storres | - Powers of two are a special case when comparing the midpoint in |
| 2453 | 205 | storres | the 0 direction.. |
| 2454 | 172 | storres | """ |
| 2455 | 183 | storres | ## Arguments filtering. |
| 2456 | 183 | storres | ## TODO: filter the first argument for consistence. |
| 2457 | 172 | storres | if targetRF is None: |
| 2458 | 172 | storres | targetRF = argument.parent() |
| 2459 | 172 | storres | ## Ditto for the real field holding the midpoints. |
| 2460 | 172 | storres | if targetPlusOnePrecRF is None: |
| 2461 | 172 | storres | targetPlusOnePrecRF = RealField(targetRF.prec()+1) |
| 2462 | 183 | storres | ## If no quasiExactField is provided, create a high accuracy RealField. |
| 2463 | 172 | storres | if quasiExactRF is None: |
| 2464 | 172 | storres | quasiExactRF = RealField(1536) |
| 2465 | 195 | storres | function = function.function(function.variables()[0]) |
| 2466 | 172 | storres | exactArgument = quasiExactRF(argument) |
| 2467 | 172 | storres | quasiExactValue = function(exactArgument) |
| 2468 | 172 | storres | roundedValue = targetRF(quasiExactValue) |
| 2469 | 172 | storres | roundedValuePrecPlusOne = targetPlusOnePrecRF(roundedValue) |
| 2470 | 172 | storres | # Upper midpoint. |
| 2471 | 172 | storres | roundedValuePrecPlusOneNext = roundedValuePrecPlusOne.nextabove() |
| 2472 | 172 | storres | # Lower midpoint. |
| 2473 | 172 | storres | roundedValuePrecPlusOnePrev = roundedValuePrecPlusOne.nextbelow() |
| 2474 | 172 | storres | binade = slz_compute_binade(roundedValue) |
| 2475 | 172 | storres | binadeCorrectedTargetAccuracy = targetAccuracy * 2^binade |
| 2476 | 172 | storres | #print "Argument:", argument |
| 2477 | 172 | storres | #print "f(x):", roundedValue, binade, floor(binade), ceil(binade) |
| 2478 | 174 | storres | #print "Binade:", binade |
| 2479 | 172 | storres | #print "binadeCorrectedTargetAccuracy:", \ |
| 2480 | 174 | storres | #binadeCorrectedTargetAccuracy.n(prec=100) |
| 2481 | 172 | storres | #print "binadeCorrectedTargetAccuracy:", \ |
| 2482 | 172 | storres | # binadeCorrectedTargetAccuracy.n(prec=100).str(base=2) |
| 2483 | 172 | storres | #print "Upper midpoint:", \ |
| 2484 | 172 | storres | # roundedValuePrecPlusOneNext.n(prec=targetPlusOnePrecRF.prec()).str(base=2) |
| 2485 | 172 | storres | #print "Rounded value :", \ |
| 2486 | 172 | storres | # roundedValuePrecPlusOne.n(prec=targetPlusOnePrecRF.prec()).str(base=2), \ |
| 2487 | 172 | storres | # roundedValuePrecPlusOne.ulp().n(prec=2).str(base=2) |
| 2488 | 172 | storres | #print "QuasiEx value :", quasiExactValue.n(prec=250).str(base=2) |
| 2489 | 172 | storres | #print "Lower midpoint:", \ |
| 2490 | 172 | storres | # roundedValuePrecPlusOnePrev.n(prec=targetPlusOnePrecRF.prec()).str(base=2) |
| 2491 | 205 | storres | ## Make quasiExactValue = 0 a special case to move it out of the way. |
| 2492 | 205 | storres | if quasiExactValue == 0: |
| 2493 | 205 | storres | return False |
| 2494 | 205 | storres | ## Begining of the general case : check with the midpoint of |
| 2495 | 172 | storres | # greatest absolute value. |
| 2496 | 172 | storres | if quasiExactValue > 0: |
| 2497 | 172 | storres | if abs(quasiExactRF(roundedValuePrecPlusOneNext) - quasiExactValue) <\ |
| 2498 | 172 | storres | binadeCorrectedTargetAccuracy: |
| 2499 | 183 | storres | #print "Too close to the upper midpoint: ", \ |
| 2500 | 174 | storres | #abs(quasiExactRF(roundedValuePrecPlusOneNext) - quasiExactValue).n(prec=100) |
| 2501 | 172 | storres | #print argument.n().str(base=16, \ |
| 2502 | 172 | storres | # no_sci=False) |
| 2503 | 172 | storres | #print "Lower midpoint:", \ |
| 2504 | 172 | storres | # roundedValuePrecPlusOnePrev.n(prec=targetPlusOnePrecRF.prec()).str(base=2) |
| 2505 | 172 | storres | #print "Value :", \ |
| 2506 | 183 | storres | # quasiExactValue.n(prec=200).str(base=2) |
| 2507 | 172 | storres | #print "Upper midpoint:", \ |
| 2508 | 172 | storres | # roundedValuePrecPlusOneNext.n(prec=targetPlusOnePrecRF.prec()).str(base=2) |
| 2509 | 172 | storres | return True |
| 2510 | 205 | storres | else: # quasiExactValue < 0, the 0 case has been previously filtered out. |
| 2511 | 172 | storres | if abs(quasiExactRF(roundedValuePrecPlusOnePrev) - quasiExactValue) < \ |
| 2512 | 172 | storres | binadeCorrectedTargetAccuracy: |
| 2513 | 172 | storres | #print "Too close to the upper midpoint: ", \ |
| 2514 | 172 | storres | # abs(quasiExactRF(roundedValuePrecPlusOneNext) - quasiExactValue).n(prec=100) |
| 2515 | 172 | storres | #print argument.n().str(base=16, \ |
| 2516 | 172 | storres | # no_sci=False) |
| 2517 | 172 | storres | #print "Lower midpoint:", \ |
| 2518 | 172 | storres | # roundedValuePrecPlusOnePrev.n(prec=targetPlusOnePrecRF.prec()).str(base=2) |
| 2519 | 172 | storres | #print "Value :", \ |
| 2520 | 172 | storres | # quasiExactValue.n(prec=200).str(base=2) |
| 2521 | 172 | storres | #print "Upper midpoint:", \ |
| 2522 | 172 | storres | # roundedValuePrecPlusOneNext.n(prec=targetPlusOnePrecRF.prec()).str(base=2) |
| 2523 | 172 | storres | |
| 2524 | 172 | storres | return True |
| 2525 | 172 | storres | #2345678901234567890123456789012345678901234567890123456789012345678901234567890 |
| 2526 | 172 | storres | ## For the midpoint of smaller absolute value, |
| 2527 | 172 | storres | # split cases with the powers of 2. |
| 2528 | 172 | storres | if sno_abs_is_power_of_two(roundedValue): |
| 2529 | 172 | storres | if quasiExactValue > 0: |
| 2530 | 172 | storres | if abs(quasiExactRF(roundedValuePrecPlusOnePrev) - quasiExactValue) <\ |
| 2531 | 172 | storres | binadeCorrectedTargetAccuracy / 2: |
| 2532 | 172 | storres | #print "Lower midpoint:", \ |
| 2533 | 172 | storres | # roundedValuePrecPlusOnePrev.n(prec=targetPlusOnePrecRF.prec()).str(base=2) |
| 2534 | 172 | storres | #print "Value :", \ |
| 2535 | 172 | storres | # quasiExactValue.n(prec=200).str(base=2) |
| 2536 | 172 | storres | #print "Upper midpoint:", \ |
| 2537 | 172 | storres | # roundedValuePrecPlusOneNext.n(prec=targetPlusOnePrecRF.prec()).str(base=2) |
| 2538 | 172 | storres | |
| 2539 | 172 | storres | return True |
| 2540 | 172 | storres | else: |
| 2541 | 172 | storres | if abs(quasiExactRF(roundedValuePrecPlusOneNext) - quasiExactValue) < \ |
| 2542 | 172 | storres | binadeCorrectedTargetAccuracy / 2: |
| 2543 | 172 | storres | #print "Lower midpoint:", \ |
| 2544 | 172 | storres | # roundedValuePrecPlusOnePrev.n(prec=targetPlusOnePrecRF.prec()).str(base=2) |
| 2545 | 172 | storres | #print "Value :", |
| 2546 | 172 | storres | # quasiExactValue.n(prec=200).str(base=2) |
| 2547 | 172 | storres | #print "Upper midpoint:", |
| 2548 | 172 | storres | # roundedValuePrecPlusOneNext.n(prec=targetPlusOnePrecRF.prec()).str(base=2) |
| 2549 | 172 | storres | |
| 2550 | 172 | storres | return True |
| 2551 | 172 | storres | #2345678901234567890123456789012345678901234567890123456789012345678901234567890 |
| 2552 | 172 | storres | else: ## Not a power of 2, regular comparison with the midpoint of |
| 2553 | 172 | storres | # smaller absolute value. |
| 2554 | 172 | storres | if quasiExactValue > 0: |
| 2555 | 172 | storres | if abs(quasiExactRF(roundedValuePrecPlusOnePrev) - quasiExactValue) < \ |
| 2556 | 172 | storres | binadeCorrectedTargetAccuracy: |
| 2557 | 172 | storres | #print "Lower midpoint:", \ |
| 2558 | 172 | storres | # roundedValuePrecPlusOnePrev.n(prec=targetPlusOnePrecRF.prec()).str(base=2) |
| 2559 | 172 | storres | #print "Value :", \ |
| 2560 | 172 | storres | # quasiExactValue.n(prec=200).str(base=2) |
| 2561 | 172 | storres | #print "Upper midpoint:", \ |
| 2562 | 172 | storres | # roundedValuePrecPlusOneNext.n(prec=targetPlusOnePrecRF.prec()).str(base=2) |
| 2563 | 172 | storres | |
| 2564 | 172 | storres | return True |
| 2565 | 172 | storres | else: # quasiExactValue <= 0 |
| 2566 | 172 | storres | if abs(quasiExactRF(roundedValuePrecPlusOneNext) - quasiExactValue) < \ |
| 2567 | 172 | storres | binadeCorrectedTargetAccuracy: |
| 2568 | 172 | storres | #print "Lower midpoint:", \ |
| 2569 | 172 | storres | # roundedValuePrecPlusOnePrev.n(prec=targetPlusOnePrecRF.prec()).str(base=2) |
| 2570 | 172 | storres | #print "Value :", \ |
| 2571 | 172 | storres | # quasiExactValue.n(prec=200).str(base=2) |
| 2572 | 172 | storres | #print "Upper midpoint:", \ |
| 2573 | 172 | storres | # roundedValuePrecPlusOneNext.n(prec=targetPlusOnePrecRF.prec()).str(base=2) |
| 2574 | 172 | storres | |
| 2575 | 172 | storres | return True |
| 2576 | 172 | storres | #print "distance to the upper midpoint:", \ |
| 2577 | 172 | storres | # abs(quasiExactRF(roundedValuePrecPlusOneNext) - quasiExactValue).n(prec=100).str(base=2) |
| 2578 | 172 | storres | #print "distance to the lower midpoint:", \ |
| 2579 | 172 | storres | # abs(quasiExactRF(roundedValuePrecPlusOnePrev) - quasiExactValue).n(prec=100).str(base=2) |
| 2580 | 172 | storres | return False |
| 2581 | 172 | storres | # End slz_is_htrn |
| 2582 | 247 | storres | # |
| 2583 | 247 | storres | def slz_pm1(): |
| 2584 | 247 | storres | """ |
| 2585 | 253 | storres | Compute a uniform RV in {-1, 1} (not zero).
|
| 2586 | 247 | storres | """ |
| 2587 | 247 | storres | ## function getrandbits(N) generates a long int with N random bits. |
| 2588 | 276 | storres | # Here it generates either 0 or 1. The multiplication by 2 and the |
| 2589 | 276 | storres | # subtraction of 1 make that: |
| 2590 | 276 | storres | # if getranbits(1) == 0 -> O * 2 - 1 = -1 |
| 2591 | 276 | storres | # else -> 1 * 1 - 1 = 1. |
| 2592 | 247 | storres | return getrandbits(1) * 2-1 |
| 2593 | 247 | storres | # End slz_pm1. |
| 2594 | 247 | storres | # |
| 2595 | 247 | storres | def slz_random_proj_pm1(r, c): |
| 2596 | 247 | storres | """ |
| 2597 | 247 | storres | r x c matrix with \pm 1 ({-1, 1}) coefficients
|
| 2598 | 247 | storres | """ |
| 2599 | 247 | storres | M = matrix(r, c) |
| 2600 | 247 | storres | for i in range(0, r): |
| 2601 | 247 | storres | for j in range (0, c): |
| 2602 | 247 | storres | M[i,j] = slz_pm1() |
| 2603 | 247 | storres | return M |
| 2604 | 247 | storres | # End random_proj_pm1. |
| 2605 | 247 | storres | # |
| 2606 | 247 | storres | def slz_random_proj_uniform(n, r, c): |
| 2607 | 247 | storres | """ |
| 2608 | 277 | storres | r x c integer matrix with uniform n-bit integer elements. |
| 2609 | 247 | storres | """ |
| 2610 | 247 | storres | M = matrix(r, c) |
| 2611 | 247 | storres | for i in range(0, r): |
| 2612 | 247 | storres | for j in range (0, c): |
| 2613 | 247 | storres | M[i,j] = slz_uniform(n) |
| 2614 | 247 | storres | return M |
| 2615 | 247 | storres | # End slz_random_proj_uniform. |
| 2616 | 247 | storres | # |
| 2617 | 80 | storres | def slz_rat_poly_of_int_to_poly_for_coppersmith(ratPolyOfInt, |
| 2618 | 80 | storres | precision, |
| 2619 | 80 | storres | targetHardnessToRound, |
| 2620 | 80 | storres | variable1, |
| 2621 | 80 | storres | variable2): |
| 2622 | 80 | storres | """ |
| 2623 | 90 | storres | Creates a new multivariate polynomial with integer coefficients for use |
| 2624 | 90 | storres | with the Coppersmith method. |
| 2625 | 80 | storres | A the same time it computes : |
| 2626 | 80 | storres | - 2^K (N); |
| 2627 | 90 | storres | - 2^k (bound on the second variable) |
| 2628 | 80 | storres | - lcm |
| 2629 | 90 | storres | |
| 2630 | 90 | storres | :param ratPolyOfInt: a polynomial with rational coefficients and integer |
| 2631 | 90 | storres | variables. |
| 2632 | 90 | storres | :param precision: the precision of the floating-point coefficients. |
| 2633 | 90 | storres | :param targetHardnessToRound: the hardness to round we want to check. |
| 2634 | 90 | storres | :param variable1: the first variable of the polynomial (an expression). |
| 2635 | 90 | storres | :param variable2: the second variable of the polynomial (an expression). |
| 2636 | 90 | storres | |
| 2637 | 90 | storres | :returns: a 4 elements tuple: |
| 2638 | 90 | storres | - the polynomial; |
| 2639 | 91 | storres | - the modulus (N); |
| 2640 | 91 | storres | - the t bound; |
| 2641 | 90 | storres | - the lcm used to compute the integral coefficients and the |
| 2642 | 90 | storres | module. |
| 2643 | 80 | storres | """ |
| 2644 | 80 | storres | # Create a new integer polynomial ring. |
| 2645 | 80 | storres | IP = PolynomialRing(ZZ, name=str(variable1) + "," + str(variable2)) |
| 2646 | 80 | storres | # Coefficients are issued in the increasing power order. |
| 2647 | 80 | storres | ratPolyCoefficients = ratPolyOfInt.coefficients() |
| 2648 | 91 | storres | # Print the reversed list for debugging. |
| 2649 | 179 | storres | |
| 2650 | 179 | storres | #print "Rational polynomial coefficients:", ratPolyCoefficients[::-1] |
| 2651 | 94 | storres | # Build the list of number we compute the lcm of. |
| 2652 | 80 | storres | coefficientDenominators = sro_denominators(ratPolyCoefficients) |
| 2653 | 179 | storres | #print "Coefficient denominators:", coefficientDenominators |
| 2654 | 80 | storres | coefficientDenominators.append(2^precision) |
| 2655 | 170 | storres | coefficientDenominators.append(2^(targetHardnessToRound)) |
| 2656 | 87 | storres | leastCommonMultiple = lcm(coefficientDenominators) |
| 2657 | 80 | storres | # Compute the expression corresponding to the new polynomial |
| 2658 | 80 | storres | coefficientNumerators = sro_numerators(ratPolyCoefficients) |
| 2659 | 91 | storres | #print coefficientNumerators |
| 2660 | 80 | storres | polynomialExpression = 0 |
| 2661 | 80 | storres | power = 0 |
| 2662 | 170 | storres | # Iterate over two lists at the same time, stop when the shorter |
| 2663 | 170 | storres | # (is this case coefficientsNumerators) is |
| 2664 | 170 | storres | # exhausted. Both lists are ordered in the order of increasing powers |
| 2665 | 170 | storres | # of variable1. |
| 2666 | 80 | storres | for numerator, denominator in \ |
| 2667 | 94 | storres | zip(coefficientNumerators, coefficientDenominators): |
| 2668 | 80 | storres | multiplicator = leastCommonMultiple / denominator |
| 2669 | 80 | storres | newCoefficient = numerator * multiplicator |
| 2670 | 80 | storres | polynomialExpression += newCoefficient * variable1^power |
| 2671 | 80 | storres | power +=1 |
| 2672 | 80 | storres | polynomialExpression += - variable2 |
| 2673 | 80 | storres | return (IP(polynomialExpression), |
| 2674 | 170 | storres | leastCommonMultiple / 2^precision, # 2^K aka N. |
| 2675 | 170 | storres | #leastCommonMultiple / 2^(targetHardnessToRound + 1), # tBound |
| 2676 | 170 | storres | leastCommonMultiple / 2^(targetHardnessToRound), # tBound |
| 2677 | 91 | storres | leastCommonMultiple) # If we want to make test computations. |
| 2678 | 80 | storres | |
| 2679 | 170 | storres | # End slz_rat_poly_of_int_to_poly_for_coppersmith |
| 2680 | 79 | storres | |
| 2681 | 79 | storres | def slz_rat_poly_of_rat_to_rat_poly_of_int(ratPolyOfRat, |
| 2682 | 79 | storres | precision): |
| 2683 | 79 | storres | """ |
| 2684 | 79 | storres | Makes a variable substitution into the input polynomial so that the output |
| 2685 | 79 | storres | polynomial can take integer arguments. |
| 2686 | 79 | storres | All variables of the input polynomial "have precision p". That is to say |
| 2687 | 103 | storres | that they are rationals with denominator == 2^(precision - 1): |
| 2688 | 103 | storres | x = y/2^(precision - 1). |
| 2689 | 79 | storres | We "incorporate" these denominators into the coefficients with, |
| 2690 | 79 | storres | respectively, the "right" power. |
| 2691 | 79 | storres | """ |
| 2692 | 79 | storres | polynomialField = ratPolyOfRat.parent() |
| 2693 | 91 | storres | polynomialVariable = ratPolyOfRat.variables()[0] |
| 2694 | 91 | storres | #print "The polynomial field is:", polynomialField |
| 2695 | 79 | storres | return \ |
| 2696 | 91 | storres | polynomialField(ratPolyOfRat.subs({polynomialVariable : \
|
| 2697 | 79 | storres | polynomialVariable/2^(precision-1)})) |
| 2698 | 79 | storres | |
| 2699 | 79 | storres | # End slz_rat_poly_of_rat_to_rat_poly_of_int |
| 2700 | 79 | storres | |
| 2701 | 177 | storres | def slz_reduce_and_test_base(matrixToReduce, |
| 2702 | 177 | storres | nAtAlpha, |
| 2703 | 177 | storres | monomialsCountSqrt): |
| 2704 | 177 | storres | """ |
| 2705 | 177 | storres | Reduces the basis, tests the Coppersmith condition and returns |
| 2706 | 177 | storres | a list of rows that comply with the condition. |
| 2707 | 177 | storres | """ |
| 2708 | 177 | storres | ccComplientRowsList = [] |
| 2709 | 177 | storres | reducedMatrix = matrixToReduce.LLL(None) |
| 2710 | 177 | storres | if not reducedMatrix.is_LLL_reduced(): |
| 2711 | 177 | storres | raise Exception("reducedMatrix is not actually reduced. Aborting!")
|
| 2712 | 177 | storres | else: |
| 2713 | 177 | storres | print "reducedMatrix is actually reduced." |
| 2714 | 177 | storres | print "N^alpha:", nAtAlpha.n() |
| 2715 | 177 | storres | rowIndex = 0 |
| 2716 | 177 | storres | for row in reducedMatrix.rows(): |
| 2717 | 177 | storres | l2Norm = row.norm(2) |
| 2718 | 177 | storres | print "L_2 norm for vector # ", rowIndex, "= ", RR(l2Norm), "*", \ |
| 2719 | 177 | storres | monomialsCountSqrt.n(), ". Is vector OK?", |
| 2720 | 177 | storres | if (l2Norm * monomialsCountSqrt < nAtAlpha): |
| 2721 | 177 | storres | ccComplientRowsList.append(row) |
| 2722 | 177 | storres | print "True" |
| 2723 | 177 | storres | else: |
| 2724 | 177 | storres | print "False" |
| 2725 | 177 | storres | # End for |
| 2726 | 177 | storres | return ccComplientRowsList |
| 2727 | 177 | storres | # End slz_reduce_and_test_base |
| 2728 | 115 | storres | |
| 2729 | 247 | storres | def slz_reduce_lll_proj(matToReduce, n): |
| 2730 | 247 | storres | """ |
| 2731 | 247 | storres | Compute the transformation matrix that realizes an LLL reduction on |
| 2732 | 247 | storres | the random uniform projected matrix. |
| 2733 | 247 | storres | Return both the initial matrix "reduced" by the transformation matrix and |
| 2734 | 247 | storres | the transformation matrix itself. |
| 2735 | 247 | storres | """ |
| 2736 | 247 | storres | ## Compute the projected matrix. |
| 2737 | 249 | storres | """ |
| 2738 | 249 | storres | # Random matrix elements {-2^(n-1),...,0,...,2^(n-1)-1}.
|
| 2739 | 247 | storres | matProjector = slz_random_proj_uniform(n, |
| 2740 | 247 | storres | matToReduce.ncols(), |
| 2741 | 247 | storres | matToReduce.nrows()) |
| 2742 | 249 | storres | """ |
| 2743 | 249 | storres | # Random matrix elements in {-1,0,1}.
|
| 2744 | 249 | storres | matProjector = slz_random_proj_pm1(matToReduce.ncols(), |
| 2745 | 249 | storres | matToReduce.nrows()) |
| 2746 | 247 | storres | matProjected = matToReduce * matProjector |
| 2747 | 247 | storres | ## Build the argument matrix for LLL in such a way that the transformation |
| 2748 | 247 | storres | # matrix is also returned. This matrix is obtained at almost no extra |
| 2749 | 247 | storres | # cost. An identity matrix must be appended to |
| 2750 | 247 | storres | # the left of the initial matrix. The transformation matrix will |
| 2751 | 247 | storres | # will be recovered at the same location from the returned matrix . |
| 2752 | 247 | storres | idMat = identity_matrix(matProjected.nrows()) |
| 2753 | 247 | storres | augmentedMatToReduce = idMat.augment(matProjected) |
| 2754 | 247 | storres | reducedProjMat = \ |
| 2755 | 247 | storres | augmentedMatToReduce.LLL(algorithm='fpLLL:wrapper') |
| 2756 | 247 | storres | ## Recover the transformation matrix (the left part of the reduced matrix). |
| 2757 | 247 | storres | # We discard the reduced matrix itself. |
| 2758 | 247 | storres | transMat = reducedProjMat.submatrix(0, |
| 2759 | 247 | storres | 0, |
| 2760 | 247 | storres | reducedProjMat.nrows(), |
| 2761 | 247 | storres | reducedProjMat.nrows()) |
| 2762 | 247 | storres | ## Return the initial matrix "reduced" and the transformation matrix tuple. |
| 2763 | 247 | storres | return (transMat * matToReduce, transMat) |
| 2764 | 247 | storres | # End slz_reduce_lll_proj. |
| 2765 | 247 | storres | # |
| 2766 | 277 | storres | def slz_reduce_lll_proj_02(matToReduce, n): |
| 2767 | 277 | storres | """ |
| 2768 | 277 | storres | Compute the transformation matrix that realizes an LLL reduction on |
| 2769 | 277 | storres | the random uniform projected matrix. |
| 2770 | 277 | storres | Return both the initial matrix "reduced" by the transformation matrix and |
| 2771 | 277 | storres | the transformation matrix itself. |
| 2772 | 277 | storres | """ |
| 2773 | 277 | storres | ## Compute the projected matrix. |
| 2774 | 277 | storres | """ |
| 2775 | 277 | storres | # Random matrix elements {-2^(n-1),...,0,...,2^(n-1)-1}.
|
| 2776 | 277 | storres | matProjector = slz_random_proj_uniform(n, |
| 2777 | 277 | storres | matToReduce.ncols(), |
| 2778 | 277 | storres | matToReduce.nrows()) |
| 2779 | 277 | storres | """ |
| 2780 | 277 | storres | # Random matrix elements in {-8,0,7} -> 4.
|
| 2781 | 277 | storres | matProjector = slz_random_proj_uniform(matToReduce.ncols(), |
| 2782 | 277 | storres | matToReduce.nrows(), |
| 2783 | 277 | storres | 4) |
| 2784 | 277 | storres | matProjected = matToReduce * matProjector |
| 2785 | 277 | storres | ## Build the argument matrix for LLL in such a way that the transformation |
| 2786 | 277 | storres | # matrix is also returned. This matrix is obtained at almost no extra |
| 2787 | 277 | storres | # cost. An identity matrix must be appended to |
| 2788 | 277 | storres | # the left of the initial matrix. The transformation matrix will |
| 2789 | 277 | storres | # will be recovered at the same location from the returned matrix . |
| 2790 | 277 | storres | idMat = identity_matrix(matProjected.nrows()) |
| 2791 | 277 | storres | augmentedMatToReduce = idMat.augment(matProjected) |
| 2792 | 277 | storres | reducedProjMat = \ |
| 2793 | 277 | storres | augmentedMatToReduce.LLL(algorithm='fpLLL:wrapper') |
| 2794 | 277 | storres | ## Recover the transformation matrix (the left part of the reduced matrix). |
| 2795 | 277 | storres | # We discard the reduced matrix itself. |
| 2796 | 277 | storres | transMat = reducedProjMat.submatrix(0, |
| 2797 | 277 | storres | 0, |
| 2798 | 277 | storres | reducedProjMat.nrows(), |
| 2799 | 277 | storres | reducedProjMat.nrows()) |
| 2800 | 277 | storres | ## Return the initial matrix "reduced" and the transformation matrix tuple. |
| 2801 | 277 | storres | return (transMat * matToReduce, transMat) |
| 2802 | 277 | storres | # End slz_reduce_lll_proj_02. |
| 2803 | 277 | storres | # |
| 2804 | 229 | storres | def slz_resultant(poly1, poly2, elimVar, debug = False): |
| 2805 | 205 | storres | """ |
| 2806 | 205 | storres | Compute the resultant for two polynomials for a given variable |
| 2807 | 205 | storres | and return the (poly1, poly2, resultant) if the resultant |
| 2808 | 205 | storres | is not 0. |
| 2809 | 205 | storres | Return () otherwise. |
| 2810 | 205 | storres | """ |
| 2811 | 205 | storres | polynomialRing0 = poly1.parent() |
| 2812 | 205 | storres | resultantInElimVar = poly1.resultant(poly2,polynomialRing0(elimVar)) |
| 2813 | 213 | storres | if resultantInElimVar is None: |
| 2814 | 229 | storres | if debug: |
| 2815 | 229 | storres | print poly1 |
| 2816 | 229 | storres | print poly2 |
| 2817 | 229 | storres | print "have GCD = ", poly1.gcd(poly2) |
| 2818 | 213 | storres | return None |
| 2819 | 205 | storres | if resultantInElimVar.is_zero(): |
| 2820 | 229 | storres | if debug: |
| 2821 | 229 | storres | print poly1 |
| 2822 | 229 | storres | print poly2 |
| 2823 | 229 | storres | print "have GCD = ", poly1.gcd(poly2) |
| 2824 | 205 | storres | return None |
| 2825 | 205 | storres | else: |
| 2826 | 229 | storres | if debug: |
| 2827 | 229 | storres | print poly1 |
| 2828 | 229 | storres | print poly2 |
| 2829 | 229 | storres | print "have resultant in t = ", resultantInElimVar |
| 2830 | 205 | storres | return resultantInElimVar |
| 2831 | 205 | storres | # End slz_resultant. |
| 2832 | 205 | storres | # |
| 2833 | 177 | storres | def slz_resultant_tuple(poly1, poly2, elimVar): |
| 2834 | 179 | storres | """ |
| 2835 | 179 | storres | Compute the resultant for two polynomials for a given variable |
| 2836 | 179 | storres | and return the (poly1, poly2, resultant) if the resultant |
| 2837 | 180 | storres | is not 0. |
| 2838 | 179 | storres | Return () otherwise. |
| 2839 | 179 | storres | """ |
| 2840 | 181 | storres | polynomialRing0 = poly1.parent() |
| 2841 | 177 | storres | resultantInElimVar = poly1.resultant(poly2,polynomialRing0(elimVar)) |
| 2842 | 180 | storres | if resultantInElimVar.is_zero(): |
| 2843 | 177 | storres | return () |
| 2844 | 177 | storres | else: |
| 2845 | 177 | storres | return (poly1, poly2, resultantInElimVar) |
| 2846 | 177 | storres | # End slz_resultant_tuple. |
| 2847 | 177 | storres | # |
| 2848 | 247 | storres | def slz_uniform(n): |
| 2849 | 247 | storres | """ |
| 2850 | 253 | storres | Compute a uniform RV in [-2^(n-1), 2^(n-1)-1] (zero is included). |
| 2851 | 247 | storres | """ |
| 2852 | 247 | storres | ## function getrandbits(n) generates a long int with n random bits. |
| 2853 | 247 | storres | return getrandbits(n) - 2^(n-1) |
| 2854 | 247 | storres | # End slz_uniform. |
| 2855 | 247 | storres | # |
| 2856 | 246 | storres | sys.stderr.write("\t...sageSLZ loaded\n")
|