Laboratoire de l'Informatique et du Parallélisme » Pobyso

r"""

Main SLZ algorithm body in several versions.

AUTHORS:

- S.T. (2015-10-10): initial version

Examples:

    TODO

"""

print "sageRationalOperations loading..."

def srs_compute_lattice_volume(inputFunction,

                    inputLowerBound,

                    inputUpperBound,

                    alpha,

                    degree,

                    precision,

                    emin,

                    emax,

                    targetHardnessToRound,

                    debug = False):

    """

    Changes from V2:

        Root search is changed:

            - we compute the resultants in i and in t;

            - we compute the roots set of each of these resultants;

            - we combine all the possible pairs between the two sets;

            - we check these pairs in polynomials for correctness.

    Changes from V1: 

        1- check for roots as soon as a resultant is computed;

        2- once a non null resultant is found, check for roots;

        3- constant resultant == no root.

    """

    if debug:

        print "Function                :", inputFunction

        print "Lower bound             :", inputLowerBound

        print "Upper bounds            :", inputUpperBound

        print "Alpha                   :", alpha

        print "Degree                  :", degree

        print "Precision               :", precision

        print "Emin                    :", emin

        print "Emax                    :", emax

        print "Target hardness-to-round:", targetHardnessToRound

        print

    ## Important constants.

    ### Stretch the interval if no error happens.

    noErrorIntervalStretch = 1 + 2^(-5)

    ### If no vector validates the Coppersmith condition, shrink the interval

    #   by the following factor.

    noCoppersmithIntervalShrink = 1/2

    ### If only (or at least) one vector validates the Coppersmith condition, 

    #   shrink the interval by the following factor.

    oneCoppersmithIntervalShrink = 3/4

    #### If only null resultants are found, shrink the interval by the 

    #    following factor.

    onlyNullResultantsShrink     = 3/4

    ## Structures.

    RRR         = RealField(precision)

    RRIF        = RealIntervalField(precision)

    ## Converting input bound into the "right" field.

    lowerBound = RRR(inputLowerBound)

    upperBound = RRR(inputUpperBound)

    ## Before going any further, check domain and image binade conditions.

    print inputFunction(1).n()

    output = slz_fix_bounds_for_binades(lowerBound, upperBound, inputFunction)

    if output is None:

        print "Invalid domain/image binades. Domain:",\

        lowerBound, upperBound, "Images:", \

        inputFunction(lowerBound), inputFunction(upperBound)

        raise Exception("Invalid domain/image binades.")

    lb = output[0] ; ub = output[1]

    if lb != lowerBound or ub != upperBound:

        print "lb:", lb, " - ub:", ub

        print "Invalid domain/image binades. Domain:",\

        lowerBound, upperBound, "Images:", \

        inputFunction(lowerBound), inputFunction(upperBound)

        raise Exception("Invalid domain/image binades.")

    #

    ## Progam initialization

    ### Approximation polynomial accuracy and hardness to round.

    polyApproxAccur           = 2^(-(targetHardnessToRound + 1))

    polyTargetHardnessToRound = targetHardnessToRound + 1

    ### Significand to integer conversion ratio.

    toIntegerFactor           = 2^(precision-1)

    print "Polynomial approximation required accuracy:", polyApproxAccur.n()

    ### Variables and rings for polynomials and root searching.

    i=var('i')

    t=var('t')

    inputFunctionVariable = inputFunction.variables()[0]

    function = inputFunction.subs({inputFunctionVariable:i})

    #  Polynomial Rings over the integers, for root finding.

    Zi = ZZ[i]

    Zt = ZZ[t]

    Zit = ZZ[i,t]

    ## Number of iterations limit.

    maxIter = 100000

    #

    ## Compute the scaled function and the degree, in their Sollya version 

    #  once for all.

    (scaledf, sdlb, sdub, silb, siub) = \

        slz_compute_scaled_function(function, lowerBound, upperBound, precision)

    print "Scaled function:", scaledf._assume_str().replace('_SAGE_VAR_', '')

    scaledfSo = sollya_lib_parse_string(scaledf._assume_str().replace('_SAGE_VAR_', ''))

    degreeSo  = pobyso_constant_from_int_sa_so(degree)

    #

    ## Compute the scaling. boundsIntervalRifSa defined out of the loops.

    domainBoundsInterval   = RRIF(lowerBound, upperBound)

    (unscalingFunction, scalingFunction) = \

        slz_interval_scaling_expression(domainBoundsInterval, i)

    #print scalingFunction, unscalingFunction

    ## Set the Sollya internal precision (with an arbitrary minimum of 192).

    internalSollyaPrec = ceil((RR('1.5') * targetHardnessToRound) / 64) * 64

    if internalSollyaPrec < 192:

        internalSollyaPrec = 192

    pobyso_set_prec_sa_so(internalSollyaPrec)

    print "Sollya internal precision:", internalSollyaPrec

    ## Some variables.

    ### General variables

    lb                  = sdlb

    ub                  = sdub

    nbw                 = 0

    intervalUlp         = ub.ulp()

    #### Will be set by slz_interval_and_polynomila_to_sage.

    ic                  = 0 

    icAsInt             = 0    # Set from ic.

    solutionsSet        = set()

    tsErrorWidth        = []

    csErrorVectors      = []

    csVectorsResultants = []

    floatP              = 0  # Taylor polynomial.

    floatPcv            = 0  # Ditto with variable change.

    intvl               = "" # Taylor interval

    terr                = 0  # Taylor error.

    iterCount = 0

    htrnSet = set()

    ### Timers and counters.

    wallTimeStart                   = 0

    cpuTimeStart                    = 0

    taylCondFailedCount             = 0

    coppCondFailedCount             = 0

    resultCondFailedCount           = 0

    coppCondFailed                  = False

    resultCondFailed                = False

    globalResultsList               = []

    basisConstructionsCount         = 0

    basisConstructionsFullTime      = 0

    basisConstructionTime           = 0

    reductionsCount                 = 0

    reductionsFullTime              = 0

    reductionTime                   = 0

    resultantsComputationsCount     = 0

    resultantsComputationsFullTime  = 0

    resultantsComputationTime       = 0

    rootsComputationsCount          = 0

    rootsComputationsFullTime       = 0

    rootsComputationTime            = 0

    print

    ## Global times are started here.

    wallTimeStart                   = walltime()

    cpuTimeStart                    = cputime()

    ## Main loop.

    while True:

        if lb >= sdub:

            print "Lower bound reached upper bound."

            break

        if iterCount == maxIter:

            print "Reached maxIter. Aborting"

            break

        iterCount += 1

        print "[", lb, ",", ub, "]", ((ub - lb) / intervalUlp).log2().n(), \

            "log2(numbers)." 

        ### Compute a Sollya polynomial that will honor the Taylor condition.

        prceSo = slz_compute_polynomial_and_interval(scaledfSo,

                                                     degreeSo,

                                                     lb,

                                                     ub,

                                                     polyApproxAccur)

        ### Convert back the data into Sage space.                         

        (floatP, floatPcv, intvl, ic, terr) = \

            slz_interval_and_polynomial_to_sage((prceSo[0], prceSo[0],

                                                 prceSo[1], prceSo[2], 

                                                 prceSo[3]))

        intvl = RRIF(intvl)

        ## Clean-up Sollya stuff.

        for elem in prceSo:

            sollya_lib_clear_obj(elem)

        #print  floatP, floatPcv, intvl, ic, terr

        #print floatP

        #print intvl.endpoints()[0].n(), \

        #      ic.n(),

        #intvl.endpoints()[1].n()

        ### Check returned data.

        #### Is approximation error OK?

        if terr > polyApproxAccur:

            exceptionErrorMess  = \

                "Approximation failed - computed error:" + \

                str(terr) + " - target error: "

            exceptionErrorMess += \

                str(polyApproxAccur) + ". Aborting!"

            raise Exception(exceptionErrorMess)

        #### Is lower bound OK?

        if lb != intvl.endpoints()[0]:

            exceptionErrorMess =  "Wrong lower bound:" + \

                               str(lb) + ". Aborting!"

            raise Exception(exceptionErrorMess)

        #### Set upper bound.

        if ub > intvl.endpoints()[1]:

            ub = intvl.endpoints()[1]

            print "[", lb, ",", ub, "]", ((ub - lb) / intervalUlp).log2().n(), \

            "log2(numbers)." 

            taylCondFailedCount += 1

        #### Is interval not degenerate?

        if lb >= ub:

            exceptionErrorMess = "Degenerate interval: " + \

                                "lowerBound(" + str(lb) +\

                                ")>= upperBound(" + str(ub) + \

                                "). Aborting!"

            raise Exception(exceptionErrorMess)

        #### Is interval center ok?

        if ic <= lb or ic >= ub:

            exceptionErrorMess =  "Invalid interval center for " + \

                                str(lb) + ',' + str(ic) + ',' +  \

                                str(ub) + ". Aborting!"

            raise Exception(exceptionErrorMess)

        ##### Current interval width and reset future interval width.

        bw = ub - lb

        nbw = 0

        icAsInt = int(ic * toIntegerFactor)

        #### The following ratio is always >= 1. In case we may want to

        #    enlarge the interval

        curTaylErrRat = polyApproxAccur / terr

        ### Make the  integral transformations.

        #### Bounds and interval center.

        intIc = int(ic * toIntegerFactor)

        intLb = int(lb * toIntegerFactor) - intIc

        intUb = int(ub * toIntegerFactor) - intIc

        #

        #### Polynomials

        basisConstructionTime         = cputime()

        ##### To a polynomial with rational coefficients with rational arguments

        ratRatP = slz_float_poly_of_float_to_rat_poly_of_rat_pow_two(floatP)

        ##### To a polynomial with rational coefficients with integer arguments

        ratIntP = \

            slz_rat_poly_of_rat_to_rat_poly_of_int(ratRatP, precision)

        #####  Ultimately a multivariate polynomial with integer coefficients  

        #      with integer arguments.

        coppersmithTuple = \

            slz_rat_poly_of_int_to_poly_for_coppersmith(ratIntP, 

                                                        precision, 

                                                        targetHardnessToRound, 

                                                        i, t)

        #### Recover Coppersmith information.

        intIntP = coppersmithTuple[0]

        N = coppersmithTuple[1]

        nAtAlpha = N^alpha

        tBound = coppersmithTuple[2]

        leastCommonMultiple = coppersmithTuple[3]

        iBound = max(abs(intLb),abs(intUb))

        basisConstructionsFullTime        += cputime(basisConstructionTime)

        basisConstructionsCount           += 1

        reductionTime                     = cputime()

        #### Compute the reduced polynomials.

        ccReducedPolynomialsList =  \

            slz_compute_coppersmith_reduced_polynomials_with_lattice_volume(intIntP, 

                                                        alpha, 

                                                        N, 

                                                        iBound, 

                                                        tBound)

        if ccReducedPolynomialsList is None:

            raise Exception("Reduction failed.")

        reductionsFullTime    += cputime(reductionTime)

        reductionsCount       += 1

        if len(ccReducedPolynomialsList) < 2:

            print "Nothing to form resultants with."

            print

            coppCondFailedCount += 1

            coppCondFailed = True

            ##### Apply a different shrink factor according to 

            #  the number of compliant polynomials.

            if len(ccReducedPolynomialsList) == 0:

                ub = lb + bw * noCoppersmithIntervalShrink

            else: # At least one compliant polynomial.

                ub = lb + bw * oneCoppersmithIntervalShrink

            if ub > sdub:

                ub = sdub

            if lb == ub:

                raise Exception("Cant shrink interval \

                anymore to get Coppersmith condition.")

            nbw = 0

            continue

        #### We have at least two polynomials.

        #    Let us try to compute resultants.

        #    For each resultant computed, go for the solutions.

        ##### Build the pairs list.

        polyPairsList           = []

        for polyOuterIndex in xrange(0, len(ccReducedPolynomialsList) - 1):

            for polyInnerIndex in xrange(polyOuterIndex+1, 

                                         len(ccReducedPolynomialsList)):

                polyPairsList.append((ccReducedPolynomialsList[polyOuterIndex],

                                      ccReducedPolynomialsList[polyInnerIndex]))

        #### Actual root search.

        rootsSet            = set()

        hasNonNullResultant = False

        for polyPair in polyPairsList:

            if hasNonNullResultant:

                break

            resultantsComputationTime   = cputime()

            currentResultantI = \

                slz_resultant(polyPair[0],

                              polyPair[1],

                              t)

            resultantsComputationsCount    += 1

            if currentResultantI is None:

                resultantsComputationsFullTime +=  \

                    cputime(resultantsComputationTime)

                print "Nul resultant"

                continue # Next polyPair.

            currentResultantT = \

                slz_resultant(polyPair[0],

                              polyPair[1],

                              i)

            resultantsComputationsFullTime += cputime(resultantsComputationTime)

            resultantsComputationsCount    += 1

            if currentResultantT is None:                    

                print "Nul resultant"

                continue # Next polyPair.

            else:

                hasNonNullResultant = True

            #### We have a non null resultants pair. From now on, whatever the

            #    root search yields, no extra root search is necessary.

            #### A constant resultant leads to no root. Root search is done.

            if currentResultantI.degree() < 1:

                print "Resultant is constant:", currentResultantI

                break # Next polyPair and should break.

            if currentResultantT.degree() < 1:

                print "Resultant is constant:", currentResultantT

                break # Next polyPair and should break.

            #### Actual roots computation.

            rootsComputationTime       = cputime()

            ##### Compute i roots

            iRootsList = Zi(currentResultantI).roots()

            rootsComputationsCount      +=  1

            if len(iRootsList) == 0:

                rootsComputationsFullTime   =   cputime(rootsComputationTime)

                print "No roots in \"i\"."

                break # No roots in i.

            tRootsList = Zt(currentResultantT).roots()

            rootsComputationsFullTime   =   cputime(rootsComputationTime)

            rootsComputationsCount      +=  1

            if len(tRootsList) == 0:

                print "No roots in \"t\"."

                break # No roots in i.

            ##### For each iRoot, get a tRoot and check against the polynomials.

            for iRoot in iRootsList:

                ####### Roots returned by roots() are (value, multiplicity) 

                #       tuples.

                #print "iRoot:", iRoot

                for tRoot in tRootsList:

                ###### Use the tRoot against each polynomial, alternatively.

                    if polyPair[0](iRoot[0],tRoot[0]) != 0:

                        continue

                    if polyPair[1](iRoot[0],tRoot[0]) != 0:

                        continue

                    rootsSet.add((iRoot[0], tRoot[0]))

            # End of roots computation.

        # End loop for polyPair in polyParsList.  Will break at next iteration.

        # since a non null resultant was found.

        #### Prepare for results for the current interval..

        intervalResultsList = []

        intervalResultsList.append((lb, ub))

        #### Check roots.

        rootsResultsList = []

        for root in rootsSet:

            specificRootResultsList = []

            failingBounds = []

            intIntPdivN = intIntP(root[0], root[1]) / N

            if int(intIntPdivN) != intIntPdivN:

                continue # Next root

            # Root qualifies for modular equation, test it for hardness to round.

            hardToRoundCaseAsFloat = RRR((icAsInt + root[0]) / toIntegerFactor)

            #print "Before unscaling:", hardToRoundCaseAsFloat.n(prec=precision)

            #print scalingFunction

            scaledHardToRoundCaseAsFloat = \

                scalingFunction(hardToRoundCaseAsFloat) 

            print "Candidate HTRNc at x =", \

                scaledHardToRoundCaseAsFloat.n().str(base=2),

            if slz_is_htrn(scaledHardToRoundCaseAsFloat,

                           function,

                           2^-(targetHardnessToRound),

                           RRR):

                print hardToRoundCaseAsFloat, "is HTRN case."

                if lb <= hardToRoundCaseAsFloat and hardToRoundCaseAsFloat <= ub:

                    print "Found in interval."

                else:

                    print "Found out of interval."

                specificRootResultsList.append(hardToRoundCaseAsFloat.n().str(base=2))

                # Check the root is in the bounds

                if abs(root[0]) > iBound or abs(root[1]) > tBound:

                    print "Root", root, "is out of bounds for modular equation."

                    if abs(root[0]) > iBound:

                        print "root[0]:", root[0]

                        print "i bound:", iBound

                        failingBounds.append('i')

                        failingBounds.append(root[0])

                        failingBounds.append(iBound)

                    if abs(root[1]) > tBound:

                        print "root[1]:", root[1]

                        print "t bound:", tBound

                        failingBounds.append('t')

                        failingBounds.append(root[1])

                        failingBounds.append(tBound)

                if len(failingBounds) > 0:

                    specificRootResultsList.append(failingBounds)

            else: # From slz_is_htrn...

                print "is not an HTRN case."

            if len(specificRootResultsList) > 0:

                rootsResultsList.append(specificRootResultsList)

        if len(rootsResultsList) > 0:

            intervalResultsList.append(rootsResultsList)

        ### Check if a non null resultant was found. If not shrink the interval.

        if not hasNonNullResultant:

            print "Only null resultants for this reduction, shrinking interval."

            resultCondFailed      =  True

            resultCondFailedCount += 1

            ### Shrink interval for next iteration.

            ub = lb + bw * onlyNullResultantsShrink

            if ub > sdub:

                ub = sdub

            nbw = 0

            continue

        #### An intervalResultsList has at least the bounds.

        globalResultsList.append(intervalResultsList)   

        #### Compute an incremented width for next upper bound, only

        #    if not Coppersmith condition nor resultant condition

        #    failed at the previous run. 

        if not coppCondFailed and not resultCondFailed:

            nbw = noErrorIntervalStretch * bw

        else:

            nbw = bw

        ##### Reset the failure flags. They will be raised

        #     again if needed.

        coppCondFailed   = False

        resultCondFailed = False

        #### For next iteration (at end of loop)

        #print "nbw:", nbw

        lb  = ub

        ub += nbw     

        if ub > sdub:

            ub = sdub

        print

    # End while True

    ## Main loop just ended.

    globalWallTime = walltime(wallTimeStart)

    globalCpuTime  = cputime(cpuTimeStart)

    ## Output results

    print ; print "Intervals and HTRNs" ; print

    for intervalResultsList in globalResultsList:

        intervalResultString = "[" + str(intervalResultsList[0][0]) +\

                               "," + str(intervalResultsList[0][1])  + "]"

        print intervalResultString,

        if len(intervalResultsList) > 1:

            rootsResultsList = intervalResultsList[1]

            specificRootResultIndex = 0

            for specificRootResultsList in rootsResultsList:

                if specificRootResultIndex == 0:

                    print "\t", specificRootResultsList[0],

                else:

                    print " " * len(intervalResultString), "\t", \

                          specificRootResultsList[0],

                if len(specificRootResultsList) > 1:

                    print specificRootResultsList[1]

                specificRootResultIndex += 1

        print ; print

    #print globalResultsList

    #

    print "Timers and counters"

    print

    print "Number of iterations:", iterCount

    print "Taylor condition failures:", taylCondFailedCount

    print "Coppersmith condition failures:", coppCondFailedCount

    print "Resultant condition failures:", resultCondFailedCount

    print "Iterations count: ", iterCount

    print "Number of intervals:", len(globalResultsList)

    print "Number of basis constructions:", basisConstructionsCount 

    print "Total CPU time spent in basis constructions:", \

        basisConstructionsFullTime

    if basisConstructionsCount != 0:

        print "Average basis construction CPU time:", \

            basisConstructionsFullTime/basisConstructionsCount

    print "Number of reductions:", reductionsCount

    print "Total CPU time spent in reductions:", reductionsFullTime

    if reductionsCount != 0:

        print "Average reduction CPU time:", \

            reductionsFullTime/reductionsCount

    print "Number of resultants computation rounds:", \

        resultantsComputationsCount

    print "Total CPU time spent in resultants computation rounds:", \

        resultantsComputationsFullTime

    if resultantsComputationsCount != 0:

        print "Average resultants computation round CPU time:", \

            resultantsComputationsFullTime/resultantsComputationsCount

    print "Number of root finding rounds:", rootsComputationsCount

    print "Total CPU time spent in roots finding rounds:", \

        rootsComputationsFullTime

    if rootsComputationsCount != 0:

        print "Average roots finding round CPU time:", \

            rootsComputationsFullTime/rootsComputationsCount

    print "Global Wall time:", globalWallTime

    print "Global CPU time:", globalCpuTime

    ## Output counters

# End srs_compute_lattice_volume

"""

SLZ runtime function.

"""

def srs_run_SLZ_v01(inputFunction,

                    inputLowerBound,

                    inputUpperBound,

                    alpha,

                    degree,

                    precision,

                    emin,

                    emax,

                    targetHardnessToRound,

                    debug = False):

    if debug:

        print "Function                :", inputFunction

        print "Lower bound             :", inputLowerBound

        print "Upper bounds            :", inputUpperBound

        print "Alpha                   :", alpha

        print "Degree                  :", degree

        print "Precision               :", precision

        print "Emin                    :", emin

        print "Emax                    :", emax

        print "Target hardness-to-round:", targetHardnessToRound

        print

    ## Important constants.

    ### Stretch the interval if no error happens.

    noErrorIntervalStretch = 1 + 2^(-5)

    ### If no vector validates the Coppersmith condition, shrink the interval

    #   by the following factor.

    noCoppersmithIntervalShrink = 1/2

    ### If only (or at least) one vector validates the Coppersmith condition, 

    #   shrink the interval by the following factor.

    oneCoppersmithIntervalShrink = 3/4

    #### If only null resultants are found, shrink the interval by the 

    #    following factor.

    onlyNullResultantsShrink     = 3/4

    ## Structures.

    RRR         = RealField(precision)

    RRIF        = RealIntervalField(precision)

    ## Converting input bound into the "right" field.

    lowerBound = RRR(inputLowerBound)

    upperBound = RRR(inputUpperBound)

    ## Before going any further, check domain and image binade conditions.

    print inputFunction(1).n()

    output = slz_fix_bounds_for_binades(lowerBound, upperBound, inputFunction)

    if output is None:

        print "Invalid domain/image binades. Domain:",\

        lowerBound, upperBound, "Images:", \

        inputFunction(lowerBound), inputFunction(upperBound)

        raise Exception("Invalid domain/image binades.")

    lb = output[0] ; ub = output[1]

    if lb is None or lb != lowerBound or ub != upperBound:

        print "lb:", lb, " - ub:", ub

        print "Invalid domain/image binades. Domain:",\

        lowerBound, upperBound, "Images:", \

        inputFunction(lowerBound), inputFunction(upperBound)

        raise Exception("Invalid domain/image binades.")

    #

    ## Progam initialization

    ### Approximation polynomial accuracy and hardness to round.

    polyApproxAccur           = 2^(-(targetHardnessToRound + 1))

    polyTargetHardnessToRound = targetHardnessToRound + 1

    ### Significand to integer conversion ratio.

    toIntegerFactor           = 2^(precision-1)

    print "Polynomial approximation required accuracy:", polyApproxAccur.n()

    ### Variables and rings for polynomials and root searching.

    i=var('i')

    t=var('t')

    inputFunctionVariable = inputFunction.variables()[0]

    function = inputFunction.subs({inputFunctionVariable:i})

    #  Polynomial Rings over the integers, for root finding.

    Zi = ZZ[i]

    Zt = ZZ[t]

    Zit = ZZ[i,t]

    ## Number of iterations limit.

    maxIter = 100000

    #

    ## Compute the scaled function and the degree, in their Sollya version 

    #  once for all.

    (scaledf, sdlb, sdub, silb, siub) = \

        slz_compute_scaled_function(function, lowerBound, upperBound, precision)

    print "Scaled function:", scaledf._assume_str().replace('_SAGE_VAR_', '')

    scaledfSo = sollya_lib_parse_string(scaledf._assume_str().replace('_SAGE_VAR_', ''))

    degreeSo  = pobyso_constant_from_int_sa_so(degree)

    #

    ## Compute the scaling. boundsIntervalRifSa defined out of the loops.

    domainBoundsInterval   = RRIF(lowerBound, upperBound)

    (unscalingFunction, scalingFunction) = \

        slz_interval_scaling_expression(domainBoundsInterval, i)

    #print scalingFunction, unscalingFunction

    ## Set the Sollya internal precision (with an arbitrary minimum of 192).

    internalSollyaPrec = ceil((RR('1.5') * targetHardnessToRound) / 64) * 64

    if internalSollyaPrec < 192:

        internalSollyaPrec = 192

    pobyso_set_prec_sa_so(internalSollyaPrec)

    print "Sollya internal precision:", internalSollyaPrec

    ## Some variables.

    ### General variables

    lb                  = sdlb

    ub                  = sdub

    nbw                 = 0

    intervalUlp         = ub.ulp()

    #### Will be set by slz_interval_and_polynomila_to_sage.

    ic                  = 0 

    icAsInt             = 0    # Set from ic.

    solutionsSet        = set()

    tsErrorWidth        = []

    csErrorVectors      = []

    csVectorsResultants = []

    floatP              = 0  # Taylor polynomial.

    floatPcv            = 0  # Ditto with variable change.

    intvl               = "" # Taylor interval

    terr                = 0  # Taylor error.

    iterCount = 0

    htrnSet = set()

    ### Timers and counters.

    wallTimeStart                   = 0

    cpuTimeStart                    = 0

    taylCondFailedCount             = 0

    coppCondFailedCount             = 0

    resultCondFailedCount           = 0

    coppCondFailed                  = False

    resultCondFailed                = False

    globalResultsList               = []

    basisConstructionsCount         = 0

    basisConstructionsFullTime      = 0

    basisConstructionTime           = 0

    reductionsCount                 = 0

    reductionsFullTime              = 0

    reductionTime                   = 0

    resultantsComputationsCount     = 0

    resultantsComputationsFullTime  = 0

    resultantsComputationTime       = 0

    rootsComputationsCount          = 0

    rootsComputationsFullTime       = 0

    rootsComputationTime            = 0

    print

    ## Global times are started here.

    wallTimeStart                   = walltime()

    cpuTimeStart                    = cputime()

    ## Main loop.

    while True:

        if lb >= sdub:

            print "Lower bound reached upper bound."

            break

        if iterCount == maxIter:

            print "Reached maxIter. Aborting"

            break

        iterCount += 1

        print "[", lb, ",", ub, "]", ((ub - lb) / intervalUlp).log2().n(), \

            "log2(numbers)." 

        ### Compute a Sollya polynomial that will honor the Taylor condition.

        prceSo = slz_compute_polynomial_and_interval(scaledfSo,

                                                     degreeSo,

                                                     lb,

                                                     ub,

                                                     polyApproxAccur)

        ### Convert back the data into Sage space.                         

        (floatP, floatPcv, intvl, ic, terr) = \

            slz_interval_and_polynomial_to_sage((prceSo[0], prceSo[0],

                                                 prceSo[1], prceSo[2], 

                                                 prceSo[3]))

        intvl = RRIF(intvl)

        ## Clean-up Sollya stuff.

        for elem in prceSo:

            sollya_lib_clear_obj(elem)

        #print  floatP, floatPcv, intvl, ic, terr

        #print floatP

        #print intvl.endpoints()[0].n(), \

        #      ic.n(),

        #intvl.endpoints()[1].n()

        ### Check returned data.

        #### Is approximation error OK?

        if terr > polyApproxAccur:

            exceptionErrorMess  = \

                "Approximation failed - computed error:" + \

                str(terr) + " - target error: "

            exceptionErrorMess += \

                str(polyApproxAccur) + ". Aborting!"

            raise Exception(exceptionErrorMess)

        #### Is lower bound OK?

        if lb != intvl.endpoints()[0]:

            exceptionErrorMess =  "Wrong lower bound:" + \

                               str(lb) + ". Aborting!"

            raise Exception(exceptionErrorMess)

        #### Set upper bound.

        if ub > intvl.endpoints()[1]:

            ub = intvl.endpoints()[1]

            print "[", lb, ",", ub, "]", ((ub - lb) / intervalUlp).log2().n(), \

            "log2(numbers)." 

            taylCondFailedCount += 1

        #### Is interval not degenerate?

        if lb >= ub:

            exceptionErrorMess = "Degenerate interval: " + \

                                "lowerBound(" + str(lb) +\

                                ")>= upperBound(" + str(ub) + \

                                "). Aborting!"

            raise Exception(exceptionErrorMess)

        #### Is interval center ok?

        if ic <= lb or ic >= ub:

            exceptionErrorMess =  "Invalid interval center for " + \

                                str(lb) + ',' + str(ic) + ',' +  \

                                str(ub) + ". Aborting!"

            raise Exception(exceptionErrorMess)

        ##### Current interval width and reset future interval width.

        bw = ub - lb

        nbw = 0

        icAsInt = int(ic * toIntegerFactor)

        #### The following ratio is always >= 1. In case we may want to

        #  enlarge the interval

        curTaylErrRat = polyApproxAccur / terr

        ## Make the  integral transformations.

        ### First for interval center and bounds.

        intIc = int(ic * toIntegerFactor)

        intLb = int(lb * toIntegerFactor) - intIc

        intUb = int(ub * toIntegerFactor) - intIc

        #

        #### For polynomials

        basisConstructionTime         = cputime()

        ##### To a polynomial with rational coefficients with rational arguments

        ratRatP = slz_float_poly_of_float_to_rat_poly_of_rat_pow_two(floatP)

        ##### To a polynomial with rational coefficients with integer arguments

        ratIntP = \

            slz_rat_poly_of_rat_to_rat_poly_of_int(ratRatP, precision)

        #####  Ultimately a polynomial with integer coefficients with integer 

        #      arguments.

        coppersmithTuple = \

            slz_rat_poly_of_int_to_poly_for_coppersmith(ratIntP, 

                                                        precision, 

                                                        targetHardnessToRound, 

                                                        i, t)

        #### Recover Coppersmith information.

        intIntP = coppersmithTuple[0]

        N = coppersmithTuple[1]

        nAtAlpha = N^alpha

        tBound = coppersmithTuple[2]

        leastCommonMultiple = coppersmithTuple[3]

        iBound = max(abs(intLb),abs(intUb))

        basisConstructionsFullTime        += cputime(basisConstructionTime)

        basisConstructionsCount           += 1

        reductionTime                     = cputime()

        # Compute the reduced polynomials.

        ccReducedPolynomialsList =  \

                slz_compute_coppersmith_reduced_polynomials(intIntP, 

                                                            alpha, 

                                                            N, 

                                                            iBound, 

                                                            tBound)

        if ccReducedPolynomialsList is None:

            raise Exception("Reduction failed.")

        reductionsFullTime    += cputime(reductionTime)

        reductionsCount       += 1

        if len(ccReducedPolynomialsList) < 2:

            print "Nothing to form resultants with."

            print

            coppCondFailedCount += 1

            coppCondFailed = True

            ##### Apply a different shrink factor according to 

            #  the number of compliant polynomials.

            if len(ccReducedPolynomialsList) == 0:

                ub = lb + bw * noCoppersmithIntervalShrink

            else: # At least one compliant polynomial.

                ub = lb + bw * oneCoppersmithIntervalShrink

            if ub > sdub:

                ub = sdub

            if lb == ub:

                raise Exception("Cant shrink interval \

                anymore to get Coppersmith condition.")

            nbw = 0

            continue

        #### We have at least two polynomials.

        #    Let us try to compute resultants.

        resultantsComputationTime      = cputime()

        resultantsInTTuplesList = []

        for polyOuterIndex in xrange(0, len(ccReducedPolynomialsList) - 1):

            for polyInnerIndex in xrange(polyOuterIndex+1, 

                                         len(ccReducedPolynomialsList)):

                resultantTuple = \

                slz_resultant_tuple(ccReducedPolynomialsList[polyOuterIndex],

                                ccReducedPolynomialsList[polyInnerIndex],

                                t)

                if len(resultantTuple) > 2:                    

                    #print resultantTuple[2]

                    resultantsInTTuplesList.append(resultantTuple)

                else:

                    print "No non nul resultant"

        print len(resultantsInTTuplesList), "resultant(s) in t tuple(s) created."

        resultantsComputationsFullTime += cputime(resultantsComputationTime)

        resultantsComputationsCount    += 1

        if len(resultantsInTTuplesList) == 0:

            print "Only null resultants, shrinking interval."

            resultCondFailed      =  True

            resultCondFailedCount += 1

            ### Shrink interval for next iteration.

            ub = lb + bw * onlyNullResultantsShrink

            if ub > sdub:

                ub = sdub

            nbw = 0

            continue

        #### Compute roots.

        rootsComputationTime       = cputime()

        reducedPolynomialsRootsSet = set()

        ##### Solve in the second variable since resultants are in the first

        #     variable.

        for resultantInTTuple in resultantsInTTuplesList:

            currentResultant = resultantInTTuple[2]

            ##### If the resultant degree is not at least 1, there are no roots.

            if currentResultant.degree() < 1:

                print "Resultant is constant:", currentResultant

                continue # Next resultantInTTuple

            ##### Compute i roots

            iRootsList = Zi(currentResultant).roots()

            ##### For each iRoot, compute the corresponding tRoots and check 

            #     them in the input polynomial.

            for iRoot in iRootsList:

                ####### Roots returned by roots() are (value, multiplicity) 

                #       tuples.

                #print "iRoot:", iRoot

                ###### Use the tRoot against each polynomial, alternatively.

                for indexInTuple in range(0,2):

                    currentPolynomial = resultantInTTuple[indexInTuple]

                    ####### If the polynomial is univariate, just drop it.

                    if len(currentPolynomial.variables()) < 2:

                        print "    Current polynomial is not in two variables."

                        continue # Next indexInTuple

                    tRootsList = \

                        Zt(currentPolynomial.subs({currentPolynomial.variables()[0]:iRoot[0]})).roots()

                    ####### The tRootsList can be empty, hence the test.

                    if len(tRootsList) == 0:

                        print "    No t root."

                        continue # Next indexInTuple

                    for tRoot in tRootsList:

                        reducedPolynomialsRootsSet.add((iRoot[0], tRoot[0]))

        # End of roots computation

        rootsComputationsFullTime   =   cputime(rootsComputationTime)

        rootsComputationsCount      +=  1

        ##### Prepare for results.

        intervalResultsList = []

        intervalResultsList.append((lb, ub))

        #### Check roots.

        rootsResultsList = []

        for root in reducedPolynomialsRootsSet:

            specificRootResultsList = []

            failingBounds = []

            intIntPdivN = intIntP(root[0], root[1]) / N

            if int(intIntPdivN) != intIntPdivN:

                continue # Next root

            # Root qualifies for modular equation, test it for hardness to round.

            hardToRoundCaseAsFloat = RRR((icAsInt + root[0]) / toIntegerFactor)

            #print "Before unscaling:", hardToRoundCaseAsFloat.n(prec=precision)

            #print scalingFunction

            scaledHardToRoundCaseAsFloat = \

                scalingFunction(hardToRoundCaseAsFloat) 

            print "Candidate HTRNc at x =", \

                scaledHardToRoundCaseAsFloat.n().str(base=2),

            if slz_is_htrn(scaledHardToRoundCaseAsFloat,

                           function,

                           2^-(targetHardnessToRound),

                           RRR):

                print hardToRoundCaseAsFloat, "is HTRN case."

                if lb <= hardToRoundCaseAsFloat and hardToRoundCaseAsFloat <= ub:

                    print "Found in interval."

                else:

                    print "Found out of interval."

                specificRootResultsList.append(hardToRoundCaseAsFloat.n().str(base=2))

                # Check the root is in the bounds

                if abs(root[0]) > iBound or abs(root[1]) > tBound:

                    print "Root", root, "is out of bounds."

                    if abs(root[0]) > iBound:

                        print "root[0]:", root[0]

                        print "i bound:", iBound

                        failingBounds.append('i')

                        failingBounds.append(root[0])

                        failingBounds.append(iBound)

                    if abs(root[1]) > tBound:

                        print "root[1]:", root[1]

                        print "t bound:", tBound

                        failingBounds.append('t')

                        failingBounds.append(root[1])

                        failingBounds.append(tBound)

                if len(failingBounds) > 0:

                    specificRootResultsList.append(failingBounds)

            else: # From slz_is_htrn...

                print "is not an HTRN case."

            if len(specificRootResultsList) > 0:

                rootsResultsList.append(specificRootResultsList)

        if len(rootsResultsList) > 0:

            intervalResultsList.append(rootsResultsList)

        #### An intervalResultsList has at least the bounds.

        globalResultsList.append(intervalResultsList)   

        #### Compute an incremented width for next upper bound, only

        #    if not Coppersmith condition nor resultant condition

        #    failed at the previous run. 

        if not coppCondFailed and not resultCondFailed:

            nbw = noErrorIntervalStretch * bw

        else:

            nbw = bw

        ##### Reset the failure flags. They will be raised

        #     again if needed.

        coppCondFailed   = False

        resultCondFailed = False

        #### For next iteration (at end of loop)

        #print "nbw:", nbw

        lb  = ub

        ub += nbw     

        if ub > sdub:

            ub = sdub

        print

    # End while True

    ## Main loop just ended.

    globalWallTime = walltime(wallTimeStart)

    globalCpuTime  = cputime(cpuTimeStart)

    ## Output results

    print ; print "Intervals and HTRNs" ; print

    for intervalResultsList in globalResultsList:

        intervalResultString = "[" + str(intervalResultsList[0][0]) +\

                               "," + str(intervalResultsList[0][1])  + "]"

        print intervalResultString,

        if len(intervalResultsList) > 1:

            rootsResultsList = intervalResultsList[1]

            specificRootResultIndex = 0

            for specificRootResultsList in rootsResultsList:

                if specificRootResultIndex == 0:

                    print "\t", specificRootResultsList[0],

                else:

                    print " " * len(intervalResultString), "\t", \

                          specificRootResultsList[0],

                if len(specificRootResultsList) > 1:

                    print specificRootResultsList[1]

                specificRootResultIndex += 1

        print ; print

    #print globalResultsList

    #

    print "Timers and counters"

    print

    print "Number of iterations:", iterCount

    print "Taylor condition failures:", taylCondFailedCount

    print "Coppersmith condition failures:", coppCondFailedCount

    print "Resultant condition failures:", resultCondFailedCount

    print "Iterations count: ", iterCount

    print "Number of intervals:", len(globalResultsList)

    print "Number of basis constructions:", basisConstructionsCount 

    print "Total CPU time spent in basis constructions:", \

        basisConstructionsFullTime

    if basisConstructionsCount != 0:

        print "Average basis construction CPU time:", \

            basisConstructionsFullTime/basisConstructionsCount

    print "Number of reductions:", reductionsCount

    print "Total CPU time spent in reductions:", reductionsFullTime

    if reductionsCount != 0:

        print "Average reduction CPU time:", \

            reductionsFullTime/reductionsCount

    print "Number of resultants computation rounds:", \

        resultantsComputationsCount

    print "Total CPU time spent in resultants computation rounds:", \

        resultantsComputationsFullTime

    if resultantsComputationsCount != 0:

        print "Average resultants computation round CPU time:", \

            resultantsComputationsFullTime/resultantsComputationsCount

    print "Number of root finding rounds:", rootsComputationsCount

    print "Total CPU time spent in roots finding rounds:", \

        rootsComputationsFullTime

    if rootsComputationsCount != 0:

        print "Average roots finding round CPU time:", \

            rootsComputationsFullTime/rootsComputationsCount

    print "Global Wall time:", globalWallTime

    print "Global CPU time:", globalCpuTime

    ## Output counters

# End srs_runSLZ-v01

def srs_run_SLZ_v02(inputFunction,

                    inputLowerBound,

                    inputUpperBound,

                    alpha,

                    degree,

                    precision,

                    emin,

                    emax,

                    targetHardnessToRound,

                    debug = False):

    """

    Changes from V1: 

        1- check for roots as soon as a resultant is computed;

        2- once a non null resultant is found, check for roots;

        3- constant resultant == no root.

    """

    if debug:

        print "Function                :", inputFunction

        print "Lower bound             :", inputLowerBound

        print "Upper bounds            :", inputUpperBound

        print "Alpha                   :", alpha

        print "Degree                  :", degree

        print "Precision               :", precision

        print "Emin                    :", emin

        print "Emax                    :", emax

        print "Target hardness-to-round:", targetHardnessToRound

        print

    ## Important constants.

    ### Stretch the interval if no error happens.

    noErrorIntervalStretch = 1 + 2^(-5)

    ### If no vector validates the Coppersmith condition, shrink the interval

    #   by the following factor.

    noCoppersmithIntervalShrink = 1/2

    ### If only (or at least) one vector validates the Coppersmith condition, 

    #   shrink the interval by the following factor.

    oneCoppersmithIntervalShrink = 3/4

    #### If only null resultants are found, shrink the interval by the 

    #    following factor.

    onlyNullResultantsShrink     = 3/4

    ## Structures.

    RRR         = RealField(precision)

    RRIF        = RealIntervalField(precision)

    ## Converting input bound into the "right" field.

    lowerBound = RRR(inputLowerBound)

    upperBound = RRR(inputUpperBound)

    ## Before going any further, check domain and image binade conditions.

    print inputFunction(1).n()

    output = slz_fix_bounds_for_binades(lowerBound, upperBound, inputFunction)

    if output is None:

        print "Invalid domain/image binades. Domain:",\

        lowerBound, upperBound, "Images:", \

        inputFunction(lowerBound), inputFunction(upperBound)

        raise Exception("Invalid domain/image binades.")

    lb = output[0] ; ub = output[1]

    if lb != lowerBound or ub != upperBound:

        print "lb:", lb, " - ub:", ub

        print "Invalid domain/image binades. Domain:",\

        lowerBound, upperBound, "Images:", \

        inputFunction(lowerBound), inputFunction(upperBound)

        raise Exception("Invalid domain/image binades.")

    #

    ## Progam initialization

    ### Approximation polynomial accuracy and hardness to round.

    polyApproxAccur           = 2^(-(targetHardnessToRound + 1))

    polyTargetHardnessToRound = targetHardnessToRound + 1

    ### Significand to integer conversion ratio.

    toIntegerFactor           = 2^(precision-1)

    print "Polynomial approximation required accuracy:", polyApproxAccur.n()

    ### Variables and rings for polynomials and root searching.

    i=var('i')

    t=var('t')

    inputFunctionVariable = inputFunction.variables()[0]

    function = inputFunction.subs({inputFunctionVariable:i})

    #  Polynomial Rings over the integers, for root finding.

    Zi = ZZ[i]

    Zt = ZZ[t]

    Zit = ZZ[i,t]

    ## Number of iterations limit.

    maxIter = 100000

    #

    ## Compute the scaled function and the degree, in their Sollya version 

    #  once for all.

    (scaledf, sdlb, sdub, silb, siub) = \

        slz_compute_scaled_function(function, lowerBound, upperBound, precision)

    print "Scaled function:", scaledf._assume_str().replace('_SAGE_VAR_', '')

    scaledfSo = sollya_lib_parse_string(scaledf._assume_str().replace('_SAGE_VAR_', ''))

    degreeSo  = pobyso_constant_from_int_sa_so(degree)

    #

    ## Compute the scaling. boundsIntervalRifSa defined out of the loops.

    domainBoundsInterval   = RRIF(lowerBound, upperBound)

    (unscalingFunction, scalingFunction) = \

        slz_interval_scaling_expression(domainBoundsInterval, i)

    #print scalingFunction, unscalingFunction

    ## Set the Sollya internal precision (with an arbitrary minimum of 192).

    internalSollyaPrec = ceil((RR('1.5') * targetHardnessToRound) / 64) * 64

    if internalSollyaPrec < 192:

        internalSollyaPrec = 192

    pobyso_set_prec_sa_so(internalSollyaPrec)

    print "Sollya internal precision:", internalSollyaPrec

    ## Some variables.

    ### General variables

    lb                  = sdlb

    ub                  = sdub

    nbw                 = 0

    intervalUlp         = ub.ulp()

    #### Will be set by slz_interval_and_polynomila_to_sage.

    ic                  = 0 

    icAsInt             = 0    # Set from ic.

    solutionsSet        = set()

    tsErrorWidth        = []

    csErrorVectors      = []

    csVectorsResultants = []

    floatP              = 0  # Taylor polynomial.

    floatPcv            = 0  # Ditto with variable change.

    intvl               = "" # Taylor interval

    terr                = 0  # Taylor error.

    iterCount = 0

    htrnSet = set()

    ### Timers and counters.

    wallTimeStart                   = 0

    cpuTimeStart                    = 0

    taylCondFailedCount             = 0

    coppCondFailedCount             = 0

    resultCondFailedCount           = 0

    coppCondFailed                  = False

    resultCondFailed                = False

    globalResultsList               = []

    basisConstructionsCount         = 0

    basisConstructionsFullTime      = 0

    basisConstructionTime           = 0

    reductionsCount                 = 0

    reductionsFullTime              = 0

    reductionTime                   = 0

    resultantsComputationsCount     = 0

    resultantsComputationsFullTime  = 0

    resultantsComputationTime       = 0

    rootsComputationsCount          = 0

    rootsComputationsFullTime       = 0

    rootsComputationTime            = 0

    print

    ## Global times are started here.

    wallTimeStart                   = walltime()

    cpuTimeStart                    = cputime()

    ## Main loop.

    while True:

        if lb >= sdub:

            print "Lower bound reached upper bound."

            break

        if iterCount == maxIter:

            print "Reached maxIter. Aborting"

            break

        iterCount += 1

        print "[", lb, ",", ub, "]", ((ub - lb) / intervalUlp).log2().n(), \

            "log2(numbers)." 

        ### Compute a Sollya polynomial that will honor the Taylor condition.

        prceSo = slz_compute_polynomial_and_interval(scaledfSo,

                                                     degreeSo,

                                                     lb,

                                                     ub,

                                                     polyApproxAccur)

        ### Convert back the data into Sage space.                         

        (floatP, floatPcv, intvl, ic, terr) = \

            slz_interval_and_polynomial_to_sage((prceSo[0], prceSo[0],

                                                 prceSo[1], prceSo[2], 

                                                 prceSo[3]))

        intvl = RRIF(intvl)

        ## Clean-up Sollya stuff.

        for elem in prceSo:

            sollya_lib_clear_obj(elem)

        #print  floatP, floatPcv, intvl, ic, terr

        #print floatP

        #print intvl.endpoints()[0].n(), \

        #      ic.n(),

        #intvl.endpoints()[1].n()

        ### Check returned data.

        #### Is approximation error OK?

        if terr > polyApproxAccur:

            exceptionErrorMess  = \

                "Approximation failed - computed error:" + \

                str(terr) + " - target error: "

            exceptionErrorMess += \

                str(polyApproxAccur) + ". Aborting!"

            raise Exception(exceptionErrorMess)

        #### Is lower bound OK?

        if lb != intvl.endpoints()[0]:

            exceptionErrorMess =  "Wrong lower bound:" + \

                               str(lb) + ". Aborting!"

            raise Exception(exceptionErrorMess)

        #### Set upper bound.

        if ub > intvl.endpoints()[1]:

            ub = intvl.endpoints()[1]

            print "[", lb, ",", ub, "]", ((ub - lb) / intervalUlp).log2().n(), \

            "log2(numbers)." 

            taylCondFailedCount += 1

        #### Is interval not degenerate?

        if lb >= ub:

            exceptionErrorMess = "Degenerate interval: " + \

                                "lowerBound(" + str(lb) +\

                                ")>= upperBound(" + str(ub) + \

                                "). Aborting!"

            raise Exception(exceptionErrorMess)

        #### Is interval center ok?

        if ic <= lb or ic >= ub:

            exceptionErrorMess =  "Invalid interval center for " + \

                                str(lb) + ',' + str(ic) + ',' +  \

                                str(ub) + ". Aborting!"

            raise Exception(exceptionErrorMess)

        ##### Current interval width and reset future interval width.

        bw = ub - lb

        nbw = 0

        icAsInt = int(ic * toIntegerFactor)

        #### The following ratio is always >= 1. In case we may want to

        #    enlarge the interval

        curTaylErrRat = polyApproxAccur / terr

        ### Make the  integral transformations.

        #### Bounds and interval center.

        intIc = int(ic * toIntegerFactor)

        intLb = int(lb * toIntegerFactor) - intIc

        intUb = int(ub * toIntegerFactor) - intIc

        #

        #### Polynomials

        basisConstructionTime         = cputime()

        ##### To a polynomial with rational coefficients with rational arguments

        ratRatP = slz_float_poly_of_float_to_rat_poly_of_rat_pow_two(floatP)

        ##### To a polynomial with rational coefficients with integer arguments

        ratIntP = \

            slz_rat_poly_of_rat_to_rat_poly_of_int(ratRatP, precision)

        #####  Ultimately a multivariate polynomial with integer coefficients  

        #      with integer arguments.

        coppersmithTuple = \

            slz_rat_poly_of_int_to_poly_for_coppersmith(ratIntP, 

                                                        precision, 

                                                        targetHardnessToRound, 

                                                        i, t)

        #### Recover Coppersmith information.

        intIntP = coppersmithTuple[0]

        N = coppersmithTuple[1]

        nAtAlpha = N^alpha

        tBound = coppersmithTuple[2]

        leastCommonMultiple = coppersmithTuple[3]

        iBound = max(abs(intLb),abs(intUb))

        basisConstructionsFullTime        += cputime(basisConstructionTime)

        basisConstructionsCount           += 1

        reductionTime                     = cputime()

        #### Compute the reduced polynomials.

        ccReducedPolynomialsList =  \

                slz_compute_coppersmith_reduced_polynomials(intIntP, 

                                                            alpha, 

                                                            N, 

                                                            iBound, 

                                                            tBound)

        if ccReducedPolynomialsList is None:

            raise Exception("Reduction failed.")

        reductionsFullTime    += cputime(reductionTime)

        reductionsCount       += 1

        if len(ccReducedPolynomialsList) < 2:

            print "Nothing to form resultants with."

            print

            coppCondFailedCount += 1

            coppCondFailed = True

            ##### Apply a different shrink factor according to 

            #  the number of compliant polynomials.

            if len(ccReducedPolynomialsList) == 0:

                ub = lb + bw * noCoppersmithIntervalShrink

            else: # At least one compliant polynomial.

                ub = lb + bw * oneCoppersmithIntervalShrink

            if ub > sdub:

                ub = sdub

            if lb == ub:

                raise Exception("Cant shrink interval \

                anymore to get Coppersmith condition.")

            nbw = 0

            continue

        #### We have at least two polynomials.

        #    Let us try to compute resultants.

        #    For each resultant computed, go for the solutions.

        ##### Build the pairs list.

        polyPairsList           = []

        for polyOuterIndex in xrange(0, len(ccReducedPolynomialsList) - 1):

            for polyInnerIndex in xrange(polyOuterIndex+1, 

                                         len(ccReducedPolynomialsList)):

                polyPairsList.append((ccReducedPolynomialsList[polyOuterIndex],

                                      ccReducedPolynomialsList[polyInnerIndex]))

        #### Actual root search.

        rootsSet            = set()

        hasNonNullResultant = False

        for polyPair in polyPairsList:

            if hasNonNullResultant:

                break

            resultantsComputationTime   = cputime()

            currentResultant = \

                slz_resultant(polyPair[0],

                              polyPair[1],

                              t)

            resultantsComputationsFullTime += cputime(resultantsComputationTime)

            resultantsComputationsCount    += 1

            if currentResultant is None:                    

                print "Nul resultant"

                continue # Next polyPair.

            else:

                hasNonNullResultant = True

            #### We have a non null resultant. From now on, whatever the

            #    root search yields, no extra root search is necessary.

            #### A constant resultant leads to no root. Root search is done.

            if currentResultant.degree() < 1:

                print "Resultant is constant:", currentResultant

                continue # Next polyPair and should break.

            #### Actual roots computation.

            rootsComputationTime       = cputime()

            ##### Compute i roots

            iRootsList = Zi(currentResultant).roots()

            ##### For each iRoot, compute the corresponding tRoots and 

            #     and build populate the .rootsSet.

            for iRoot in iRootsList:

                ####### Roots returned by roots() are (value, multiplicity) 

                #       tuples.

                #print "iRoot:", iRoot

                ###### Use the tRoot against each polynomial, alternatively.

                for indexInPair in range(0,2):

                    currentPolynomial = polyPair[indexInPair]

                    ####### If the polynomial is univariate, just drop it.

                    if len(currentPolynomial.variables()) < 2:

                        print "    Current polynomial is not in two variables."

                        continue # Next indexInPair

                    tRootsList = \

                        Zt(currentPolynomial.subs({currentPolynomial.variables()[0]:iRoot[0]})).roots()

                    ####### The tRootsList can be empty, hence the test.

                    if len(tRootsList) == 0:

                        print "    No t root."

                        continue # Next indexInPair

                    for tRoot in tRootsList:

                        rootsSet.add((iRoot[0], tRoot[0]))

            # End of roots computation.

            rootsComputationsFullTime   =   cputime(rootsComputationTime)

            rootsComputationsCount      +=  1

        # End loop for polyPair in polyParsList.  Will break at next iteration.

        # since a non null resultant was found.

        #### Prepare for results for the current interval..

        intervalResultsList = []

        intervalResultsList.append((lb, ub))

        #### Check roots.

        rootsResultsList = []

        for root in rootsSet:

            specificRootResultsList = []

            failingBounds = []

            intIntPdivN = intIntP(root[0], root[1]) / N

            if int(intIntPdivN) != intIntPdivN:

                continue # Next root

            # Root qualifies for modular equation, test it for hardness to round.

            hardToRoundCaseAsFloat = RRR((icAsInt + root[0]) / toIntegerFactor)

            #print "Before unscaling:", hardToRoundCaseAsFloat.n(prec=precision)

            #print scalingFunction

            scaledHardToRoundCaseAsFloat = \

                scalingFunction(hardToRoundCaseAsFloat) 

            print "Candidate HTRNc at x =", \

                scaledHardToRoundCaseAsFloat.n().str(base=2),

            if slz_is_htrn(scaledHardToRoundCaseAsFloat,

                           function,

                           2^-(targetHardnessToRound),

                           RRR):

                print hardToRoundCaseAsFloat, "is HTRN case."

                if lb <= hardToRoundCaseAsFloat and hardToRoundCaseAsFloat <= ub:

                    print "Found in interval."

                else:

                    print "Found out of interval."

                specificRootResultsList.append(hardToRoundCaseAsFloat.n().str(base=2))

                # Check the root is in the bounds

                if abs(root[0]) > iBound or abs(root[1]) > tBound:

                    print "Root", root, "is out of bounds for modular equation."

                    if abs(root[0]) > iBound:

                        print "root[0]:", root[0]

                        print "i bound:", iBound

                        failingBounds.append('i')

                        failingBounds.append(root[0])

                        failingBounds.append(iBound)

                    if abs(root[1]) > tBound:

                        print "root[1]:", root[1]

                        print "t bound:", tBound

                        failingBounds.append('t')

                        failingBounds.append(root[1])

                        failingBounds.append(tBound)

                if len(failingBounds) > 0:

                    specificRootResultsList.append(failingBounds)

            else: # From slz_is_htrn...

                print "is not an HTRN case."

            if len(specificRootResultsList) > 0:

                rootsResultsList.append(specificRootResultsList)

        if len(rootsResultsList) > 0:

            intervalResultsList.append(rootsResultsList)

        ### Check if a non null resultant was found. If not shrink the interval.

        if not hasNonNullResultant:

            print "Only null resultants for this reduction, shrinking interval."

            resultCondFailed      =  True

            resultCondFailedCount += 1

            ### Shrink interval for next iteration.

            ub = lb + bw * onlyNullResultantsShrink

            if ub > sdub:

                ub = sdub

            nbw = 0

            continue

        #### An intervalResultsList has at least the bounds.

        globalResultsList.append(intervalResultsList)   

        #### Compute an incremented width for next upper bound, only

        #    if not Coppersmith condition nor resultant condition

        #    failed at the previous run. 

        if not coppCondFailed and not resultCondFailed:

            nbw = noErrorIntervalStretch * bw

        else:

            nbw = bw

        ##### Reset the failure flags. They will be raised

        #     again if needed.

        coppCondFailed   = False

        resultCondFailed = False

        #### For next iteration (at end of loop)

        #print "nbw:", nbw

        lb  = ub

        ub += nbw     

        if ub > sdub:

            ub = sdub

        print

    # End while True

    ## Main loop just ended.

    globalWallTime = walltime(wallTimeStart)

    globalCpuTime  = cputime(cpuTimeStart)

    ## Output results

    print ; print "Intervals and HTRNs" ; print

    for intervalResultsList in globalResultsList:

        intervalResultString = "[" + str(intervalResultsList[0][0]) +\

                               "," + str(intervalResultsList[0][1])  + "]"

        print intervalResultString,

        if len(intervalResultsList) > 1:

            rootsResultsList = intervalResultsList[1]

            specificRootResultIndex = 0

            for specificRootResultsList in rootsResultsList:

                if specificRootResultIndex == 0:

                    print "\t", specificRootResultsList[0],

                else:

                    print " " * len(intervalResultString), "\t", \

                          specificRootResultsList[0],

                if len(specificRootResultsList) > 1:

                    print specificRootResultsList[1]

                specificRootResultIndex += 1

        print ; print

    #print globalResultsList

    #

    print "Timers and counters"

    print

    print "Number of iterations:", iterCount

    print "Taylor condition failures:", taylCondFailedCount

    print "Coppersmith condition failures:", coppCondFailedCount

    print "Resultant condition failures:", resultCondFailedCount

    print "Iterations count: ", iterCount

    print "Number of intervals:", len(globalResultsList)

    print "Number of basis constructions:", basisConstructionsCount 

    print "Total CPU time spent in basis constructions:", \

        basisConstructionsFullTime

    if basisConstructionsCount != 0:

        print "Average basis construction CPU time:", \

            basisConstructionsFullTime/basisConstructionsCount

    print "Number of reductions:", reductionsCount

    print "Total CPU time spent in reductions:", reductionsFullTime

    if reductionsCount != 0:

        print "Average reduction CPU time:", \

            reductionsFullTime/reductionsCount

    print "Number of resultants computation rounds:", \

        resultantsComputationsCount

    print "Total CPU time spent in resultants computation rounds:", \

        resultantsComputationsFullTime

    if resultantsComputationsCount != 0:

        print "Average resultants computation round CPU time:", \

            resultantsComputationsFullTime/resultantsComputationsCount

    print "Number of root finding rounds:", rootsComputationsCount

    print "Total CPU time spent in roots finding rounds:", \

        rootsComputationsFullTime

    if rootsComputationsCount != 0:

        print "Average roots finding round CPU time:", \

            rootsComputationsFullTime/rootsComputationsCount

    print "Global Wall time:", globalWallTime

    print "Global CPU time:", globalCpuTime

    ## Output counters

# End srs_runSLZ-v02

def srs_run_SLZ_v03(inputFunction,

                    inputLowerBound,

                    inputUpperBound,

                    alpha,

                    degree,

                    precision,

                    emin,

                    emax,

                    targetHardnessToRound,

                    debug = False):

    """

    Changes from V2:

        Root search is changed:

            - we compute the resultants in i and in t;

            - we compute the roots set of each of these resultants;

            - we combine all the possible pairs between the two sets;

            - we check these pairs in polynomials for correctness.

    Changes from V1: 

        1- check for roots as soon as a resultant is computed;

        2- once a non null resultant is found, check for roots;

        3- constant resultant == no root.

    """

    if debug:

        print "Function                :", inputFunction

        print "Lower bound             :", inputLowerBound

        print "Upper bounds            :", inputUpperBound

        print "Alpha                   :", alpha

        print "Degree                  :", degree

        print "Precision               :", precision

        print "Emin                    :", emin

        print "Emax                    :", emax

        print "Target hardness-to-round:", targetHardnessToRound

        print

    ## Important constants.

    ### Stretch the interval if no error happens.

    noErrorIntervalStretch = 1 + 2^(-5)

    ### If no vector validates the Coppersmith condition, shrink the interval

    #   by the following factor.

    noCoppersmithIntervalShrink = 1/2

    ### If only (or at least) one vector validates the Coppersmith condition, 

    #   shrink the interval by the following factor.

    oneCoppersmithIntervalShrink = 3/4

    #### If only null resultants are found, shrink the interval by the 

    #    following factor.

    onlyNullResultantsShrink     = 3/4

    ## Structures.

    RRR         = RealField(precision)

    RRIF        = RealIntervalField(precision)

    ## Converting input bound into the "right" field.

    lowerBound = RRR(inputLowerBound)

    upperBound = RRR(inputUpperBound)

    ## Before going any further, check domain and image binade conditions.

    print inputFunction(1).n()

    output = slz_fix_bounds_for_binades(lowerBound, upperBound, inputFunction)

    if output is None:

        print "Invalid domain/image binades. Domain:",\

        lowerBound, upperBound, "Images:", \

        inputFunction(lowerBound), inputFunction(upperBound)

        raise Exception("Invalid domain/image binades.")

    lb = output[0] ; ub = output[1]

    if lb != lowerBound or ub != upperBound:

        print "lb:", lb, " - ub:", ub

        print "Invalid domain/image binades. Domain:",\

        lowerBound, upperBound, "Images:", \

        inputFunction(lowerBound), inputFunction(upperBound)

        raise Exception("Invalid domain/image binades.")

    #

    ## Progam initialization

    ### Approximation polynomial accuracy and hardness to round.

    polyApproxAccur           = 2^(-(targetHardnessToRound + 1))

    polyTargetHardnessToRound = targetHardnessToRound + 1

    ### Significand to integer conversion ratio.

    toIntegerFactor           = 2^(precision-1)

    print "Polynomial approximation required accuracy:", polyApproxAccur.n()

    ### Variables and rings for polynomials and root searching.

    i=var('i')

    t=var('t')

    inputFunctionVariable = inputFunction.variables()[0]

    function = inputFunction.subs({inputFunctionVariable:i})

    #  Polynomial Rings over the integers, for root finding.

    Zi = ZZ[i]

    Zt = ZZ[t]

    Zit = ZZ[i,t]

    ## Number of iterations limit.

    maxIter = 100000

    #

    ## Compute the scaled function and the degree, in their Sollya version 

    #  once for all.

    (scaledf, sdlb, sdub, silb, siub) = \

        slz_compute_scaled_function(function, lowerBound, upperBound, precision)

    print "Scaled function:", scaledf._assume_str().replace('_SAGE_VAR_', '')

    scaledfSo = sollya_lib_parse_string(scaledf._assume_str().replace('_SAGE_VAR_', ''))

    degreeSo  = pobyso_constant_from_int_sa_so(degree)

    #

    ## Compute the scaling. boundsIntervalRifSa defined out of the loops.

    domainBoundsInterval   = RRIF(lowerBound, upperBound)

    (unscalingFunction, scalingFunction) = \

        slz_interval_scaling_expression(domainBoundsInterval, i)

    #print scalingFunction, unscalingFunction

    ## Set the Sollya internal precision (with an arbitrary minimum of 192).

    internalSollyaPrec = ceil((RR('1.5') * targetHardnessToRound) / 64) * 64

    if internalSollyaPrec < 192:

        internalSollyaPrec = 192

    pobyso_set_prec_sa_so(internalSollyaPrec)

    print "Sollya internal precision:", internalSollyaPrec

    ## Some variables.

    ### General variables

    lb                  = sdlb

    ub                  = sdub

    nbw                 = 0

    intervalUlp         = ub.ulp()

    #### Will be set by slz_interval_and_polynomila_to_sage.

    ic                  = 0 

    icAsInt             = 0    # Set from ic.

    solutionsSet        = set()

    tsErrorWidth        = []

    csErrorVectors      = []

    csVectorsResultants = []