static void DivideGaussian(BigInteger *real, BigInteger *imag)
{
BigInteger Tmp, norm, realNum, imagNum;
CopyBigInt(&Tmp, real);
Tmp.sign = SIGN_POSITIVE;
BigIntMultiply(real, real, &norm);
BigIntMultiply(imag, imag, &Tmp);
BigIntAdd(&norm, &Tmp, &norm);
BigIntMultiply(&ReValue, real, &realNum);
BigIntMultiply(&ImValue, imag, &Tmp);
BigIntAdd(&realNum, &Tmp, &realNum);
BigIntMultiply(&ImValue, real, &imagNum);
BigIntMultiply(&ReValue, imag, &Tmp);
BigIntSubt(&imagNum, &Tmp, &imagNum);
BigIntRemainder(&realNum, &norm, &Tmp);
if (Tmp.nbrLimbs == 1 && Tmp.limbs[0].x == 0)
{
BigIntRemainder(&imagNum, &norm, &Tmp);
if (Tmp.nbrLimbs == 1 && Tmp.limbs[0].x == 0)
{
BigIntDivide(&realNum, &norm, &ReValue);
BigIntDivide(&imagNum, &norm, &ImValue);
w("<li>");
showNumber(real, imag);
w("</li>");
}
}
}
void GaussianFactorization(void)
{
BigInteger prime, q, r, M1, M2, Tmp;
struct sFactors *pstFactor;
BigIntMultiply(&ReValue, &ReValue, &tofactor);
BigIntMultiply(&ImValue, &ImValue, &Tmp);
BigIntAdd(&tofactor, &Tmp, &tofactor);
NbrFactorsNorm = 0;
#ifdef __EMSCRIPTEN__
originalTenthSecond = tenths();
#endif
if (tofactor.nbrLimbs == 1 && tofactor.limbs[0].x == 0)
{ // Norm is zero.
w("<ul><li>Any gaussian prime divides this number</li></ul>");
return;
}
w("<ul>");
if (tofactor.nbrLimbs > 1 || tofactor.limbs[0].x > 1)
{ // norm greater than 1. Factor norm.
int index, index2;
char *ptrFactorDec = tofactorDec;
NumberLength = tofactor.nbrLimbs;
CompressBigInteger(nbrToFactor, &tofactor);
strcpy(ptrFactorDec, "Re² + Im² = ");
ptrFactorDec += strlen(ptrFactorDec);
Bin2Dec(ReValue.limbs, ptrFactorDec, ReValue.nbrLimbs, groupLen);
ptrFactorDec += strlen(ptrFactorDec);
strcpy(ptrFactorDec, "² + ");
ptrFactorDec += strlen(ptrFactorDec);
Bin2Dec(ImValue.limbs, ptrFactorDec, ImValue.nbrLimbs, groupLen);
ptrFactorDec += strlen(ptrFactorDec);
strcpy(ptrFactorDec, "²");
ptrFactorDec += strlen(ptrFactorDec);
factor(&tofactor, nbrToFactor, factorsNorm, astFactorsNorm, NULL);
NbrFactorsNorm = astFactorsNorm[0].multiplicity;
pstFactor = &astFactorsNorm[1];
for (index = 0; index < NbrFactorsNorm; index++)
{
int *ptrPrime = pstFactor->ptrFactor;
NumberLength = *ptrPrime;
UncompressBigInteger(ptrPrime, &prime);
prime.sign = SIGN_POSITIVE;
if (prime.nbrLimbs == 1 && prime.limbs[0].x == 2)
{ // Prime factor is 2.
for (index2 = 0; index2 < pstFactor->multiplicity; index2++)
{
M1.nbrLimbs = M2.nbrLimbs = 1;
M1.limbs[0].x = M2.limbs[0].x = 1;
M1.sign = SIGN_POSITIVE;
M2.sign = SIGN_NEGATIVE;
DivideGaussian(&M1, &M1); // Divide by 1+i
DivideGaussian(&M1, &M2); // Divide by 1-i
}
}
if ((prime.limbs[0].x & 2) == 0)
{ // Prime is congruent to 1 (mod 4)
CopyBigInt(&q, &prime);
NumberLength = prime.nbrLimbs;
memcpy(&TestNbr, prime.limbs, NumberLength * sizeof(limb));
TestNbr[NumberLength].x = 0;
GetMontgomeryParms(NumberLength);
subtractdivide(&q, 1, 4); // q = (prime-1)/4
memset(&K, 0, NumberLength * sizeof(limb));
memset(minusOneMont, 0, NumberLength * sizeof(limb));
SubtBigNbrModN(minusOneMont, MontgomeryMultR1, minusOneMont, TestNbr, NumberLength);
K[0].x = 1;
do
{ // Loop that finds mult1 = sqrt(-1) mod prime in Montgomery notation.
K[0].x++;
modPow(K, q.limbs, q.nbrLimbs, mult1.limbs);
} while (!memcmp(mult1.limbs, MontgomeryMultR1, NumberLength * sizeof(limb)) ||
!memcmp(mult1.limbs, minusOneMont, NumberLength * sizeof(limb)));
K[0].x = 1;
modmult(mult1.limbs, K, mult1.limbs); // Convert mult1 to standard notation.
UncompressLimbsBigInteger(mult1.limbs, &mult1); // Convert to Big Integer.
mult2.nbrLimbs = 1; // mult2 <- 1
mult2.limbs[0].x = 1;
mult2.sign = SIGN_POSITIVE;
for (;;)
{
// norm <- (mult1^2 + mult2^2) / prime
BigIntMultiply(&mult1, &mult1, &tofactor);
BigIntMultiply(&mult2, &mult2, &Tmp);
BigIntAdd(&tofactor, &Tmp, &Tmp);
BigIntDivide(&Tmp, &prime, &tofactor);
if (tofactor.nbrLimbs == 1 && tofactor.limbs[0].x == 1)
{ // norm equals 1.
break;
}
BigIntRemainder(&mult1, &tofactor, &M1);
BigIntRemainder(&mult2, &tofactor, &M2);
BigIntAdd(&M1, &M1, &Tmp);
BigIntSubt(&tofactor, &Tmp, &Tmp);
if (Tmp.sign == SIGN_NEGATIVE)
{
BigIntSubt(&M1, &tofactor, &M1);
}
BigIntAdd(&M2, &M2, &Tmp);
BigIntSubt(&tofactor, &Tmp, &Tmp);
if (Tmp.sign == SIGN_NEGATIVE)
{
BigIntSubt(&M2, &tofactor, &M2);
}
// Compute q <- (mult1*M1 + mult2*M2) / norm
BigIntMultiply(&mult1, &M1, &q);
BigIntMultiply(&mult2, &M2, &Tmp);
BigIntAdd(&q, &Tmp, &Tmp);
BigIntDivide(&Tmp, &tofactor, &q);
// Compute Mult2 <- (mult1*M2 - mult2*M1) / tofactor
BigIntMultiply(&mult1, &M2, &r);
BigIntMultiply(&mult2, &M1, &Tmp);
BigIntSubt(&r, &Tmp, &Tmp);
BigIntDivide(&Tmp, &tofactor, &mult2);
CopyBigInt(&mult1, &q);
mult1.sign = SIGN_POSITIVE; // mult1 <- abs(mult1)
mult2.sign = SIGN_POSITIVE; // mult2 <- abs(mult2)
} /* end while */
CopyBigInt(&M1, &mult1);
CopyBigInt(&M2, &mult2);
BigIntSubt(&M1, &M2, &Tmp);
if (Tmp.sign == SIGN_NEGATIVE)
{
CopyBigInt(&Tmp, &mult1);
CopyBigInt(&mult1, &mult2);
CopyBigInt(&mult2, &Tmp);
}
for (index2 = 0; index2 < pstFactor->multiplicity; index2++)
{
DivideGaussian(&mult1, &mult2);
BigIntNegate(&mult2, &Tmp);
DivideGaussian(&mult1, &Tmp);
}
} // end p = 1 (mod 4)
else
{ // if p = 3 (mod 4)
q.nbrLimbs = 1; // q <- 0
q.limbs[0].x = 0;
q.sign = SIGN_POSITIVE;
for (index2 = 0; index2 < pstFactor->multiplicity; index2++)
{
DivideGaussian(&prime, &q);
} // end p = 3 (mod 4)
}
pstFactor++;
}
}
// Process units: 1, -1, i, -i.
if (ReValue.nbrLimbs == 1 && ReValue.limbs[0].x == 1)
{
if (ReValue.sign == SIGN_POSITIVE)
{ // Value is 1.
if (NbrFactorsNorm == 0)
{
w("No gaussian prime divides this number");
}
}
else
{ // Value is -1.
w("<li>-1</li>");
}
}
else if (ImValue.sign == SIGN_POSITIVE)
{
w("<li>i</li>");
}
else
{
w("<li>-i</li>");
}
w("</ul>");
}
BigIntRef BigIntDivide (BigIntRef bi, BigIntRef bi2) {
if (bi2 == bi) {
// create a copy of bi2, call, and release the copy
BigIntRef nbi2 = BigIntCreateCopy(bi2, 0);
BigIntRef answer = BigIntDivide(bi, nbi2);
BigIntRelease(nbi2);
return answer;
}
UInt8 neg1 = bi->flags & kBigIntFlagNegative;
UInt8 neg2 = bi2->flags & kBigIntFlagNegative;
if (neg1) bi->flags ^= kBigIntFlagNegative;
if (neg2) bi2->flags ^= kBigIntFlagNegative;
if (BigIntOperatorCheck(bi2, kBigIntGreaterThan, bi)) {
if (neg1) bi->flags |= kBigIntFlagNegative;
if (neg2) bi2->flags |= kBigIntFlagNegative;
return BigIntNew(0);
}
BitBuffer quotientBuffer = BitBufferNew(bi->bits->byteCount);
BigIntRef remainingBits = BigIntNew(0);
int lastIntIndex = bi->binDigits;
int remainingBitsI = bi->binDigits;
while (BBYes) {
// is nextNumber big enough?
if (BigIntOperatorCheck(remainingBits, kBigIntLessThan, bi2)) {
// pull down the next number, or free
if (remainingBitsI <= 0) {
break;
} else {
// we will make remainingBits larger
remainingBitsI--;
BigIntRef newInt = BigIntCreateCopy(remainingBits, 1);
BitBufferSetBit(newInt->bits, BitBufferGetBit(bi->bits, remainingBitsI),
0);
int doIt = 1;
if (!BitBufferGetBit(bi->bits, remainingBitsI)) {
if (remainingBits->binDigits == 0) {
doIt = 0;
}
}
if (doIt) {
BigIntRelease(remainingBits);
remainingBits = newInt;
} else BigIntRelease(newInt);
}
} else {
// create a new big number
if (remainingBitsI < 0) break;
BitBufferSetBit(quotientBuffer, 1, remainingBitsI);
BigIntRef newInt = BigIntSubtract(remainingBits, bi2);
BigIntRelease(remainingBits);
remainingBits = newInt;
lastIntIndex = remainingBitsI;
//remainingBitsI --;
}
}
BigIntRelease(remainingBits);
if (neg1) bi->flags |= kBigIntFlagNegative;
if (neg2) bi2->flags |= kBigIntFlagNegative;
BigIntRef quotient = BigIntNew(0);
BigIntRemoveDecimalString(quotient);
BitBufferFree(quotient->bits, 1);
quotient->bits = quotientBuffer;
quotient->binDigits = quotientBuffer->bitCount;
quotient->flags = kBigIntFlagNegative & (neg1 ^ neg2);
return quotient;
}
void DiscreteLogarithm(void)
{
BigInteger groupOrder, subGroupOrder, powSubGroupOrder, powSubGroupOrderBak;
BigInteger Exponent, runningExp, baseExp, mod;
BigInteger logar, logarMult, runningExpBase;
BigInteger currentExp;
int indexBase, indexExp;
int index, expon;
limb addA, addB, addA2, addB2;
limb mult1, mult2;
double magnitude, firstLimit, secondLimit;
long long brentK, brentR;
unsigned char EndPollardBrentRho;
int nbrLimbs;
struct sFactors *pstFactors;
enum eLogMachineState logMachineState;
char *ptr;
#ifdef __EMSCRIPTEN__
lModularMult = 0;
#endif
NumberLength = modulus.nbrLimbs;
if (!TestBigNbrEqual(&LastModulus, &modulus))
{
CompressBigInteger(nbrToFactor, &modulus);
Bin2Dec(modulus.limbs, tofactorDec, modulus.nbrLimbs, groupLen);
factor(&modulus, nbrToFactor, factorsMod, astFactorsMod, NULL);
NbrFactorsMod = astFactorsMod[0].multiplicity;
}
intToBigInteger(&DiscreteLog, 0); // DiscreteLog <- 0
intToBigInteger(&DiscreteLogPeriod, 1); // DiscreteLogPeriod <- 1
for (index = 1; index <= NbrFactorsMod; index++)
{
int mostSignificantDword, leastSignificantDword;
int NbrFactors;
int *ptrPrime;
int multiplicity;
ptrPrime = astFactorsMod[index].ptrFactor;
NumberLength = *ptrPrime;
UncompressBigInteger(ptrPrime, &groupOrder);
groupOrder.sign = SIGN_POSITIVE;
BigIntRemainder(&base, &groupOrder, &tmpBase);
if (tmpBase.nbrLimbs == 1 && tmpBase.limbs[0].x == 0)
{ // modulus and base are not relatively prime.
int ctr;
multiplicity = astFactorsMod[index].multiplicity;
CopyBigInt(&bigNbrA, &power);
for (ctr = multiplicity; ctr > 0; ctr--)
{
BigIntRemainder(&bigNbrA, &groupOrder, &bigNbrB);
if (bigNbrB.nbrLimbs != 1 || bigNbrB.limbs[0].x != 0)
{ // Exit loop if integer division cannot be performed
break;
}
BigIntDivide(&bigNbrA, &groupOrder, &bigNbrB);
CopyBigInt(&bigNbrA, &bigNbrB);
}
if (ctr == 0)
{ // Power is multiple of prime^exp.
continue;
}
// Compute prime^mutliplicity.
BigIntPowerIntExp(&groupOrder, multiplicity, &tmp2);
BigIntRemainder(&base, &tmp2, &tmpBase);
// Get tentative exponent.
ctr = multiplicity - ctr;
intToBigInteger(&bigNbrB, ctr); // Convert exponent to big integer.
NumberLength = tmp2.nbrLimbs;
memcpy(TestNbr, tmp2.limbs, (NumberLength + 1) * sizeof(limb));
GetMontgomeryParms(NumberLength);
BigIntModularPower(&tmpBase, &bigNbrB, &bigNbrA);
BigIntRemainder(&power, &tmp2, &bigNbrB);
BigIntSubt(&bigNbrA, &bigNbrB, &bigNbrA);
if (bigNbrA.nbrLimbs == 1 && bigNbrA.limbs[0].x == 0)
{
intToBigInteger(&DiscreteLog, ctr); // DiscreteLog <- exponent
intToBigInteger(&DiscreteLogPeriod, 0); // DiscreteLogPeriod <- 0
break;
}
showText("There is no discrete logarithm");
DiscreteLogPeriod.sign = SIGN_NEGATIVE;
return;
}
else
{ // modulus and base are relatively prime.
BigIntRemainder(&power, &groupOrder, &bigNbrB);
if (bigNbrB.nbrLimbs == 1 && bigNbrB.limbs[0].x == 0)
{ // power is multiple of prime. Error.
showText("There is no discrete logarithm");
DiscreteLogPeriod.sign = SIGN_NEGATIVE;
return;
}
}
CompressLimbsBigInteger(baseMontg, &tmpBase);
BigIntRemainder(&power, &groupOrder, &tmpBase);
CompressLimbsBigInteger(powerMontg, &tmpBase);
// Compute group order as the prime minus 1.
groupOrder.limbs[0].x--;
showText("Computing discrete logarithm...");
CompressBigInteger(nbrToFactor, &groupOrder);
factor(&groupOrder, nbrToFactor, factorsGO, astFactorsGO, NULL); // factor groupOrder.
NbrFactors = astFactorsGO[0].multiplicity;
NumberLength = *ptrPrime;
UncompressBigInteger(ptrPrime, &mod);
intToBigInteger(&logar, 0); // logar <- 0
intToBigInteger(&logarMult, 1); // logarMult <- 1
NumberLength = mod.nbrLimbs;
memcpy(TestNbr, mod.limbs, NumberLength * sizeof(limb));
TestNbr[NumberLength].x = 0;
// yieldFreq = 1000000 / (NumberLength*NumberLength);
GetMontgomeryParms(NumberLength);
#if 0
char *ptrText = textExp;
strcpy(ptrText, "<p>NumberLength (2) = ");
ptrText = ptrText + strlen(ptrText);
int2dec(&ptrText, NumberLength);
strcpy(ptrText, "</p>");
DiscreteLogPeriod.sign = SIGN_NEGATIVE;
return;
#endif
// Convert base and power to Montgomery notation.
modmult(baseMontg, MontgomeryMultR2, baseMontg);
modmult(powerMontg, MontgomeryMultR2, powerMontg);
mostSignificantDword = NumberLength - 1;
if (NumberLength == 1)
{
leastSignificantDword = NumberLength - 1;
firstLimit = (double)TestNbr[leastSignificantDword].x / 3;
}
else
{
leastSignificantDword = NumberLength - 2;
firstLimit = ((double)TestNbr[mostSignificantDword].x * LIMB_RANGE +
TestNbr[leastSignificantDword].x) / 3;
}
secondLimit = firstLimit * 2;
for (indexBase = 0; indexBase < NbrFactors; indexBase++)
{
NumberLength = *astFactorsGO[indexBase + 1].ptrFactor;
UncompressBigInteger(astFactorsGO[indexBase + 1].ptrFactor, &subGroupOrder);
subGroupOrder.sign = SIGN_POSITIVE;
strcpy(textExp, "Computing discrete logarithm in subgroup of ");
Bin2Dec(subGroupOrder.limbs, textExp + strlen(textExp), subGroupOrder.nbrLimbs, groupLen);
ptr = textExp + strlen(textExp);
if (astFactorsGO[indexBase + 1].multiplicity > 1)
{
*ptr++ = '<';
*ptr++ = 's';
*ptr++ = 'u';
*ptr++ = 'p';
*ptr++ = '>';
int2dec(&ptr, astFactorsGO[indexBase + 1].multiplicity);
*ptr++ = '<';
*ptr++ = '/';
*ptr++ = 's';
*ptr++ = 'u';
*ptr++ = 'p';
*ptr++ = '>';
}
strcpy(ptr, " elements.");
showText(textExp);
NumberLength = mod.nbrLimbs;
memcpy(TestNbr, mod.limbs, NumberLength * sizeof(limb));
NumberLengthOther = subGroupOrder.nbrLimbs;
memcpy(TestNbrOther, subGroupOrder.limbs, NumberLengthOther * sizeof(limb));
TestNbr[NumberLength].x = 0;
GetMontgomeryParms(NumberLength);
nbrLimbs = subGroupOrder.nbrLimbs;
dN = (double)subGroupOrder.limbs[nbrLimbs - 1].x;
if (nbrLimbs > 1)
{
dN += (double)subGroupOrder.limbs[nbrLimbs - 2].x / LIMB_RANGE;
if (nbrLimbs > 2)
{
dN += (double)subGroupOrder.limbs[nbrLimbs - 3].x / LIMB_RANGE / LIMB_RANGE;
}
}
CopyBigInt(&baseExp, &groupOrder);
// Check whether base is primitive root.
BigIntDivide(&groupOrder, &subGroupOrder, &tmpBase);
modPow(baseMontg, tmpBase.limbs, tmpBase.nbrLimbs, primRootPwr);
if (!memcmp(primRootPwr, MontgomeryMultR1, NumberLength * sizeof(limb)))
{ // Power is one, so it is not a primitive root.
logMachineState = CALC_LOG_BASE;
// Find primitive root
primRoot[0].x = 1;
if (NumberLength > 1)
{
memset(&primRoot[1], 0, (NumberLength - 1) * sizeof(limb));
}
do
{
primRoot[0].x++;
modPow(primRoot, tmpBase.limbs, tmpBase.nbrLimbs, primRootPwr);
} while (!memcmp(primRootPwr, MontgomeryMultR1, NumberLength * sizeof(limb)));
}
else
{ // Power is not 1, so the base is a primitive root.
logMachineState = BASE_PRIMITIVE_ROOT;
memcpy(primRoot, baseMontg, NumberLength * sizeof(limb));
}
for (;;)
{ // Calculate discrete logarithm in subgroup.
runningExp.nbrLimbs = 1; // runningExp <- 0
runningExp.limbs[0].x = 0;
runningExp.sign = SIGN_POSITIVE;
powSubGroupOrder.nbrLimbs = 1; // powSubGroupOrder <- 1
powSubGroupOrder.limbs[0].x = 1;
powSubGroupOrder.sign = SIGN_POSITIVE;
CopyBigInt(¤tExp, &groupOrder);
if (logMachineState == BASE_PRIMITIVE_ROOT)
{
memcpy(basePHMontg, baseMontg, NumberLength * sizeof(limb));
memcpy(currPowerMontg, powerMontg, NumberLength * sizeof(limb));
}
else if (logMachineState == CALC_LOG_BASE)
{
memcpy(basePHMontg, primRoot, NumberLength * sizeof(limb));
memcpy(currPowerMontg, baseMontg, NumberLength * sizeof(limb));
}
else
{ // logMachineState == CALC_LOG_POWER
memcpy(primRoot, basePHMontg, NumberLength * sizeof(limb));
memcpy(currPowerMontg, powerMontg, NumberLength * sizeof(limb));
}
for (indexExp = 0; indexExp < astFactorsGO[indexBase + 1].multiplicity; indexExp++)
{
/* PH below comes from Pohlig-Hellman algorithm */
BigIntDivide(¤tExp, &subGroupOrder, ¤tExp);
modPow(currPowerMontg, currentExp.limbs, currentExp.nbrLimbs, powerPHMontg);
BigIntDivide(&baseExp, &subGroupOrder, &baseExp);
if (subGroupOrder.nbrLimbs == 1 && subGroupOrder.limbs[0].x < 20)
{ // subGroupOrder less than 20.
if (!ComputeDLogModSubGroupOrder(indexBase, indexExp, &Exponent, &subGroupOrder))
{
return;
}
}
else
{ // Use Pollard's rho method with Brent's modification
memcpy(nbrPower, powerPHMontg, NumberLength * sizeof(limb));
memcpy(nbrBase, primRootPwr, NumberLength * sizeof(limb));
memcpy(nbrR2, nbrBase, NumberLength * sizeof(limb));
memset(nbrA2, 0, NumberLength * sizeof(limb));
memset(nbrB2, 0, NumberLength * sizeof(limb));
nbrB2[0].x = 1;
addA2.x = addB2.x = 0;
mult2.x = 1;
brentR = 1;
brentK = 0;
EndPollardBrentRho = FALSE;
do
{
memcpy(nbrR, nbrR2, NumberLength * sizeof(limb));
memcpy(nbrA, nbrA2, NumberLength * sizeof(limb));
memcpy(nbrB, nbrB2, NumberLength * sizeof(limb));
addA = addA2;
addB = addB2;
mult1 = mult2;
brentR *= 2;
do
{
brentK++;
if (NumberLength == 1)
{
magnitude = (double)nbrR2[leastSignificantDword].x;
}
else
{
magnitude = (double)nbrR2[mostSignificantDword].x * LIMB_RANGE +
nbrR2[leastSignificantDword].x;
}
if (magnitude < firstLimit)
{
modmult(nbrR2, nbrPower, nbrROther);
addA2.x++;
}
else if (magnitude < secondLimit)
{
modmult(nbrR2, nbrR2, nbrROther);
mult2.x *= 2;
addA2.x *= 2;
addB2.x *= 2;
}
else
{
modmult(nbrR2, nbrBase, nbrROther);
addB2.x++;
}
// Exchange nbrR2 and nbrROther
memcpy(nbrTemp, nbrR2, NumberLength * sizeof(limb));
memcpy(nbrR2, nbrROther, NumberLength * sizeof(limb));
memcpy(nbrROther, nbrTemp, NumberLength * sizeof(limb));
if (addA2.x >= (int)(LIMB_RANGE / 2) || addB2.x >= (int)(LIMB_RANGE / 2) ||
mult2.x >= (int)(LIMB_RANGE / 2))
{
// nbrA2 <- (nbrA2 * mult2 + addA2) % subGroupOrder
AdjustExponent(nbrA2, mult2, addA2, &subGroupOrder);
// nbrB2 <- (nbrB2 * mult2 + addB2) % subGroupOrder
AdjustExponent(nbrB2, mult2, addB2, &subGroupOrder);
mult2.x = 1;
addA2.x = addB2.x = 0;
}
if (!memcmp(nbrR, nbrR2, NumberLength * sizeof(limb)))
{
EndPollardBrentRho = TRUE;
break;
}
} while (brentK < brentR);
} while (EndPollardBrentRho == FALSE);
ExchangeMods(); // TestNbr <- subGroupOrder
// nbrA <- (nbrA * mult1 + addA) % subGroupOrder
AdjustExponent(nbrA, mult1, addA, &subGroupOrder);
// nbrB <- (nbrB * mult1 + addB) % subGroupOrder
AdjustExponent(nbrB, mult1, addB, &subGroupOrder);
// nbrA2 <- (nbrA * mult2 + addA2) % subGroupOrder
AdjustExponent(nbrA2, mult2, addA2, &subGroupOrder);
// nbrB2 <- (nbrA * mult2 + addB2) % subGroupOrder
AdjustExponent(nbrB2, mult2, addB2, &subGroupOrder);
// nbrB <- (nbrB2 - nbrB) % subGroupOrder
SubtBigNbrMod(nbrB2, nbrB, nbrB);
SubtBigNbrMod(nbrA, nbrA2, nbrA);
if (BigNbrIsZero(nbrA))
{ // Denominator is zero, so rho does not work.
ExchangeMods(); // TestNbr <- modulus
if (!ComputeDLogModSubGroupOrder(indexBase, indexExp, &Exponent, &subGroupOrder))
{
return; // Cannot compute discrete logarithm.
}
}
else
{
// Exponent <- (nbrB / nbrA) (mod subGroupOrder)
UncompressLimbsBigInteger(nbrA, &bigNbrA);
UncompressLimbsBigInteger(nbrB, &bigNbrB);
BigIntModularDivisionSaveTestNbr(&bigNbrB, &bigNbrA, &subGroupOrder, &Exponent);
Exponent.sign = SIGN_POSITIVE;
ExchangeMods(); // TestNbr <- modulus
}
}
modPow(primRoot, Exponent.limbs, Exponent.nbrLimbs, tmpBase.limbs);
ModInvBigNbr(tmpBase.limbs, tmpBase.limbs, TestNbr, NumberLength);
modmult(tmpBase.limbs, currPowerMontg, currPowerMontg);
BigIntMultiply(&Exponent, &powSubGroupOrder, &tmpBase);
BigIntAdd(&runningExp, &tmpBase, &runningExp);
BigIntMultiply(&powSubGroupOrder, &subGroupOrder, &powSubGroupOrder);
modPow(primRoot, subGroupOrder.limbs, subGroupOrder.nbrLimbs, tmpBase.limbs);
memcpy(primRoot, tmpBase.limbs, NumberLength * sizeof(limb));
}
if (logMachineState == BASE_PRIMITIVE_ROOT)
{ // Discrete logarithm was determined for this subgroup.
ExponentsGOComputed[indexBase] = astFactorsGO[indexBase + 1].multiplicity;
break;
}
if (logMachineState == CALC_LOG_BASE)
{
CopyBigInt(&runningExpBase, &runningExp);
logMachineState = CALC_LOG_POWER;
}
else
{ // Set powSubGroupOrderBak to powSubGroupOrder.
// if runningExpBase is not multiple of subGroupOrder,
// discrete logarithm is runningExp/runningExpBase mod powSubGroupOrderBak.
// Otherwise if runningExp is multiple of subGroupOrder, there is no logarithm.
// Otherwise, divide runningExp, runnignExpBase and powSubGroupOrderBak by subGroupOrder and repeat.
ExponentsGOComputed[indexBase] = astFactorsGO[indexBase + 1].multiplicity;
CopyBigInt(&powSubGroupOrderBak, &powSubGroupOrder);
do
{
BigIntRemainder(&runningExpBase, &subGroupOrder, &tmpBase);
if (tmpBase.nbrLimbs > 1 || tmpBase.limbs[0].x != 0)
{ // runningExpBase is not multiple of subGroupOrder
BigIntModularDivisionSaveTestNbr(&runningExp, &runningExpBase, &powSubGroupOrderBak, &tmpBase);
CopyBigInt(&runningExp, &tmpBase);
break;
}
BigIntRemainder(&runningExp, &subGroupOrder, &tmpBase);
if (tmpBase.nbrLimbs > 1 || tmpBase.limbs[0].x != 0)
{ // runningExpBase is not multiple of subGroupOrder
showText("There is no discrete logarithm");
DiscreteLogPeriod.sign = SIGN_NEGATIVE;
return;
}
BigIntDivide(&runningExp, &subGroupOrder, &tmpBase);
CopyBigInt(&runningExp, &tmpBase);
BigIntDivide(&runningExpBase, &subGroupOrder, &tmpBase);
CopyBigInt(&runningExpBase, &tmpBase);
BigIntDivide(&powSubGroupOrderBak, &subGroupOrder, &tmpBase);
CopyBigInt(&powSubGroupOrderBak, &tmpBase);
ExponentsGOComputed[indexBase]--;
if (tmpBase.nbrLimbs == 1 && tmpBase.limbs[0].x == 1)
{
break;
}
BigIntRemainder(&runningExpBase, &subGroupOrder, &tmpBase);
} while (tmpBase.nbrLimbs == 1 && tmpBase.limbs[0].x == 0);
CopyBigInt(&powSubGroupOrder, &powSubGroupOrderBak);
// The logarithm is runningExp / runningExpBase mod powSubGroupOrder
// When powSubGroupOrder is even, we cannot use Montgomery.
if (powSubGroupOrder.limbs[0].x & 1)
{ // powSubGroupOrder is odd.
BigIntModularDivisionSaveTestNbr(&runningExp, &runningExpBase, &powSubGroupOrder, &tmpBase);
CopyBigInt(&runningExp, &tmpBase);
}
else
{ // powSubGroupOrder is even (power of 2).
NumberLength = powSubGroupOrder.nbrLimbs;
CompressLimbsBigInteger(nbrB, &runningExpBase);
ComputeInversePower2(nbrB, nbrA, nbrB2); // nbrB2 is auxiliary var.
CompressLimbsBigInteger(nbrB, &runningExp);
multiply(nbrA, nbrB, nbrA, NumberLength, NULL); // nbrA <- quotient.
UncompressLimbsBigInteger(nbrA, &runningExp);
}
break;
}
}
CopyBigInt(&nbrV[indexBase], &runningExp);
NumberLength = powSubGroupOrder.nbrLimbs;
memcpy(TestNbr, powSubGroupOrder.limbs, NumberLength * sizeof(limb));
TestNbr[NumberLength].x = 0;
GetMontgomeryParms(NumberLength);
for (indexExp = 0; indexExp < indexBase; indexExp++)
{
// nbrV[indexBase] <- (nbrV[indexBase] - nbrV[indexExp])*
// modinv(PrimesGO[indexExp]^(ExponentsGO[indexExp]),
// powSubGroupOrder)
NumberLength = mod.nbrLimbs;
BigIntSubt(&nbrV[indexBase], &nbrV[indexExp], &nbrV[indexBase]);
BigIntRemainder(&nbrV[indexBase], &powSubGroupOrder, &nbrV[indexBase]);
if (nbrV[indexBase].sign == SIGN_NEGATIVE)
{
BigIntAdd(&nbrV[indexBase], &powSubGroupOrder, &nbrV[indexBase]);
}
pstFactors = &astFactorsGO[indexExp + 1];
UncompressBigInteger(pstFactors->ptrFactor, &tmpBase);
BigIntPowerIntExp(&tmpBase, ExponentsGOComputed[indexExp], &tmpBase);
BigIntRemainder(&tmpBase, &powSubGroupOrder, &tmpBase);
NumberLength = powSubGroupOrder.nbrLimbs;
CompressLimbsBigInteger(tmp2.limbs, &tmpBase);
modmult(tmp2.limbs, MontgomeryMultR2, tmp2.limbs);
if (NumberLength > 1 || TestNbr[0].x != 1)
{ // If TestNbr != 1 ...
ModInvBigNbr(tmp2.limbs, tmp2.limbs, TestNbr, NumberLength);
}
tmpBase.limbs[0].x = 1;
memset(&tmpBase.limbs[1], 0, (NumberLength - 1) * sizeof(limb));
modmult(tmpBase.limbs, tmp2.limbs, tmp2.limbs);
UncompressLimbsBigInteger(tmp2.limbs, &tmpBase);
BigIntMultiply(&tmpBase, &nbrV[indexBase], &nbrV[indexBase]);
}
BigIntRemainder(&nbrV[indexBase], &powSubGroupOrder, &nbrV[indexBase]);
BigIntMultiply(&nbrV[indexBase], &logarMult, &tmpBase);
BigIntAdd(&logar, &tmpBase, &logar);
BigIntMultiply(&logarMult, &powSubGroupOrder, &logarMult);
}
multiplicity = astFactorsMod[index].multiplicity;
UncompressBigInteger(ptrPrime, &bigNbrB);
expon = 1;
if (bigNbrB.nbrLimbs == 1 && bigNbrB.limbs[0].x == 2)
{ // Prime factor is 2. Base and power are odd at this moment.
int lsbBase = base.limbs[0].x;
int lsbPower = power.limbs[0].x;
if (multiplicity > 1)
{
int mask = (multiplicity == 2? 3 : 7);
expon = (multiplicity == 2 ? 2 : 3);
if ((lsbPower & mask) == 1)
{
intToBigInteger(&logar, 0);
intToBigInteger(&logarMult, (lsbBase == 1 ? 1 : 2));
}
else if (((lsbPower - lsbBase) & mask) == 0)
{
intToBigInteger(&logar, 1);
intToBigInteger(&logarMult, 2);
}
else
{
showText("There is no discrete logarithm");
DiscreteLogPeriod.sign = SIGN_NEGATIVE;
return;
}
}
}
for (; expon < multiplicity; expon++)
{ // Repeated factor.
// L = logar, LM = logarMult
// B = base, P = power, p = prime
// B^n = P (mod p^(k+1)) -> n = L + m*LM m = ?
// B^(L + m*LM) = P
// (B^LM) ^ m = P*B^(-L)
// B^LM = r*p^k + 1, P*B^(-L) = s*p^k + 1
// (r*p^k + 1)^m = s*p^k + 1
// From binomial theorem: m = s / r (mod p)
// If r = 0 and s != 0 there is no solution.
// If r = 0 and s = 0 do not change LM.
BigIntPowerIntExp(&bigNbrB, expon + 1, &bigNbrA);
NumberLength = bigNbrA.nbrLimbs;
memcpy(TestNbr, bigNbrA.limbs, NumberLength * sizeof(limb));
GetMontgomeryParms(NumberLength);
BigIntRemainder(&base, &bigNbrA, &tmpBase);
CompressLimbsBigInteger(baseMontg, &tmpBase);
modmult(baseMontg, MontgomeryMultR2, baseMontg);
modPow(baseMontg, logarMult.limbs, logarMult.nbrLimbs, primRootPwr); // B^LM
tmpBase.limbs[0].x = 1; // Convert from Montgomery to standard notation.
memset(&tmpBase.limbs[1], 0, (NumberLength - 1) * sizeof(limb));
modmult(primRootPwr, tmpBase.limbs, primRootPwr); // B^LM
ModInvBigNbr(baseMontg, tmpBase.limbs, TestNbr, NumberLength); // B^(-1)
modPow(tmpBase.limbs, logar.limbs, logar.nbrLimbs, primRoot); // B^(-L)
BigIntRemainder(&power, &bigNbrA, &tmpBase);
CompressLimbsBigInteger(tmp2.limbs, &tmpBase);
modmult(primRoot, tmp2.limbs, primRoot); // P*B^(-L)
BigIntDivide(&bigNbrA, &bigNbrB, &tmpBase);
UncompressLimbsBigInteger(primRootPwr, &tmp2);
BigIntDivide(&tmp2, &tmpBase, &bigNbrA); // s
UncompressLimbsBigInteger(primRoot, &baseModGO); // Use baseMontGO as temp var.
BigIntDivide(&baseModGO, &tmpBase, &tmp2); // r
if (bigNbrA.nbrLimbs == 1 && bigNbrA.limbs[0].x == 0)
{ // r equals zero.
if (tmp2.nbrLimbs != 1 || tmp2.limbs[0].x != 0)
{ // s does not equal zero.
showText("There is no discrete logarithm");
DiscreteLogPeriod.sign = SIGN_NEGATIVE;
return;
}
}
else
{ // r does not equal zero.
BigIntModularDivisionSaveTestNbr(&tmp2, &bigNbrA, &bigNbrB, &tmpBase); // m
BigIntMultiply(&tmpBase, &logarMult, &tmp2);
BigIntAdd(&logar, &tmp2, &logar);
BigIntMultiply(&logarMult, &bigNbrB, &logarMult);
}
}
// Based on logar and logarMult, compute DiscreteLog and DiscreteLogPeriod
// using the following formulas, that can be deduced from the Chinese
// Remainder Theorem:
// L = logar, LM = logarMult, DL = DiscreteLog, DLP = DiscreteLogPeriod.
// The modular implementation does not allow operating with even moduli.
//
// g <- gcd(LM, DLP)
// if (L%g != DL%g) there is no discrete logarithm, so go out.
// h <- LM / g
// if h is odd:
// t <- (L - DL) / DLP (mod h)
// t <- DLP * t + DL
// else
// i <- DLP / g
// t <- (DL - L) / LM (mod i)
// t <- LM * t + L
// endif
// DLP <- DLP * h
// DL <- t % DLP
BigIntGcd(&logarMult, &DiscreteLogPeriod, &tmpBase);
BigIntRemainder(&logar, &tmpBase, &bigNbrA);
BigIntRemainder(&DiscreteLog, &tmpBase, &bigNbrB);
if (!TestBigNbrEqual(&bigNbrA, &bigNbrB))
{
showText("There is no discrete logarithm");
DiscreteLogPeriod.sign = SIGN_NEGATIVE;
return;
}
BigIntDivide(&logarMult, &tmpBase, &tmp2);
if (tmp2.limbs[0].x & 1)
{ // h is odd.
BigIntSubt(&logar, &DiscreteLog, &tmpBase);
BigIntModularDivisionSaveTestNbr(&tmpBase, &DiscreteLogPeriod, &tmp2, &bigNbrA);
BigIntMultiply(&DiscreteLogPeriod, &bigNbrA, &tmpBase);
BigIntAdd(&tmpBase, &DiscreteLog, &tmpBase);
}
else
{ // h is even.
BigIntDivide(&DiscreteLogPeriod, &tmpBase, &bigNbrB);
BigIntSubt(&DiscreteLog, &logar, &tmpBase);
BigIntModularDivisionSaveTestNbr(&tmpBase, &logarMult, &bigNbrB, &bigNbrA);
BigIntMultiply(&logarMult, &bigNbrA, &tmpBase);
BigIntAdd(&tmpBase, &logar, &tmpBase);
}
BigIntMultiply(&DiscreteLogPeriod, &tmp2, &DiscreteLogPeriod);
BigIntRemainder(&tmpBase, &DiscreteLogPeriod, &DiscreteLog);
}
#if 0
textExp.setText(DiscreteLog.toString());
textPeriod.setText(DiscreteLogPeriod.toString());
long t = OldTimeElapsed / 1000;
labelStatus.setText("Time elapsed: " +
t / 86400 + "d " + (t % 86400) / 3600 + "h " + ((t % 3600) / 60) + "m " + (t % 60) +
"s mod mult: " + lModularMult);
#endif
}
BigIntRef BigIntAdd (BigIntRef bi, BigIntRef bi2) {
if (bi2 == bi) {
// create a copy of bi2, call, and release the copy
BigIntRef nbi2 = BigIntCreateCopy(bi2, 0);
BigIntRef answer = BigIntDivide(bi, nbi2);
BigIntRelease(nbi2);
return answer;
}
if (BigIntFlagIsSet(bi, kBigIntFlagNegative)) {
if (BigIntFlagIsSet(bi2, kBigIntFlagNegative)) {
// subtract a positive from a negative
bi2->flags ^= kBigIntFlagNegative;
BigIntRef answer = BigIntSubtract(bi, bi2);
bi2->flags |= kBigIntFlagNegative;
return answer;
} else {
// we need to subtract bi from bi2
bi->flags ^= kBigIntFlagNegative;
// run some DNA tests on that sucker
BigIntRef answer = BigIntSubtract(bi2, bi);
bi->flags |= kBigIntFlagNegative;
return answer;
}
} else if (BigIntFlagIsSet(bi2, kBigIntFlagNegative)) {
// subtract
bi2->flags ^= kBigIntFlagNegative;
BigIntRef answer = BigIntSubtract(bi, bi2);
bi2->flags |= kBigIntFlagNegative;
return answer;
}
BigIntRef newInt = (BigIntRef)malloc(sizeof(_BigInt));
// create a bitbuffer
newInt->bits = BitBufferNew((bi->binDigits + bi2->binDigits) / 8 + 1);
newInt->flags = 0;
newInt->b10string = NULL;
newInt->binDigits = 0;
newInt->referenceCount = 1;
BigBool carry = 0;
// maximum answer length
BigIntDWORD maxLength = bi->binDigits;
if (maxLength < bi2->binDigits) maxLength = bi2->binDigits;
for (int i = 0; i < maxLength + 1; i++) {
UInt8 bit1 = 0;
UInt8 bit2 = 0;
UInt8 newBit = 0;
//int i1 = (maxLength - bi2->binDigits) - (i + 1);
//int i2 = (maxLength - bi2->binDigits) - (i + 1);
int i1 = i;
int i2 = i;
if (i1 >= 0 && i1 < bi->bits->bitCount) bit1 = BitBufferGetBit(bi->bits, i1);
if (i2 >= 0 && i2 < bi2->bits->bitCount) bit2 = BitBufferGetBit(bi2->bits, i2);
// check for carry flag
if (bit1 && bit2) {
if (!carry) newBit = 0, carry = 1;
else carry = 1, newBit = 1;
} else if (bit1 || bit2) {
// if we are not carrying,
// make the next bit 1
if (!carry) newBit = 1;
} else {
if (carry) {
newBit = 1;
carry = 0;
}
}
if (newBit == 0 && i >= bi->bits->bitCount && i >= bi2->bits->bitCount) {
// we have to break
break;
} else {
BitBufferAddBit(newInt->bits, newBit);
}
if (newBit) {
newInt->binDigits = i + 1;
}
}
// precalculate the base10 string.
// this will be removed at a later date.
// if (bi->b10string && bi2->b10string)
// newInt->b10string = addDecString(bi->b10string, bi2->b10string);
return newInt;
}
