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ecm.c
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//
// This file is part of Alpertron Calculators.
//
// Copyright 2015-2021 Dario Alejandro Alpern
//
// Alpertron Calculators is free software: you can redistribute it and/or modify
// it under the terms of the GNU General Public License as published by
// the Free Software Foundation, either version 3 of the License, or
// (at your option) any later version.
//
// Alpertron Calculators is distributed in the hope that it will be useful,
// but WITHOUT ANY WARRANTY; without even the implied warranty of
// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
// GNU General Public License for more details.
//
// You should have received a copy of the GNU General Public License
// along with Alpertron Calculators. If not, see <http://www.gnu.org/licenses/>.
//
#include <string.h>
#include <stdint.h>
#include <math.h>
#include "bignbr.h"
#include "expression.h"
#include "factor.h"
#include "commonstruc.h"
#include "skiptest.h"
#if MAX_PRIME_SIEVE == 11
#define MAX_SIEVE_PRIME 5000
#else
#define MAX_SIEVE_PRIME (10 * SIEVE_SIZE)
#endif
#ifdef __EMSCRIPTEN__
extern int nbrPrimes;
extern int indexPrimes;
extern char* ptrLowerText;
extern char lowerText[MAX_LEN * 16];
#endif
extern int StepECM;
extern int maxIndexM;
extern int indexM;
static int SmallPrime[670]; /* Primes < 5000 */
static int boundStep1;
static int boundStep2;
static int sqrtBoundStep1;
static int NumberSizeBytes;
/* ECM limits for 30, 35, ..., 95 digits */
static int limits[] = { 10, 10, 10, 10, 10, 15, 22, 26, 35, 50, 100, 150, 250 };
static void add3(limb* x3, limb* z3, const limb* x2, const limb* z2,
const limb* x1, const limb* z1, const limb* x, const limb* z);
static void duplicate(limb* x2, limb* z2, const limb* x1, const limb* z1);
/******************************************************/
/* Start of code adapted from Paul Zimmermann's ECM4C */
/******************************************************/
#define ADD 6 /* number of multiplications in an addition */
#define DUP 5 /* number of multiplications in a duplicate */
/* returns the number of modular multiplications */
static int lucas_cost(int multiplier, double v)
{
int nbrMultiplications;
int e;
int d = multiplier;
double dr = ((double)d / v) + 0.5;
int r = (int)dr;
if (r >= multiplier)
{
return (ADD * multiplier);
}
d = multiplier - r;
e = (2 * r) - multiplier;
nbrMultiplications = DUP + ADD; /* initial duplicate and final addition */
while (d != e)
{
if (d < e)
{
r = d;
d = e;
e = r;
}
if (((4 * d) <= (5 * e)) && (((d + e) % 3) == 0))
{ /* condition 1 */
r = ((2 * d) - e) / 3;
e = ((2 * e) - d) / 3;
d = r;
nbrMultiplications += 3 * ADD; /* 3 additions */
}
else if (((4 * d) <= (5 * e)) && (((d - e) % 6) == 0))
{ /* condition 2 */
d = (d - e) / 2;
nbrMultiplications += ADD + DUP; /* one addition, one duplicate */
}
else if (d <= (4 * e))
{ /* condition 3 */
d -= e;
nbrMultiplications += ADD; /* one addition */
}
else if (((d + e) % 2) == 0)
{ /* condition 4 */
d = (d - e) / 2;
nbrMultiplications += ADD + DUP; /* one addition, one duplicate */
}
else if ((d % 2) == 0)
{ /* condition 5 */
d /= 2;
nbrMultiplications += ADD + DUP; /* one addition, one duplicate */
}
else if ((d % 3) == 0)
{ /* condition 6 */
d = (d / 3) - e;
nbrMultiplications += (3 * ADD) + DUP; /* three additions, one duplicate */
}
else if (((d + e) % 3) == 0)
{ /* condition 7 */
d = (d - (2 * e)) / 3;
nbrMultiplications += (3 * ADD) + DUP; /* three additions, one duplicate */
}
else if (((d - e) % 3) == 0)
{ /* condition 8 */
d = (d - e) / 3;
nbrMultiplications += (3 * ADD) + DUP; /* three additions, one duplicate */
}
else if ((e % 2) == 0)
{ /* condition 9 */
e /= 2;
nbrMultiplications += ADD + DUP; /* one addition, one duplicate */
}
else
{ /* no more conditions */
}
}
return nbrMultiplications;
}
/* computes nP from P=(x:z) and puts the result in (x:z). Assumes n>2. */
static void prac(int multiplier, limb* x, limb *z)
{
int d;
int e;
int r;
int i;
double dr;
limb* t;
limb* xA = x;
limb* zA = z;
limb* xB = common.ecm.Aux1;
limb* zB = common.ecm.Aux2;
limb* xC = common.ecm.Aux3;
limb* zC = common.ecm.Aux4;
limb* xT = common.ecm.W1;
limb* zT = common.ecm.W2;
limb* xT2 = common.ecm.W3;
limb* zT2 = common.ecm.W4;
const double v[] =
{
1.61803398875,
1.72360679775,
1.618347119656,
1.617914406529,
1.612429949509,
1.632839806089,
1.620181980807,
1.580178728295,
1.617214616534,
1.38196601125 };
/* chooses the best value of v */
r = lucas_cost(multiplier, v[0]);
i = 0;
for (d = 1; d < 10; d++)
{
e = lucas_cost(multiplier, v[d]);
if (e < r)
{
r = e;
i = d;
}
}
d = multiplier;
dr = ((double)d / v[i]) + 0.5;
r = (int)dr;
/* first iteration always begins by Condition 3, then a swap */
d = multiplier - r;
e = (2 * r) - multiplier;
(void)memcpy(xB, xA, NumberSizeBytes); // B <- A
(void)memcpy(zB, zA, NumberSizeBytes);
(void)memcpy(xC, xA, NumberSizeBytes); // C <- A
(void)memcpy(zC, zA, NumberSizeBytes);
duplicate(xA, zA, xA, zA); /* A=2*A */
while (d != e)
{
if (d < e)
{
r = d;
d = e;
e = r;
t = xA;
xA = xB;
xB = t;
t = zA;
zA = zB;
zB = t;
}
/* do the first line of Table 4 whose condition qualifies */
if (((4 * d) <= (5 * e)) && (((d + e) % 3) == 0))
{ /* condition 1 */
r = ((2 * d) - e) / 3;
e = ((2 * e) - d) / 3;
d = r;
add3(xT, zT, xA, zA, xB, zB, xC, zC); /* T = f(A,B,C) */
add3(xT2, zT2, xT, zT, xA, zA, xB, zB); /* T2 = f(T,A,B) */
add3(xB, zB, xB, zB, xT, zT, xA, zA); /* B = f(B,T,A) */
t = xA;
xA = xT2;
xT2 = t;
t = zA;
zA = zT2;
zT2 = t; /* swap A and T2 */
}
else if (((4 * d) <= (5 * e)) && (((d - e) % 6) == 0))
{ /* condition 2 */
d = (d - e) / 2;
add3(xB, zB, xA, zA, xB, zB, xC, zC); /* B = f(A,B,C) */
duplicate(xA, zA, xA, zA); /* A = 2*A */
}
else if (d <= (4 * e))
{ /* condition 3 */
d -= e;
add3(xT, zT, xB, zB, xA, zA, xC, zC); /* T = f(B,A,C) */
t = xB;
xB = xT;
xT = xC;
xC = t;
t = zB;
zB = zT;
zT = zC;
zC = t; /* circular permutation (B,T,C) */
}
else if (((d + e) % 2) == 0)
{ /* condition 4 */
d = (d - e) / 2;
add3(xB, zB, xB, zB, xA, zA, xC, zC); /* B = f(B,A,C) */
duplicate(xA, zA, xA, zA); /* A = 2*A */
}
else if ((d % 2) == 0)
{ /* condition 5 */
d /= 2;
add3(xC, zC, xC, zC, xA, zA, xB, zB); /* C = f(C,A,B) */
duplicate(xA, zA, xA, zA); /* A = 2*A */
}
else if ((d % 3) == 0)
{ /* condition 6 */
d = (d / 3) - e;
duplicate(xT, zT, xA, zA); /* T1 = 2*A */
add3(xT2, zT2, xA, zA, xB, zB, xC, zC); /* T2 = f(A,B,C) */
add3(xA, zA, xT, zT, xA, zA, xA, zA); /* A = f(T1,A,A) */
add3(xT, zT, xT, zT, xT2, zT2, xC, zC); /* T1 = f(T1,T2,C) */
t = xC;
xC = xB;
xB = xT;
xT = t;
t = zC;
zC = zB;
zB = zT;
zT = t; /* circular permutation (C,B,T) */
}
else if (((d + e) % 3) == 0)
{ /* condition 7 */
d = (d - (2 * e)) / 3;
add3(xT, zT, xA, zA, xB, zB, xC, zC); /* T1 = f(A,B,C) */
add3(xB, zB, xT, zT, xA, zA, xB, zB); /* B = f(T1,A,B) */
duplicate(xT, zT, xA, zA);
add3(xA, zA, xA, zA, xT, zT, xA, zA); /* A = 3*A */
}
else if (((d - e) % 3) == 0)
{ /* condition 8 */
d = (d - e) / 3;
add3(xT, zT, xA, zA, xB, zB, xC, zC); /* T1 = f(A,B,C) */
add3(xC, zC, xC, zC, xA, zA, xB, zB); /* C = f(A,C,B) */
t = xB;
xB = xT;
xT = t;
t = zB;
zB = zT;
zT = t; /* swap B and T */
duplicate(xT, zT, xA, zA);
add3(xA, zA, xA, zA, xT, zT, xA, zA); /* A = 3*A */
}
else if ((e % 2) == 0)
{ /* condition 9 */
e /= 2;
add3(xC, zC, xC, zC, xB, zB, xA, zA); /* C = f(C,B,A) */
duplicate(xB, zB, xB, zB); /* B = 2*B */
}
else
{
/* no more conditions */
}
}
add3(x, z, xA, zA, xB, zB, xC, zC);
}
/* adds Q=(x2:z2) and R=(x1:z1) and puts the result in (x3:z3),
using 5/6 mul, 6 add/sub and 6 mod. One assumes that Q-R=P or R-Q=P where P=(x:z).
Uses the following global variables:
- n : number to factor
- x, z : coordinates of P
- u, v, w : auxiliary variables
Modifies: x3, z3, u, v, w.
(x3,z3) may be identical to (x2,z2) and to (x,z)
*/
static void add3(limb* x3, limb* z3, const limb* x2, const limb* z2,
const limb* x1, const limb* z1, const limb* x, const limb* z)
{
limb* t = common.ecm.fieldTX;
limb* u = common.ecm.fieldTZ;
limb* v = common.ecm.fieldUX;
limb* w = common.ecm.fieldUZ;
SubtBigNbrModN(x2, z2, v, TestNbr, NumberLength); // v = x2-z2
AddBigNbrModN(x1, z1, w, TestNbr, NumberLength); // w = x1+z1
modmult(v, w, u); // u = (x2-z2)*(x1+z1)
AddBigNbrModN(x2, z2, w, TestNbr, NumberLength); // w = x2+z2
SubtBigNbrModN(x1, z1, t, TestNbr, NumberLength); // t = x1-z1
modmult(t, w, v); // v = (x2+z2)*(x1-z1)
AddBigNbrModN(u, v, t, TestNbr, NumberLength); // t = 2*(x1*x2-z1*z2)
modmult(t, t, w); // w = 4*(x1*x2-z1*z2)^2
SubtBigNbrModN(u, v, t, TestNbr, NumberLength); // t = 2*(x2*z1-x1*z2)
modmult(t, t, v); // v = 4*(x2*z1-x1*z2)^2
if (!memcmp(x, x3, NumberSizeBytes))
{
(void)memcpy(u, x, NumberSizeBytes);
(void)memcpy(t, w, NumberSizeBytes);
modmult(z, t, w);
modmult(v, u, z3);
(void)memcpy(x3, w, NumberSizeBytes);
}
else
{
modmult(w, z, x3); // x3 = 4*z*(x1*x2-z1*z2)^2
modmult(x, v, z3); // z3 = 4*x*(x2*z1-x1*z2)^2
}
}
/* computes 2P=(x2:z2) from P=(x1:z1), with 5 mul, 4 add/sub, 5 mod.
Uses the following global variables:
- n : number to factor
- b : (a+2)/4 mod n
- u, v, w : auxiliary variables
Modifies: x2, z2, u, v, w
*/
static void duplicate(limb* x2, limb* z2, const limb* x1, const limb* z1)
{
limb* u = common.ecm.fieldUZ;
limb* v = common.ecm.fieldTX;
limb* w = common.ecm.fieldTZ;
AddBigNbrModN(x1, z1, w, TestNbr, NumberLength); // w = x1+z1
modmult(w, w, u); // u = (x1+z1)^2
SubtBigNbrModN(x1, z1, w, TestNbr, NumberLength); // w = x1-z1
modmult(w, w, v); // v = (x1-z1)^2
modmult(u, v, x2); // x2 = u*v = (x1^2 - z1^2)^2
SubtBigNbrModN(u, v, w, TestNbr, NumberLength); // w = u-v = 4*x1*z1
modmult(common.ecm.fieldAA, w, u);
AddBigNbrModN(u, v, u, TestNbr, NumberLength); // u = (v+b*w)
modmult(w, u, z2); // z2 = (w*u)
}
/* End of code adapted from Paul Zimmermann's ECM4C */
static int gcdIsOne(const limb* value)
{
UncompressLimbsBigInteger(value, &common.ecm.Temp1);
UncompressLimbsBigInteger(TestNbr, &common.ecm.Temp2);
// Return zero if value is zero or both numbers are equal.
if (BigIntIsZero(&common.ecm.Temp1))
{
return 0;
}
BigIntSubt(&common.ecm.Temp1, &common.ecm.Temp2, &common.ecm.Temp3);
if (BigIntIsZero(&common.ecm.Temp3))
{
return 0;
}
BigIntGcd(&common.ecm.Temp1, &common.ecm.Temp2, &common.ecm.Temp3);
CompressLimbsBigInteger(common.ecm.GD, &common.ecm.Temp3);
if ((common.ecm.Temp3.nbrLimbs == 1) && (common.ecm.Temp3.limbs[0].x < 2))
{
return common.ecm.Temp3.limbs[0].x; // GCD is less than 2.
}
return 2; // GCD is greater than one.
}
static void GenerateSieve(int initial)
{
int i;
int j;
int Q;
int initModQ;
for (i = 0; i < (10 * SIEVE_SIZE); i += SIEVE_SIZE)
{
(void)memcpy(&common.ecm.sieve[i], common.ecm.sieve2310, SIEVE_SIZE);
}
#if MAX_PRIME_SIEVE == 11
j = 5;
Q = 13; /* Point to prime 13 */
#else
j = 4;
Q = 11; /* Point to prime 11 */
#endif
do
{
if (initial > (Q * Q))
{
initModQ = initial % Q;
if ((initModQ & 1) != 0)
{ // initModQ is odd
i = (Q - initModQ) / 2;
}
else if (initModQ == 0)
{
i = 0;
}
else
{ // initModQ is even
i = Q - (initModQ / 2);
}
for (; i < (10 * SIEVE_SIZE); i += Q)
{
common.ecm.sieve[i] = 1; /* Composite */
}
}
else
{
i = (Q * Q) - initial;
if (i < (20 * SIEVE_SIZE))
{
for (i = i / 2; i < (10 * SIEVE_SIZE); i += Q)
{
common.ecm.sieve[i] = 1; /* Composite */
}
}
else
{
break;
}
}
j++;
Q = SmallPrime[j];
} while (Q < MAX_SIEVE_PRIME);
}
/*******************************/
/* First step of ECM algorithm */
/*******************************/
static enum eEcmResult ecmStep1(void)
{
int I;
int P;
int i;
int u;
int retcode;
int bufSize = (NumberLength + 1) * (int)sizeof(limb);
(void)memcpy(common.ecm.Xaux, common.ecm.X, bufSize);
(void)memcpy(common.ecm.Zaux, common.ecm.Z, bufSize);
(void)memcpy(common.ecm.GcdAccumulated, MontgomeryMultR1, bufSize);
for (int pass = 0; pass < 2; pass++)
{
/* For powers of 2 */
#ifdef __EMSCRIPTEN__
indexPrimes = 0;
#endif
StepECM = 1;
for (I = 1; I <= boundStep1; I *= 2)
{
duplicate(common.ecm.X, common.ecm.Z, common.ecm.X, common.ecm.Z);
if (pass == 0)
{
modmult(common.ecm.GcdAccumulated, common.ecm.Z, common.ecm.GcdAccumulated);
}
else
{
retcode = gcdIsOne(common.ecm.Z);
if (retcode == 0)
{
return FACTOR_NOT_FOUND_GCD;
}
if (retcode > 1)
{
return FACTOR_FOUND;
}
}
}
for (I = 3; I <= boundStep1; I *= 3)
{
duplicate(common.ecm.W1, common.ecm.W2, common.ecm.X, common.ecm.Z);
add3(common.ecm.X, common.ecm.Z, common.ecm.X, common.ecm.Z, common.ecm.W1, common.ecm.W2, common.ecm.X, common.ecm.Z);
if (pass == 0)
{
modmult(common.ecm.GcdAccumulated, common.ecm.Z, common.ecm.GcdAccumulated);
}
else
{
retcode = gcdIsOne(common.ecm.Z);
if (retcode == 0)
{
return FACTOR_NOT_FOUND_GCD;
}
if (retcode > 1)
{
return FACTOR_FOUND;
}
}
}
/* for powers of odd primes */
indexM = 1;
do
{
#ifdef __EMSCRIPTEN__
indexPrimes++;
#endif
P = SmallPrime[indexM];
for (int64_t IP = P; IP <= boundStep1; IP *= P)
{
prac(P, common.ecm.X, common.ecm.Z);
}
indexM++;
if (pass == 0)
{
modmult(common.ecm.GcdAccumulated, common.ecm.Z, common.ecm.GcdAccumulated);
}
else
{
retcode = gcdIsOne(common.ecm.Z);
if (retcode == 0)
{
return FACTOR_NOT_FOUND_GCD;
}
if (retcode > 1)
{
return FACTOR_FOUND;
}
}
} while (SmallPrime[indexM - 1] <= sqrtBoundStep1);
P += 2;
/* Initialize sieve2310[n]: 1 if gcd(P+2n,2310) > 1, 0 otherwise */
u = P;
for (i = 0; i < SIEVE_SIZE; i++)
{
common.ecm.sieve2310[i] =
((((u % 3) == 0)
|| ((u % 5) == 0)
|| ((u % 7) == 0)
#if MAX_PRIME_SIEVE == 11
|| ((u % 11) == 0)
#endif
) ? (unsigned char)1 : (unsigned char)0);
u += 2;
}
do
{
/* Generate sieve */
GenerateSieve(P);
/* Walk through sieve */
for (i = 0; i < (10 * SIEVE_SIZE); i++)
{
if (common.ecm.sieve[i] != 0U)
{
continue; /* Do not process composites */
}
if ((P + (2 * i)) > boundStep1)
{
break;
}
#ifdef __EMSCRIPTEN__
indexPrimes++;
#endif
prac(P + (2 * i), common.ecm.X, common.ecm.Z);
if (pass == 0)
{
modmult(common.ecm.GcdAccumulated, common.ecm.Z, common.ecm.GcdAccumulated);
}
else
{
retcode = gcdIsOne(common.ecm.Z);
if (retcode == 0)
{
return FACTOR_NOT_FOUND_GCD;
}
if (retcode > 1)
{
return FACTOR_FOUND;
}
}
}
P += 20 * SIEVE_SIZE;
} while (P < boundStep1);
if (pass == 0)
{
int result = gcdIsOne(common.ecm.GcdAccumulated);
if (result == 1)
{
break; // GCD is 1 so factor is not found.
}
if (result == 2)
{ // GCD greater than zero. Factor found.
return FACTOR_FOUND;
}
// Factor could not be found because GCD is zero.
// Repeat this curve but performing GCD on every step.
(void)memcpy(common.ecm.X, common.ecm.Xaux, bufSize);
(void)memcpy(common.ecm.Z, common.ecm.Zaux, bufSize);
}
} /* end for Pass */
return FACTOR_NOT_FOUND;
}
/******************************************************/
/* Second step (using improved standard continuation) */
/******************************************************/
static enum eEcmResult ecmStep2(void)
{
int j;
StepECM = 2;
j = 0;
for (int u = 1; u < SIEVE_SIZE; u += 2)
{
if (((u % 3) == 0) || ((u % 5) == 0) || ((u % 7) == 0)
#if MAX_PRIME_SIEVE == 11
|| ((u % 11) == 0)
#endif
)
{
common.ecm.sieve2310[u / 2] = 1U;
}
else
{
common.ecm.sieveidx[j] = u / 2;
common.ecm.sieve2310[common.ecm.sieveidx[j]] = 0U;
j++;
}
}
(void)memcpy(&common.ecm.sieve2310[HALF_SIEVE_SIZE], &common.ecm.sieve2310[0], HALF_SIEVE_SIZE);
for (int pass = 0; pass < 2; pass++)
{
int Qaux;
int J;
(void)memcpy(common.ecm.Xaux, common.ecm.X, NumberSizeBytes); // (X:Z) -> Q (output
(void)memcpy(common.ecm.Zaux, common.ecm.Z, NumberSizeBytes); // from step 1)
(void)memcpy(common.ecm.GcdAccumulated, MontgomeryMultR1, NumberSizeBytes);
(void)memcpy(common.ecm.UX, common.ecm.X, NumberSizeBytes);
(void)memcpy(common.ecm.UZ, common.ecm.Z, NumberSizeBytes); // (UX:UZ) -> Q
(void)ModInvBigNbr(common.ecm.Z, common.ecm.Aux1, TestNbr, NumberLength);
modmult(common.ecm.Aux1, common.ecm.X, common.ecm.root[0]); // root[0] <- X/Z (Q)
J = 0;
AddBigNbrModN(common.ecm.X, common.ecm.Z, common.ecm.Aux1, TestNbr, NumberLength);
modmult(common.ecm.Aux1, common.ecm.Aux1, common.ecm.W1);
SubtBigNbrModN(common.ecm.X, common.ecm.Z, common.ecm.Aux1, TestNbr, NumberLength);
modmult(common.ecm.Aux1, common.ecm.Aux1, common.ecm.W2);
modmult(common.ecm.W1, common.ecm.W2, common.ecm.TX);
SubtBigNbrModN(common.ecm.W1, common.ecm.W2, common.ecm.Aux1, TestNbr, NumberLength);
modmult(common.ecm.Aux1, common.ecm.AA, common.ecm.Aux2);
AddBigNbrModN(common.ecm.Aux2, common.ecm.W2, common.ecm.Aux3, TestNbr, NumberLength);
modmult(common.ecm.Aux1, common.ecm.Aux3, common.ecm.TZ); // (TX:TZ) -> 2Q
SubtBigNbrModN(common.ecm.X, common.ecm.Z, common.ecm.Aux1, TestNbr, NumberLength);
AddBigNbrModN(common.ecm.TX, common.ecm.TZ, common.ecm.Aux2, TestNbr, NumberLength);
modmult(common.ecm.Aux1, common.ecm.Aux2, common.ecm.W1);
AddBigNbrModN(common.ecm.X, common.ecm.Z, common.ecm.Aux1, TestNbr, NumberLength);
SubtBigNbrModN(common.ecm.TX, common.ecm.TZ, common.ecm.Aux2, TestNbr, NumberLength);
modmult(common.ecm.Aux1, common.ecm.Aux2, common.ecm.W2);
AddBigNbrModN(common.ecm.W1, common.ecm.W2, common.ecm.Aux1, TestNbr, NumberLength);
modmult(common.ecm.Aux1, common.ecm.Aux1, common.ecm.Aux2);
modmult(common.ecm.Aux2, common.ecm.UZ, common.ecm.X);
SubtBigNbrModN(common.ecm.W1, common.ecm.W2, common.ecm.Aux1, TestNbr, NumberLength);
modmult(common.ecm.Aux1, common.ecm.Aux1, common.ecm.Aux2);
modmult(common.ecm.Aux2, common.ecm.UX, common.ecm.Z); // (X:Z) -> 3Q
for (int I = 5; I < SIEVE_SIZE; I += 2)
{
(void)memcpy(common.ecm.WX, common.ecm.X, NumberSizeBytes);
(void)memcpy(common.ecm.WZ, common.ecm.Z, NumberSizeBytes);
SubtBigNbrModN(common.ecm.X, common.ecm.Z, common.ecm.Aux1, TestNbr, NumberLength);
AddBigNbrModN(common.ecm.TX, common.ecm.TZ, common.ecm.Aux2, TestNbr, NumberLength);
modmult(common.ecm.Aux1, common.ecm.Aux2, common.ecm.W1);
AddBigNbrModN(common.ecm.X, common.ecm.Z, common.ecm.Aux1, TestNbr, NumberLength);
SubtBigNbrModN(common.ecm.TX, common.ecm.TZ, common.ecm.Aux2, TestNbr, NumberLength);
modmult(common.ecm.Aux1, common.ecm.Aux2, common.ecm.W2);
AddBigNbrModN(common.ecm.W1, common.ecm.W2, common.ecm.Aux1, TestNbr, NumberLength);
modmult(common.ecm.Aux1, common.ecm.Aux1, common.ecm.Aux2);
modmult(common.ecm.Aux2, common.ecm.UZ, common.ecm.X);
SubtBigNbrModN(common.ecm.W1, common.ecm.W2, common.ecm.Aux1, TestNbr, NumberLength);
modmult(common.ecm.Aux1, common.ecm.Aux1, common.ecm.Aux2);
modmult(common.ecm.Aux2, common.ecm.UX, common.ecm.Z); // (X:Z) -> 5Q, 7Q, ...
if (pass == 0)
{
modmult(common.ecm.GcdAccumulated, common.ecm.Aux1, common.ecm.Aux2);
(void)memcpy(common.ecm.GcdAccumulated, common.ecm.Aux2, NumberSizeBytes);
}
else
{
if (gcdIsOne(common.ecm.Aux1) > 1)
{
return FACTOR_FOUND;
}
}
if (I == HALF_SIEVE_SIZE)
{
(void)memcpy(common.ecm.DX, common.ecm.X, NumberSizeBytes);
(void)memcpy(common.ecm.DZ, common.ecm.Z, NumberSizeBytes); // (DX:DZ) -> HALF_SIEVE_SIZE*Q
}
if (((I % 3) != 0) && ((I % 5) != 0) && ((I % 7) != 0)
#if MAX_PRIME_SIEVE == 11
&& ((I % 11) != 0)
#endif
)
{
J++;
(void)ModInvBigNbr(common.ecm.Z, common.ecm.Aux1, TestNbr, NumberLength);
modmult(common.ecm.Aux1, common.ecm.X, common.ecm.root[J]); // root[J] <- X/Z
}
(void)memcpy(common.ecm.UX, common.ecm.WX, NumberSizeBytes); // (UX:UZ) <-
(void)memcpy(common.ecm.UZ, common.ecm.WZ, NumberSizeBytes); // Previous (X:Z)
} /* end for I */
AddBigNbrModN(common.ecm.DX, common.ecm.DZ, common.ecm.Aux1, TestNbr, NumberLength);
modmult(common.ecm.Aux1, common.ecm.Aux1, common.ecm.W1);
SubtBigNbrModN(common.ecm.DX, common.ecm.DZ, common.ecm.Aux1, TestNbr, NumberLength);
modmult(common.ecm.Aux1, common.ecm.Aux1, common.ecm.W2);
modmult(common.ecm.W1, common.ecm.W2, common.ecm.X);
SubtBigNbrModN(common.ecm.W1, common.ecm.W2, common.ecm.Aux1, TestNbr, NumberLength);
modmult(common.ecm.Aux1, common.ecm.AA, common.ecm.Aux2);
AddBigNbrModN(common.ecm.Aux2, common.ecm.W2, common.ecm.Aux3, TestNbr, NumberLength);
modmult(common.ecm.Aux1, common.ecm.Aux3, common.ecm.Z);
(void)memcpy(common.ecm.UX, common.ecm.X, NumberSizeBytes);
(void)memcpy(common.ecm.UZ, common.ecm.Z, NumberSizeBytes); // (UX:UZ) -> SIEVE_SIZE*Q
AddBigNbrModN(common.ecm.X, common.ecm.Z, common.ecm.Aux1, TestNbr, NumberLength);
modmult(common.ecm.Aux1, common.ecm.Aux1, common.ecm.W1);
SubtBigNbrModN(common.ecm.X, common.ecm.Z, common.ecm.Aux1, TestNbr, NumberLength);
modmult(common.ecm.Aux1, common.ecm.Aux1, common.ecm.W2);
modmult(common.ecm.W1, common.ecm.W2, common.ecm.TX);
SubtBigNbrModN(common.ecm.W1, common.ecm.W2, common.ecm.Aux1, TestNbr, NumberLength);
modmult(common.ecm.Aux1, common.ecm.AA, common.ecm.Aux2);
AddBigNbrModN(common.ecm.Aux2, common.ecm.W2, common.ecm.Aux3, TestNbr, NumberLength);
modmult(common.ecm.Aux1, common.ecm.Aux3, common.ecm.TZ); // (TX:TZ) -> 2*SIEVE_SIZE*Q
SubtBigNbrModN(common.ecm.X, common.ecm.Z, common.ecm.Aux1, TestNbr, NumberLength);
AddBigNbrModN(common.ecm.TX, common.ecm.TZ, common.ecm.Aux2, TestNbr, NumberLength);
modmult(common.ecm.Aux1, common.ecm.Aux2, common.ecm.W1);
AddBigNbrModN(common.ecm.X, common.ecm.Z, common.ecm.Aux1, TestNbr, NumberLength);
SubtBigNbrModN(common.ecm.TX, common.ecm.TZ, common.ecm.Aux2, TestNbr, NumberLength);
modmult(common.ecm.Aux1, common.ecm.Aux2, common.ecm.W2);
AddBigNbrModN(common.ecm.W1, common.ecm.W2, common.ecm.Aux1, TestNbr, NumberLength);
modmult(common.ecm.Aux1, common.ecm.Aux1, common.ecm.Aux2);
modmult(common.ecm.Aux2, common.ecm.UZ, common.ecm.X);
SubtBigNbrModN(common.ecm.W1, common.ecm.W2, common.ecm.Aux1, TestNbr, NumberLength);
modmult(common.ecm.Aux1, common.ecm.Aux1, common.ecm.Aux2);
modmult(common.ecm.Aux2, common.ecm.UX, common.ecm.Z); // (X:Z) -> 3*SIEVE_SIZE*Q
Qaux = boundStep1 / (2 * SIEVE_SIZE);
maxIndexM = boundStep2 / (2 * SIEVE_SIZE);
for (indexM = 0; indexM <= maxIndexM; indexM++)
{
if (indexM >= Qaux)
{ // If inside step 2 range...
if (indexM == 0)
{
bool rc = ModInvBigNbr(common.ecm.UZ, common.ecm.Aux3, TestNbr, NumberLength);
if (rc == false)
{
(void)memcpy(common.ecm.GD, common.ecm.UZ, NumberSizeBytes);
return FACTOR_FOUND;
}
modmult(common.ecm.UX, common.ecm.Aux3, common.ecm.Aux1); // Aux1 <- X/Z (SIEVE_SIZE*Q)
}
else
{
bool rc = ModInvBigNbr(common.ecm.Z, common.ecm.Aux3, TestNbr, NumberLength);
if (rc == false)
{
(void)memcpy(common.ecm.GD, common.ecm.Z, NumberSizeBytes);
return FACTOR_FOUND;
}
// Compute Aux as X/Z for 3, 5, SIEVE_SIZE * Q)
modmult(common.ecm.X, common.ecm.Aux3, common.ecm.Aux1);
}
/* Generate sieve */
if (((indexM % 10) == 0) || (indexM == Qaux))
{
GenerateSieve(indexM / 10 * (20 * SIEVE_SIZE) + 1);
}
/* Walk through sieve */
J = HALF_SIEVE_SIZE + (indexM % 10) * SIEVE_SIZE;
for (int i = 0; i < GROUP_SIZE; i++)
{
j = common.ecm.sieveidx[i]; // 0 < J < HALF_SIEVE_SIZE
if ((common.ecm.sieve[J + j] != 0U) && (common.ecm.sieve[J - 1 - j] != 0U))
{
continue; // Do not process if both are composite numbers.
}
SubtBigNbrModN(common.ecm.Aux1, common.ecm.root[i], common.ecm.M, TestNbr, NumberLength);
modmult(common.ecm.GcdAccumulated, common.ecm.M, common.ecm.Aux2);
(void)memcpy(common.ecm.GcdAccumulated, common.ecm.Aux2, NumberSizeBytes);
}
if (pass != 0)
{
if (BigNbrIsZero(common.ecm.GcdAccumulated))
{ // This curve cannot factor the number.
return FACTOR_NOT_FOUND_GCD;
}
if (gcdIsOne(common.ecm.GcdAccumulated) > 1)
{
return FACTOR_FOUND;
}
}
} // End for.
if (indexM != 0)
{ // Update (X:Z)
(void)memcpy(common.ecm.WX, common.ecm.X, NumberSizeBytes);
(void)memcpy(common.ecm.WZ, common.ecm.Z, NumberSizeBytes);
SubtBigNbrModN(common.ecm.X, common.ecm.Z, common.ecm.Aux1, TestNbr, NumberLength);
AddBigNbrModN(common.ecm.TX, common.ecm.TZ, common.ecm.Aux2, TestNbr, NumberLength);
modmult(common.ecm.Aux1, common.ecm.Aux2, common.ecm.W1);
AddBigNbrModN(common.ecm.X, common.ecm.Z, common.ecm.Aux1, TestNbr, NumberLength);
SubtBigNbrModN(common.ecm.TX, common.ecm.TZ, common.ecm.Aux2, TestNbr, NumberLength);
modmult(common.ecm.Aux1, common.ecm.Aux2, common.ecm.W2);
AddBigNbrModN(common.ecm.W1, common.ecm.W2, common.ecm.Aux1, TestNbr, NumberLength);
modmult(common.ecm.Aux1, common.ecm.Aux1, common.ecm.Aux2);
modmult(common.ecm.Aux2, common.ecm.UZ, common.ecm.X);
SubtBigNbrModN(common.ecm.W1, common.ecm.W2, common.ecm.Aux1, TestNbr, NumberLength);
modmult(common.ecm.Aux1, common.ecm.Aux1, common.ecm.Aux2);
modmult(common.ecm.Aux2, common.ecm.UX, common.ecm.Z);
(void)memcpy(common.ecm.UX, common.ecm.WX, NumberSizeBytes);
(void)memcpy(common.ecm.UZ, common.ecm.WZ, NumberSizeBytes);
}
} // end for Q
if (pass == 0)
{
int rc;
if (BigNbrIsZero(common.ecm.GcdAccumulated))
{ // If GcdAccumulated is zero
(void)memcpy(common.ecm.X, common.ecm.Xaux, NumberSizeBytes);
(void)memcpy(common.ecm.Z, common.ecm.Zaux, NumberSizeBytes);
continue; // multiple of TestNbr, continue.
}
rc = gcdIsOne(common.ecm.GcdAccumulated);
if (rc == 1)
{
break; // GCD is one, so this curve does not find a factor.
}
if (rc == 0)
{
continue;
}
// Compute GD as GCD(GcdAccumulated, TestNbr)
if (memcmp(common.ecm.GD, TestNbr, NumberSizeBytes) != 0)
{ // GCD is not 1 or TestNbr
return FACTOR_FOUND;
}
}
} /* end for Pass */
return FACTOR_NOT_FOUND;
}
static void initSmallPrimeArray(void)
{
int potentialPrime = 3;
int numPrimes = (int)sizeof(SmallPrime) / (int)sizeof(SmallPrime[0]);
SmallPrime[0] = 2;
for (indexM = 1; indexM < numPrimes; indexM++)
{ // Loop that fills the SmallPrime array.
int divisor;
SmallPrime[indexM] = potentialPrime; /* Store prime */
do
{
potentialPrime += 2;
for (divisor = 3; (divisor * divisor) <= potentialPrime; divisor += 2)
{ /* Check if potentialPrime is prime */
if ((potentialPrime % divisor) == 0)
{
break; /* Composite */
}
}
} while ((divisor * divisor) <= potentialPrime);
}
}
enum eEcmResult ecmCurve(int *pEC, int *pNextEC)
{
int EC = *pEC;
int NextEC = *pNextEC;
enum eEcmResult result;
NumberSizeBytes = NumberLength * (int)sizeof(limb);
#ifdef __EMSCRIPTEN__
char text[20];
#endif
if (SmallPrime[0] != 2)
{ // Not initialized yet.
initSmallPrimeArray();
}
EC %= 50000000; // Convert to curve number.
do
{
#ifdef __EMSCRIPTEN__
char* ptrText;
#endif
int nbrDigits;
if (NextEC > 0)
{
EC = NextEC;
NextEC = -1;
if (EC >= TYP_SIQS)
{
int bufSize = (NumberLength - 1) * (int)sizeof(limb);
common.ecm.GD[0].x = 1; // Set GD to 1.
(void)memset(&common.ecm.GD[1], 0, bufSize);
*pEC = EC;
*pNextEC = NextEC;
return FACTOR_FOUND;
}
}
else
{
EC++;
#ifdef __EMSCRIPTEN__
text[0] = '7';
ptrText = &text[1];
int2dec(&ptrText, EC);
*ptrText = 0; // Add string terminator.
databack(text);
#endif
nbrDigits = NumberLength * 9; // Get number of digits.
if ((NextEC == 0) && (nbrDigits >= 30) && (nbrDigits <= 110))
{ // Force switch to SIQS and number not too large.
EC += TYP_SIQS;
*pEC = EC;
*pNextEC = NextEC;
(void)memcpy(common.ecm.GD, TestNbr, NumberSizeBytes);
return CHANGE_TO_SIQS;
}
if ((nbrDigits > 30) && (nbrDigits <= 90)) // If between 30 and 90 digits...
{ // Switch to SIQS.
int limit = limits[(nbrDigits - 31) / 5];
if ((EC % 50000000) >= limit)
{ // Switch to SIQS.
EC += TYP_SIQS;
*pEC = EC;
*pNextEC = NextEC;
(void)memcpy(common.ecm.GD, TestNbr, NumberSizeBytes);
return CHANGE_TO_SIQS;
}
}
}
if (EC < 26)
{
boundStep1 = 2000;
boundStep2 = 200000;
sqrtBoundStep1 = 45;
#ifdef __EMSCRIPTEN__
nbrPrimes = 303; /* Number of primes less than 2000 */
#endif
}
else if (EC < 326)
{
boundStep1 = 50000;
boundStep2 = 5000000;
sqrtBoundStep1 = 224;
#ifdef __EMSCRIPTEN__
nbrPrimes = 5133; /* Number of primes less than 50000 */
#endif
}
else if (EC < 2000)
{
boundStep1 = 1000000;
boundStep2 = 100000000;
sqrtBoundStep1 = 1001;
#ifdef __EMSCRIPTEN__
nbrPrimes = 78498; /* Number of primes less than 1000000 */
#endif
}
else
{
boundStep1 = 11000000;
boundStep2 = 1100000000;
sqrtBoundStep1 = 3316;
#ifdef __EMSCRIPTEN__
nbrPrimes = 726517; /* Number of primes less than 11000000 */
#endif
}
#ifdef __EMSCRIPTEN__
ptrText = ptrLowerText; // Point after number that is being factored.
copyStr(&ptrText, lang ? "<p>Curva " : "<p>Curve ");
int2dec(&ptrText, EC); // Show curve number.
copyStr(&ptrText, lang ? " usando límites B1=" : " using bounds B1=");
int2dec(&ptrText, boundStep1); // Show first bound.
copyStr(&ptrText, lang ? " y B2=" : " and B2=");
int2dec(&ptrText, boundStep2); // Show second bound.
copyStr(&ptrText, "</p>");