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factor.c
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#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <math.h>
#define FUNC_isqrt 1
#define FUNC_icbrt 1
#define FUNC_gcd_ui 1
#define FUNC_is_perfect_square 1
#define FUNC_clz 1
#include "ptypes.h"
#include "factor.h"
#include "sieve.h"
#include "util.h"
#include "mulmod.h"
#include "cache.h"
#include "primality.h"
#include "lucas_seq.h"
#include "montmath.h"
static int holf32(uint32_t n, UV *factors, uint32_t rounds);
/*
* You need to remember to use UV for unsigned and IV for signed types that
* are large enough to hold our data.
* If you use int, that's 32-bit on LP64 and LLP64 machines. You lose.
* If you use long, that's 32-bit on LLP64 machines. You lose.
* If you use long long, you may be too large which isn't so bad, but some
* compilers may not understand the type at all.
* perl.h already figured all this out, and provided us with these types which
* match the native integer type used inside our Perl, so just use those.
*/
static const unsigned short primes_small[] =
{0,2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,
101,103,107,109,113,127,131,137,139,149,151,157,163,167,173,179,181,191,
193,197,199,211,223,227,229,233,239,241,251,257,263,269,271,277,281,283,
293,307,311,313,317,331,337,347,349,353,359,367,373,379,383,389,397,401,
409,419,421,431,433,439,443,449,457,461,463,467,479,487,491,499,503,509,
521,523,541,547,557,563,569,571,577,587,593,599,601,607,613,617,619,631,
641,643,647,653,659,661,673,677,683,691,701,709,719,727,733,739,743,751,
757,761,769,773,787,797,809,811,821,823,827,829,839,853,857,859,863,877,
881,883,887,907,911,919,929,937,941,947,953,967,971,977,983,991,997,1009,
1013,1019,1021,1031,1033,1039,1049,1051,1061,1063,1069,1087,1091,1093,
1097,1103,1109,1117,1123,1129,1151,1153,1163,1171,1181,1187,1193,1201,
1213,1217,1223,1229,1231,1237,1249,1259,1277,1279,1283,1289,1291,1297,
1301,1303,1307,1319,1321,1327,1361,1367,1373,1381,1399,1409,1423,1427,
1429,1433,1439,1447,1451,1453,1459,1471,1481,1483,1487,1489,1493,1499,
1511,1523,1531,1543,1549,1553,1559,1567,1571,1579,1583,1597,1601,1607,
1609,1613,1619,1621,1627,1637,1657,1663,1667,1669,1693,1697,1699,1709,
1721,1723,1733,1741,1747,1753,1759,1777,1783,1787,1789,1801,1811,1823,
1831,1847,1861,1867,1871,1873,1877,1879,1889,1901,1907,1913,1931,1933,
1949,1951,1973,1979,1987,1993,1997,1999,2003,2011};
#define NPRIMES_SMALL (sizeof(primes_small)/sizeof(primes_small[0]))
static int _small_trial_factor(UV n, UV *factors, UV *newn, uint32_t *lastf)
{
int nfactors = 0;
uint32_t f = 7;
if (n > 1) {
while ( (n & 1) == 0 ) { factors[nfactors++] = 2; n /= 2; }
while ( (n % 3) == 0 ) { factors[nfactors++] = 3; n /= 3; }
while ( (n % 5) == 0 ) { factors[nfactors++] = 5; n /= 5; }
}
if (f*f <= n) {
uint32_t const lastsp = 83;
uint32_t sp = 4;
/* Trial division from 7 to 421. Use 32-bit if possible. */
if (n <= 4294967295U) {
uint32_t un = n;
while (sp < lastsp) {
while ( (un%f) == 0 ) {
factors[nfactors++] = f;
un /= f;
}
f = primes_small[++sp];
if (f*f > un) break;
}
n = un;
} else {
while (sp < lastsp) {
while ( (n%f) == 0 ) {
factors[nfactors++] = f;
n /= f;
}
f = primes_small[++sp];
if (f*f > n) break;
}
}
/* If n is small and still composite, finish it here */
if (n < 2011*2011 && f*f <= n) { /* Trial division from 431 to 2003 */
uint32_t un = n;
while (sp < NPRIMES_SMALL) {
while ( (un%f) == 0 ) {
factors[nfactors++] = f;
un /= f;
}
f = primes_small[++sp];
if (f*f > un) break;
}
n = un;
}
}
if (f*f > n && n != 1) {
factors[nfactors++] = n;
n = 1;
}
if (newn) *newn = n;
if (lastf) *lastf = f;
return nfactors;
}
static int _power_factor(UV n, UV *factors)
{
int nfactors, i, j, k = powerof(n);
if (k > 1) {
UV p = rootint(n, k);
nfactors = factor(p, factors);
for (i = nfactors; i >= 0; i--)
for (j = 0; j < k; j++)
factors[k*i+j] = factors[i];
return k*nfactors;
}
factors[0] = n;
return 1;
}
/* Find one factor of an input n. */
int factor_one(UV n, UV *factors, bool primality, bool trial)
{
int nfactors;
if (n < 4) {
factors[0] = n;
return (n == 1) ? 0 : 1;
}
/* TODO: deal with small n */
if (trial) {
uint32_t sp, f;
if (!(n&1)) { factors[0] = 2; factors[1] = n >> 1; return 2; }
if (!(n%3)) { factors[0] = 3; factors[1] = n / 3; return 2; }
if (!(n%5)) { factors[0] = 5; factors[1] = n / 5; return 2; }
for (sp = 4; (f = primes_small[sp]) < 2011; sp++) {
if ( (n % f) == 0 )
{ factors[0] = f; factors[1] = n/f; return 2; }
}
if (n < f*f) { factors[0] = n; return 1; }
}
if (primality && is_prime(n)) {
factors[0] = n;
return 1;
}
#if 0 /* Simple solution, just fine on x86_64 */
nfactors = (n < 1073741824UL) ? holf32(n, factors, 1000000)
: pbrent_factor(n, factors, 500000, 1);
if (nfactors < 2) croak("factor_one failed on %lu\n", n);
#endif
{
/* Adjust the number of rounds based on the number size and speed */
UV const nbits = BITS_PER_WORD - clz(n);
#if USE_MONTMATH
UV const br_rounds = 8000 + (9000 * ((nbits <= 45) ? 0 : (nbits-45)));
UV const sq_rounds = 200000;
#elif MULMODS_ARE_FAST
UV const br_rounds = 100 + ( 100 * ((nbits <= 45) ? 0 : (nbits-45)));
UV const sq_rounds = 100000;
#else
UV const br_rounds = (nbits >= 63) ? 120000 : (nbits >= 58) ? 500 : 0;
UV const sq_rounds = 200000;
#endif
#if BITS_PER_WORD == 64
/* For small semiprimes the fastest solution is HOLF under 32, then
* Lehman (no trial) under 38. However on random inputs, HOLF is
* best only under 28-30 bits, and adding Lehman is always slower. */
if (nbits <= 30) { /* This should always succeed */
nfactors = holf32(n, factors, 1000000);
if (nfactors > 1) return nfactors;
}
#endif
/* Almost all inputs are factored here */
if (br_rounds > 0) {
nfactors = pbrent_factor(n, factors, br_rounds, 1);
if (nfactors > 1) return nfactors;
}
#if USE_MONTMATH
nfactors = pbrent_factor(n, factors, 2*br_rounds, 3);
if (nfactors > 1) return nfactors;
#endif
/* Random 64-bit inputs at this point:
* About 3.1% are small enough that we did with HOLF.
* montmath: 96.89% pbrent, 0.01% pbrent2
* fast: 73.43% pbrent, 21.97% squfof, 1.09% p-1, 0.49% prho, long
* slow: 75.34% squfof, 19.47% pbrent, 0.20% p-1, 0.06% prho
*/
/* SQUFOF with these parameters gets 99.9% of everything left */
if (nbits <= 62) {
nfactors = squfof_factor(n, factors, sq_rounds);
if (nfactors > 1) return nfactors;
}
/* At this point we should only have 16+ digit semiprimes. */
nfactors = pminus1_factor(n, factors, 8000, 120000);
if (nfactors > 1) return nfactors;
/* Get the stragglers */
nfactors = cheb_factor(n, factors, 4000, 0);
if (nfactors > 1) return nfactors;
nfactors = prho_factor(n, factors, 120000);
if (nfactors > 1) return nfactors;
nfactors = pbrent_factor(n, factors, 500000, 5);
if (nfactors > 1) return nfactors;
nfactors = cheb_factor(n, factors, 1000000, 0);
if (nfactors > 1) return nfactors;
croak("factor_one failed on %lu\n", n);
}
return nfactors;
}
/******************************************************************************/
/* Main factor loop */
/* */
/* Puts factors in factors[] and returns the number found. */
/******************************************************************************/
int factor(UV n, UV *factors)
{
UV tofac_stack[MPU_MAX_FACTORS+1];
int nsmallfactors, npowerfactors, nfactors, i, j, ntofac = 0;
uint32_t f;
nfactors = _small_trial_factor(n, factors, &n, &f);
if (n == 1) return nfactors;
#if BITS_PER_WORD == 64
/* For small values less than f^3, use simple factor to split semiprime */
if (n < 100000000 && n < f*f*f) {
if (MR32(n)) factors[nfactors++] = n;
else nfactors += holf32(n, factors+nfactors, 10000);
return nfactors;
}
#endif
nsmallfactors = nfactors;
/* Perfect powers. Factor root only once. */
npowerfactors = _power_factor(n, factors+nsmallfactors);
if (npowerfactors > 1) return nsmallfactors + npowerfactors;
/* loop over each remaining factor, until ntofac == 0 */
do {
while ( (n >= f*f) && (!is_def_prime(n)) ) {
int split_success = factor_one(n, tofac_stack+ntofac, 0, 0) - 1;
if (split_success != 1 || tofac_stack[ntofac] == 1 || tofac_stack[ntofac] == n)
croak("internal: factor_one failed to factor %"UVuf"\n", n);
ntofac++; /* Leave one on the to-be-factored stack */
n = tofac_stack[ntofac]; /* Set n to the other one */
}
/* n is now prime (or 1), so add to already-factored stack */
if (n != 1) factors[nfactors++] = n;
/* Pop the next number off the to-factor stack */
if (ntofac > 0) n = tofac_stack[ntofac-1];
} while (ntofac-- > 0);
/* Sort the non-small factors */
for (i = nsmallfactors+1; i < nfactors; i++) {
UV fi = factors[i];
for (j = i; j > 0 && factors[j-1] > fi; j--)
factors[j] = factors[j-1];
factors[j] = fi;
}
return nfactors;
}
int factor_exp(UV n, UV *factors, UV* exponents)
{
int i = 1, j = 1, nfactors;
if (n == 1) return 0;
nfactors = factor(n, factors);
if (exponents == 0) {
for (; i < nfactors; i++)
if (factors[i] != factors[i-1])
factors[j++] = factors[i];
} else {
exponents[0] = 1;
for (; i < nfactors; i++) {
if (factors[i] != factors[i-1]) {
exponents[j] = 1;
factors[j++] = factors[i];
} else {
exponents[j-1]++;
}
}
}
return j;
}
int prime_bigomega(UV n)
{
UV factors[MPU_MAX_FACTORS+1];
return factor(n, factors);
}
int prime_omega(UV n)
{
UV factors[MPU_MAX_FACTORS+1];
int i, j, nfactors;
if (n <= 1) return (n==0);
nfactors = factor(n, factors);
for (i = 1, j = 1; i < nfactors; i++)
if (factors[i] != factors[i-1])
j++;
return j;
}
int trial_factor(UV n, UV *factors, UV f, UV last)
{
int sp, nfactors = 0;
if (f < 2) f = 2;
if (last == 0 || last*last > n) last = UV_MAX;
if (n < 4 || last < f) {
factors[0] = n;
return (n == 1) ? 0 : 1;
}
/* possibly do uint32_t specific code here */
if (f < primes_small[NPRIMES_SMALL-1]) {
while ( (n & 1) == 0 ) { factors[nfactors++] = 2; n >>= 1; }
if (3<=last) while ( (n % 3) == 0 ) { factors[nfactors++] = 3; n /= 3; }
if (5<=last) while ( (n % 5) == 0 ) { factors[nfactors++] = 5; n /= 5; }
for (sp = 4; sp < (int)NPRIMES_SMALL; sp++) {
f = primes_small[sp];
if (f*f > n || f > last) break;
while ( (n%f) == 0 ) {
factors[nfactors++] = f;
n /= f;
}
}
}
/* Trial division using a mod-30 wheel for larger values */
if (f*f <= n && f <= last) {
UV m, newlimit, limit = isqrt(n);
if (limit > last) limit = last;
m = f % 30;
while (f <= limit) {
if ( (n%f) == 0 ) {
do {
factors[nfactors++] = f;
n /= f;
} while ( (n%f) == 0 );
newlimit = isqrt(n);
if (newlimit < limit) limit = newlimit;
}
f += wheeladvance30[m];
m = nextwheel30[m];
}
}
/* All done! */
if (n != 1)
factors[nfactors++] = n;
return nfactors;
}
static UV _divisors_from_factors(UV nfactors, UV* fp, UV* fe, UV* res, UV k) {
UV s, t, count = 1;
res[0] = 1;
for (s = 0; s < nfactors; s++) {
UV i, j, scount = count, p = fp[s], e = fe[s], mult = 1;
for (j = 0; j < e; j++) {
mult *= p;
for (i = 0; i < scount; i++) {
t = res[i] * mult;
if (t <= k)
res[count++] = t;
}
}
}
return count;
}
UV* divisor_list(UV n, UV *num_divisors, UV maxd)
{
UV factors[MPU_MAX_FACTORS+1];
UV exponents[MPU_MAX_FACTORS+1];
UV* divs;
int i, nfactors, ndivisors;
if (n == 0 || maxd == 0) {
*num_divisors = 0;
return 0;
} else if (n == 1 || maxd == 1) {
New(0, divs, 1, UV);
divs[0] = 1;
*num_divisors = 1;
return divs;
}
if (maxd > n) maxd = n;
/* Factor and convert to factor/exponent pair */
nfactors = factor_exp(n, factors, exponents);
/* Calculate number of divisors, allocate space, fill with divisors */
ndivisors = exponents[0] + 1;
for (i = 1; i < nfactors; i++)
ndivisors *= (exponents[i] + 1);
New(0, divs, ndivisors, UV);
ndivisors = _divisors_from_factors(nfactors, factors, exponents, divs, maxd);
/* Sort divisors (numeric ascending) */
sort_uv_array(divs, ndivisors);
/* Return number of divisors and list */
*num_divisors = ndivisors;
return divs;
}
/* The usual method, on OEIS for instance, is:
* (p^(k*(e+1))-1) / (p^k-1)
* but that overflows quicky. Instead we rearrange as:
* 1 + p^k + p^k^2 + ... p^k^e
* Return 0 if the result overflowed.
*/
static const UV sigma_overflow[11] =
#if BITS_PER_WORD == 64
{UVCONST(3000000000000000000),UVCONST(3000000000),2487240,64260,7026,
1622, 566, 256, 139, 85, 57};
#else
{UVCONST(845404560), 52560, 1548, 252, 84, 41, 24, 16, 12, 10, 8};
#endif
UV divisor_sum(UV n, UV k)
{
UV factors[MPU_MAX_FACTORS+1];
int nfac, i, j;
UV product = 1;
if (k > 11 || (k > 0 && n >= sigma_overflow[k-1])) return 0;
/* divisors(0) = [] divisors(1) = [1] */
if (n <= 1) return n;
nfac = factor(n,factors);
if (k == 0) {
for (i = 0; i < nfac; i++) {
UV e = 1, f = factors[i];
while (i+1 < nfac && f == factors[i+1]) { e++; i++; }
product *= (e+1);
}
} else if (k == 1) {
for (i = 0; i < nfac; i++) {
UV f = factors[i];
UV pke = f, fmult = 1 + f;
while (i+1 < nfac && f == factors[i+1]) {
pke *= f;
fmult += pke;
i++;
}
product *= fmult;
}
} else {
for (i = 0; i < nfac; i++) {
UV f = factors[i];
UV fmult, pke, pk = f;
for (j = 1; j < (int)k; j++) pk *= f;
fmult = 1 + pk;
pke = pk;
while (i+1 < nfac && f == factors[i+1]) {
pke *= pk;
fmult += pke;
i++;
}
product *= fmult;
}
}
return product;
}
/******************************************************************************/
/******************************************************************************/
/******************************************************************************/
static int found_factor(UV n, UV f, UV* factors)
{
UV f2 = n/f;
int i = f > f2;
if (f == 1 || f2 == 1) {
factors[0] = n;
return 1;
}
factors[i] = f;
factors[1-i] = f2;
MPUassert( factors[0] * factors[1] == n , "incorrect factoring");
return 2;
}
static int no_factor(UV n, UV* factors)
{
factors[0] = n;
return 1;
}
/* Knuth volume 2, algorithm C.
* Can't compete with HOLF, SQUFOF, pbrent, etc.
*/
int fermat_factor(UV n, UV *factors, UV rounds)
{
IV sqn, x, y, r;
MPUassert( (n >= 3) && ((n%2) != 0) , "bad n in fermat_factor");
sqn = isqrt(n);
x = 2 * sqn + 1;
y = 1;
r = (sqn*sqn) - n;
while (r != 0) {
if (rounds-- == 0) return no_factor(n,factors);
r += x;
x += 2;
do {
r -= y;
y += 2;
} while (r > 0);
}
r = (x-y)/2;
return found_factor(n, r, factors);
}
/* Hart's One Line Factorization. */
int holf_factor(UV n, UV *factors, UV rounds)
{
UV i, s, m, f;
uint32_t root;
MPUassert( (n >= 3) && ((n%2) != 0) , "bad n in holf_factor");
/* We skip the perfect-square test for s in the loop, so we
* will never succeed if n is a perfect square. Test that now. */
if (is_perfect_square_ret(n,&root))
return found_factor(n, root, factors);
if (n <= (UV_MAX >> 6)) { /* Try with premultiplier first */
UV npre = n * ( (n <= (UV_MAX >> 13)) ? 720 :
(n <= (UV_MAX >> 11)) ? 480 :
(n <= (UV_MAX >> 10)) ? 360 :
(n <= (UV_MAX >> 8)) ? 60 : 30 );
UV ni = npre;
while (rounds--) {
s = 1 + (UV)isqrt(ni);
m = (s*s) - ni;
if (is_perfect_square_ret(m, &root)) {
f = gcd_ui(n, s - root);
if (f > 1 && f < n)
return found_factor(n, f, factors);
}
if (ni >= (ni+npre)) break;
ni += npre;
}
}
for (i = 1; i <= rounds; i++) {
s = (UV) sqrt( (double)n * (double)i );
/* Assume s^2 isn't a perfect square. We're rapidly losing precision
* so we won't be able to accurately detect it anyway. */
s++; /* s = ceil(sqrt(n*i)) */
m = sqrmod(s, n);
if (is_perfect_square_ret(m, &root)) {
f = gcd_ui( (s>root) ? s-root : root-s, n);
/* This should always succeed, but with overflow concerns.... */
return found_factor(n, f, factors);
}
}
return no_factor(n,factors);
}
static int holf32(uint32_t n, UV *factors, uint32_t rounds) {
UV npre, ni; /* These should be 64-bit */
uint32_t s, m, f;
if (n < 3) return no_factor(n,factors);
if (!(n&1)) { factors[0] = 2; factors[1] = n/2; return 2; }
if (is_perfect_square_ret(n,&f)) { factors[0] = factors[1] = f; return 2; }
ni = npre = (UV) n * ((BITS_PER_WORD == 64) ? 5040 : 1);
while (rounds--) {
s = 1 + isqrt(ni);
m = ((UV)s*(UV)s) - ni;
if (is_perfect_square_ret(m, &f)) {
f = gcd_ui(n, s - f);
if (f > 1 && f < n)
return found_factor(n, f, factors);
}
if (ni >= (ni+npre)) break; /* We've overflowed */
ni += npre;
}
return no_factor(n,factors);
}
#define ABSDIFF(x,y) (x>y) ? x-y : y-x
#if USE_MONTMATH
/* Pollard Rho with Brent's updates, using Montgomery reduction. */
int pbrent_factor(UV n, UV *factors, UV rounds, UV a)
{
UV const nbits = BITS_PER_WORD - clz(n);
const UV inner = (nbits <= 31) ? 32 : (nbits <= 35) ? 64 : (nbits <= 40) ? 160 : (nbits <= 52) ? 256 : 320;
UV f, m, r, rleft, Xi, Xm, Xs;
int irounds, fails = 6;
const uint64_t npi = mont_inverse(n), mont1 = mont_get1(n);
MPUassert( (n >= 3) && ((n%2) != 0) , "bad n in pbrent_factor");
r = f = 1;
Xi = Xm = Xs = mont1;
a = mont_geta(a,n);
while (rounds > 0) {
rleft = (r > rounds) ? rounds : r;
Xm = Xi;
/* Do rleft rounds, inner at a time */
while (rleft > 0) {
irounds = (rleft > (UV)inner) ? inner : rleft;
rleft -= irounds;
rounds -= irounds;
Xs = Xi;
if (n < (1ULL << 63)) {
Xi = mont_mulmod63(Xi,Xi+a,n);
m = ABSDIFF(Xi,Xm);
while (--irounds > 0) {
Xi = mont_mulmod63(Xi,Xi+a,n);
f = ABSDIFF(Xi,Xm);
m = mont_mulmod63(m, f, n);
}
} else if (a == mont1) {
Xi = mont_mulmod64(Xi,Xi+a,n);
m = ABSDIFF(Xi,Xm);
while (--irounds > 0) {
Xi = mont_mulmod64(Xi,Xi+a,n);
f = ABSDIFF(Xi,Xm);
m = mont_mulmod64(m, f, n);
}
} else {
Xi = addmod(mont_mulmod64(Xi,Xi,n), a, n);
m = ABSDIFF(Xi,Xm);
while (--irounds > 0) {
Xi = addmod(mont_mulmod64(Xi,Xi,n), a, n);
f = ABSDIFF(Xi,Xm);
m = mont_mulmod64(m, f, n);
}
}
f = gcd_ui(m, n);
if (f != 1)
break;
}
/* If f == 1, then we didn't find a factor. Move on. */
if (f == 1) {
r *= 2;
continue;
}
if (f == n) { /* back up, with safety */
Xi = Xs;
do {
if (n < (1ULL << 63) || a == mont1)
Xi = mont_mulmod(Xi,Xi+a,n);
else
Xi = addmod(mont_mulmod(Xi,Xi,n),a,n);
m = ABSDIFF(Xi,Xm);
f = gcd_ui(m, n);
} while (f == 1 && r-- != 0);
}
if (f == 0 || f == n) {
if (fails-- <= 0) break;
Xi = Xm = mont1;
a = addmod(a, mont_geta(11,n), n);
continue;
}
return found_factor(n, f, factors);
}
return no_factor(n,factors);
}
#else
/* Pollard Rho with Brent's updates. */
int pbrent_factor(UV n, UV *factors, UV rounds, UV a)
{
UV f, m, r, Xi, Xm;
const UV inner = (n <= 4000000000UL) ? 32 : 160;
int fails = 6;
MPUassert( (n >= 3) && ((n%2) != 0) , "bad n in pbrent_factor");
r = f = Xi = Xm = 1;
while (rounds > 0) {
UV rleft = (r > rounds) ? rounds : r;
UV saveXi = Xi;
/* Do rleft rounds, inner at a time */
while (rleft > 0) {
UV dorounds = (rleft > inner) ? inner : rleft;
saveXi = Xi;
rleft -= dorounds;
rounds -= dorounds;
Xi = sqraddmod(Xi, a, n); /* First iteration, no mulmod needed */
m = ABSDIFF(Xi,Xm);
while (--dorounds > 0) { /* Now do inner-1=63 more iterations */
Xi = sqraddmod(Xi, a, n);
f = ABSDIFF(Xi,Xm);
m = mulmod(m, f, n);
}
f = gcd_ui(m, n);
if (f != 1)
break;
}
/* If f == 1, then we didn't find a factor. Move on. */
if (f == 1) {
r *= 2;
Xm = Xi;
continue;
}
if (f == n) { /* back up, with safety */
Xi = saveXi;
do {
Xi = sqraddmod(Xi, a, n);
f = gcd_ui( ABSDIFF(Xi,Xm), n);
} while (f == 1 && r-- != 0);
}
if (f == 0 || f == n) {
if (fails-- <= 0) break;
Xm = addmod(Xm, 11, n);
Xi = Xm;
a++;
continue;
}
return found_factor(n, f, factors);
}
return no_factor(n,factors);
}
#endif
/* Pollard's Rho. */
int prho_factor(UV n, UV *factors, UV rounds)
{
UV a, f, i, m, oldU, oldV;
const UV inner = 64;
UV U = 7;
UV V = 7;
int fails = 3;
MPUassert( (n >= 3) && ((n%2) != 0) , "bad n in prho_factor");
/* We could just as well say a = 1 */
switch (n%8) {
case 1: a = 1; break;
case 3: a = 2; break;
case 5: a = 3; break;
case 7: a = 5; break;
default: a = 7; break;
}
rounds = (rounds + inner - 1) / inner;
while (rounds-- > 0) {
m = 1; oldU = U; oldV = V;
for (i = 0; i < inner; i++) {
U = sqraddmod(U, a, n);
V = sqraddmod(V, a, n);
V = sqraddmod(V, a, n);
f = (U > V) ? U-V : V-U;
m = mulmod(m, f, n);
}
f = gcd_ui(m, n);
if (f == 1)
continue;
if (f == n) { /* back up to find a factor*/
U = oldU; V = oldV;
i = inner;
do {
U = sqraddmod(U, a, n);
V = sqraddmod(V, a, n);
V = sqraddmod(V, a, n);
f = gcd_ui( (U > V) ? U-V : V-U, n);
} while (f == 1 && i-- != 0);
}
if (f == 0 || f == n) {
if (fails-- <= 0) break;
U = addmod(U,2,n);
V = U;
a++;
continue;
}
return found_factor(n, f, factors);
}
return no_factor(n,factors);
}
/* Pollard's P-1 */
int pminus1_factor(UV n, UV *factors, UV B1, UV B2)
{
UV f, k, kmin;
UV a = 2, q = 2;
UV savea = 2, saveq = 2;
UV j = 1;
UV sqrtB1 = isqrt(B1);
#if USE_MONTMATH
const uint64_t npi = mont_inverse(n), mont1 = mont_get1(n);
UV ma = mont_geta(a,n);
#define PMINUS1_APPLY_POWER ma = mont_powmod(ma, k, n)
#define PMINUS1_RECOVER_A a = mont_recover(ma,n)
#else
#define PMINUS1_APPLY_POWER a = powmod(a, k, n)
#define PMINUS1_RECOVER_A
#endif
MPUassert( (n >= 3) && ((n%2) != 0) , "bad n in pminus1_factor");
if (B1 <= primes_small[NPRIMES_SMALL-2]) {
UV i;
for (i = 1; primes_small[i] <= B1; i++) {
q = k = primes_small[i];
if (q <= sqrtB1) {
k = q*q; kmin = B1/q;
while (k <= kmin) k *= q;
}
PMINUS1_APPLY_POWER;
if ( (j++ % 32) == 0) {
PMINUS1_RECOVER_A;
if (a == 0 || gcd_ui(a-1, n) != 1)
break;
savea = a; saveq = q;
}
}
PMINUS1_RECOVER_A;
} else {
START_DO_FOR_EACH_PRIME(2, B1) {
q = k = p;
if (q <= sqrtB1) {
k = q*q; kmin = B1/q;
while (k <= kmin) k *= q;
}
PMINUS1_APPLY_POWER;
if ( (j++ % 32) == 0) {
PMINUS1_RECOVER_A;
if (a == 0 || gcd_ui(a-1, n) != 1)
break;
savea = a; saveq = q;
}
} END_DO_FOR_EACH_PRIME
PMINUS1_RECOVER_A;
}
if (a == 0) return no_factor(n,factors);
f = gcd_ui(a-1, n);
/* If we found more than one factor in stage 1, backup and single step */
if (f == n) {
a = savea;
START_DO_FOR_EACH_PRIME(saveq, B1) {
k = p; kmin = B1/p;
while (k <= kmin) k *= p;
a = powmod(a, k, n);
f = gcd_ui(a-1, n);
q = p;
if (f != 1)
break;
} END_DO_FOR_EACH_PRIME
/* If f == n again, we could do:
* for (savea = 3; f == n && savea < 100; savea = next_prime(savea)) {
* a = savea;
* for (q = 2; q <= B1; q = next_prime(q)) {
* ...
* }
* }
* but this could be a huge time sink if B1 is large, so just fail.
*/
}
/* STAGE 2 */
if (f == 1 && B2 > B1) {
UV bm = a;
UV b = 1;
UV bmdiff;
UV precomp_bm[111] = {0}; /* Enough for B2 = 189M */
/* calculate (a^q)^2, (a^q)^4, etc. */
bmdiff = sqrmod(bm, n);
precomp_bm[0] = bmdiff;
for (j = 1; j < 20; j++) {
bmdiff = mulmod(bmdiff,bm,n);
bmdiff = mulmod(bmdiff,bm,n);
precomp_bm[j] = bmdiff;
}
a = powmod(a, q, n);
j = 1;
START_DO_FOR_EACH_PRIME( q+1, B2 ) {
UV lastq = q;
UV qdiff;
q = p;
/* compute a^q = a^lastq * a^(q-lastq) */
qdiff = (q - lastq) / 2 - 1;
if (qdiff >= 111) {
bmdiff = powmod(bm, q-lastq, n); /* Big gap */
} else {
bmdiff = precomp_bm[qdiff];
if (bmdiff == 0) {
if (precomp_bm[qdiff-1] != 0)
bmdiff = mulmod(mulmod(precomp_bm[qdiff-1],bm,n),bm,n);
else
bmdiff = powmod(bm, q-lastq, n);
precomp_bm[qdiff] = bmdiff;
}
}
a = mulmod(a, bmdiff, n);
if (a == 0) break;
b = mulmod(b, a-1, n); /* if b == 0, we found multiple factors */
if ( (j++ % 64) == 0 ) {
f = gcd_ui(b, n);
if (f != 1)
break;
}
} END_DO_FOR_EACH_PRIME
f = gcd_ui(b, n);
}
return found_factor(n, f, factors);
}
/* Simple Williams p+1 */
static void pp1_pow(UV *cX, UV exp, UV n)
{
UV X0 = *cX;
UV X = *cX;
UV Y = mulsubmod(X, X, 2, n);
UV bit = UVCONST(1) << (clz(exp)-1);
while (bit) {
UV T = mulsubmod(X, Y, X0, n);
if ( exp & bit ) {
X = T;
Y = mulsubmod(Y, Y, 2, n);
} else {
Y = T;
X = mulsubmod(X, X, 2, n);
}
bit >>= 1;
}
*cX = X;
}
int pplus1_factor(UV n, UV *factors, UV B1)
{
UV X1, X2, f;
UV sqrtB1 = isqrt(B1);
MPUassert( (n >= 3) && ((n%2) != 0) , "bad n in pplus1_factor");
X1 = 7 % n;
X2 = 11 % n;
f = 1;
START_DO_FOR_EACH_PRIME(2, B1) {
UV k = p;
if (p < sqrtB1) {
UV kmin = B1/p;
while (k <= kmin)
k *= p;
}
pp1_pow(&X1, k, n);
if (X1 != 2) {
f = gcd_ui( submod(X1, 2, n) , n);
if (f != 1 && f != n) break;
}
pp1_pow(&X2, k, n);
if (X2 != 2) {
f = gcd_ui( submod(X2, 2, n) , n);
if (f != 1 && f != n) break;
}
} END_DO_FOR_EACH_PRIME
return found_factor(n, f, factors);
}
/* SQUFOF, based on Ben Buhrow's racing version. */
#if 1
/* limit to 62-bit inputs, use 32-bit types, faster */
#define SQUFOF_TYPE uint32_t
#define SQUFOF_MAX (UV_MAX >> 2)
#else
/* All 64-bit inputs possible, though we severely limit multipliers */
#define SQUFOF_TYPE UV
#define SQUFOF_MAX UV_MAX
#endif
typedef struct
{
int valid;
SQUFOF_TYPE P;
SQUFOF_TYPE bn;
SQUFOF_TYPE Qn;
SQUFOF_TYPE Q0;
SQUFOF_TYPE b0;
SQUFOF_TYPE it;
SQUFOF_TYPE imax;
SQUFOF_TYPE mult;
} mult_t;
/* N < 2^63 (or 2^31). Returns 0 or a factor */
static UV squfof_unit(UV n, mult_t* mult_save)