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factor.c
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factor.c
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/* factor -- print prime factors of n.
Copyright (C) 1986-2014 Free Software Foundation, Inc.
This program 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.
This program 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 this program. If not, see <http://www.gnu.org/licenses/>. */
/* Originally written by Paul Rubin <[email protected]>.
Adapted for GNU, fixed to factor UINT_MAX by Jim Meyering.
Arbitrary-precision code adapted by James Youngman from Torbjörn
Granlund's factorize.c, from GNU MP version 4.2.2.
In 2012, the core was rewritten by Torbjörn Granlund and Niels Möller.
Contains code from GNU MP. */
//#include <config.h>
#include <getopt.h>
#include <stdio.h>
#include <sys/time.h>
#include <assert.h>
#include <limits.h>
#include <error.h>
typedef unsigned uintmax_t;
typedef int intmax_t;
typedef int bool;
#define false 0
#define true 1
#define UINTMAX_MAX (UINT_MAX)
# define W_TYPE_SIZE (8 * sizeof (uintmax_t))
# define __ll_B ((uintmax_t) 1 << (W_TYPE_SIZE / 2))
# define __ll_lowpart(t) ((uintmax_t) (t) & (__ll_B - 1))
# define __ll_highpart(t) ((uintmax_t) (t) >> (W_TYPE_SIZE / 2))
#if !defined __clz_tab && !defined UHWtype
/* Without this seemingly useless conditional, gcc -Wunused-macros
warns that each of the two tested macros is unused on Fedora 18.
FIXME: this is just an ugly band-aid. Fix it properly. */
#endif
enum alg_type { ALG_POLLARD_RHO = 1, ALG_SQUFOF = 2 };
static enum alg_type alg;
/* 2*3*5*7*11...*101 is 128 bits, and has 26 prime factors */
#define MAX_NFACTS 26
struct factors
{
uintmax_t plarge[2]; /* Can have a single large factor */
uintmax_t p[MAX_NFACTS];
unsigned char e[MAX_NFACTS];
unsigned char nfactors;
};
static void factor (uintmax_t, struct factors *);
#ifndef umul_ppmm
# define umul_ppmm(w1, w0, u, v) \
do { \
uintmax_t __x0, __x1, __x2, __x3; \
unsigned long int __ul, __vl, __uh, __vh; \
uintmax_t __u = (u), __v = (v); \
\
__ul = __ll_lowpart (__u); \
__uh = __ll_highpart (__u); \
__vl = __ll_lowpart (__v); \
__vh = __ll_highpart (__v); \
\
__x0 = (uintmax_t) __ul * __vl; \
__x1 = (uintmax_t) __ul * __vh; \
__x2 = (uintmax_t) __uh * __vl; \
__x3 = (uintmax_t) __uh * __vh; \
\
__x1 += __ll_highpart (__x0);/* this can't give carry */ \
__x1 += __x2; /* but this indeed can */ \
if (__x1 < __x2) /* did we get it? */ \
__x3 += __ll_B; /* yes, add it in the proper pos. */ \
\
(w1) = __x3 + __ll_highpart (__x1); \
(w0) = (__x1 << W_TYPE_SIZE / 2) + __ll_lowpart (__x0); \
} while (0)
#endif
#if !defined udiv_qrnnd || defined UDIV_NEEDS_NORMALIZATION
/* Define our own, not needing normalization. This function is
currently not performance critical, so keep it simple. Similar to
the mod macro below. */
# undef udiv_qrnnd
# define udiv_qrnnd(q, r, n1, n0, d) \
do { \
uintmax_t __d1, __d0, __q, __r1, __r0; \
\
assert ((n1) < (d)); \
__d1 = (d); __d0 = 0; \
__r1 = (n1); __r0 = (n0); \
__q = 0; \
for (unsigned int __i = W_TYPE_SIZE; __i > 0; __i--) \
{ \
rsh2 (__d1, __d0, __d1, __d0, 1); \
__q <<= 1; \
if (ge2 (__r1, __r0, __d1, __d0)) \
{ \
__q++; \
sub_ddmmss (__r1, __r0, __r1, __r0, __d1, __d0); \
} \
} \
(r) = __r0; \
(q) = __q; \
} while (0)
#endif
#if !defined add_ssaaaa
# define add_ssaaaa(sh, sl, ah, al, bh, bl) \
do { \
uintmax_t _add_x; \
_add_x = (al) + (bl); \
(sh) = (ah) + (bh) + (_add_x < (al)); \
(sl) = _add_x; \
} while (0)
#endif
#define rsh2(rh, rl, ah, al, cnt) \
do { \
(rl) = ((ah) << (W_TYPE_SIZE - (cnt))) | ((al) >> (cnt)); \
(rh) = (ah) >> (cnt); \
} while (0)
#define lsh2(rh, rl, ah, al, cnt) \
do { \
(rh) = ((ah) << cnt) | ((al) >> (W_TYPE_SIZE - (cnt))); \
(rl) = (al) << (cnt); \
} while (0)
#define ge2(ah, al, bh, bl) \
((ah) > (bh) || ((ah) == (bh) && (al) >= (bl)))
#define gt2(ah, al, bh, bl) \
((ah) > (bh) || ((ah) == (bh) && (al) > (bl)))
#ifndef sub_ddmmss
# define sub_ddmmss(rh, rl, ah, al, bh, bl) \
do { \
uintmax_t _cy; \
_cy = (al) < (bl); \
(rl) = (al) - (bl); \
(rh) = (ah) - (bh) - _cy; \
} while (0)
#endif
#ifndef count_leading_zeros
# define count_leading_zeros(count, x) do { \
uintmax_t __clz_x = (x); \
unsigned int __clz_c; \
for (__clz_c = 0; \
(__clz_x & ((uintmax_t) 0xff << (W_TYPE_SIZE - 8))) == 0; \
__clz_c += 8) \
__clz_x <<= 8; \
for (; (intmax_t)__clz_x >= 0; __clz_c++) \
__clz_x <<= 1; \
(count) = __clz_c; \
} while (0)
#endif
#ifndef count_trailing_zeros
# define count_trailing_zeros(count, x) do { \
uintmax_t __ctz_x = (x); \
unsigned int __ctz_c = 0; \
while ((__ctz_x & 1) == 0) \
{ \
__ctz_x >>= 1; \
__ctz_c++; \
} \
(count) = __ctz_c; \
} while (0)
#endif
/* Requires that a < n and b <= n */
#define submod(r,a,b,n) \
do { \
uintmax_t _t = - (uintmax_t) (a < b); \
(r) = ((n) & _t) + (a) - (b); \
} while (0)
#define addmod(r,a,b,n) \
submod ((r), (a), ((n) - (b)), (n))
/* Modular two-word addition and subtraction. For performance reasons, the
most significant bit of n1 must be clear. The destination variables must be
distinct from the mod operand. */
#define addmod2(r1, r0, a1, a0, b1, b0, n1, n0) \
do { \
add_ssaaaa ((r1), (r0), (a1), (a0), (b1), (b0)); \
if (ge2 ((r1), (r0), (n1), (n0))) \
sub_ddmmss ((r1), (r0), (r1), (r0), (n1), (n0)); \
} while (0)
#define submod2(r1, r0, a1, a0, b1, b0, n1, n0) \
do { \
sub_ddmmss ((r1), (r0), (a1), (a0), (b1), (b0)); \
if ((intmax_t) (r1) < 0) \
add_ssaaaa ((r1), (r0), (r1), (r0), (n1), (n0)); \
} while (0)
#define HIGHBIT_TO_MASK(x) \
(((intmax_t)-1 >> 1) < 0 \
? (uintmax_t)((intmax_t)(x) >> (W_TYPE_SIZE - 1)) \
: ((x) & ((uintmax_t) 1 << (W_TYPE_SIZE - 1)) \
? UINTMAX_MAX : (uintmax_t) 0))
/* Compute r = a mod d, where r = <*t1,retval>, a = <a1,a0>, d = <d1,d0>.
Requires that d1 != 0. */
static uintmax_t
mod2 (uintmax_t *r1, uintmax_t a1, uintmax_t a0, uintmax_t d1, uintmax_t d0)
{
int cntd, cnta;
assert (d1 != 0);
if (a1 == 0)
{
*r1 = 0;
return a0;
}
count_leading_zeros (cntd, d1);
count_leading_zeros (cnta, a1);
int cnt = cntd - cnta;
lsh2 (d1, d0, d1, d0, cnt);
for (int i = 0; i < cnt; i++)
{
if (ge2 (a1, a0, d1, d0))
sub_ddmmss (a1, a0, a1, a0, d1, d0);
rsh2 (d1, d0, d1, d0, 1);
}
*r1 = a1;
return a0;
}
static uintmax_t
gcd_odd (uintmax_t a, uintmax_t b)
{
if ( (b & 1) == 0)
{
uintmax_t t = b;
b = a;
a = t;
}
if (a == 0)
return b;
/* Take out least significant one bit, to make room for sign */
b >>= 1;
for (;;)
{
uintmax_t t;
uintmax_t bgta;
while ((a & 1) == 0)
a >>= 1;
a >>= 1;
t = a - b;
if (t == 0)
return (a << 1) + 1;
bgta = HIGHBIT_TO_MASK (t);
/* b <-- min (a, b) */
b += (bgta & t);
/* a <-- |a - b| */
a = (t ^ bgta) - bgta;
}
}
static uintmax_t
gcd2_odd (uintmax_t *r1, uintmax_t a1, uintmax_t a0, uintmax_t b1, uintmax_t b0)
{
while ((a0 & 1) == 0)
rsh2 (a1, a0, a1, a0, 1);
while ((b0 & 1) == 0)
rsh2 (b1, b0, b1, b0, 1);
for (;;)
{
if ((b1 | a1) == 0)
{
*r1 = 0;
return gcd_odd (b0, a0);
}
if (gt2 (a1, a0, b1, b0))
{
sub_ddmmss (a1, a0, a1, a0, b1, b0);
do
rsh2 (a1, a0, a1, a0, 1);
while ((a0 & 1) == 0);
}
else if (gt2 (b1, b0, a1, a0))
{
sub_ddmmss (b1, b0, b1, b0, a1, a0);
do
rsh2 (b1, b0, b1, b0, 1);
while ((b0 & 1) == 0);
}
else
break;
}
*r1 = a1;
return a0;
}
static void
factor_insert_multiplicity (struct factors *factors,
uintmax_t prime, unsigned int m)
{
unsigned int nfactors = factors->nfactors;
uintmax_t *p = factors->p;
unsigned char *e = factors->e;
/* Locate position for insert new or increment e. */
int i;
for (i = nfactors - 1; i >= 0; i--)
{
if (p[i] <= prime)
break;
}
if (i < 0 || p[i] != prime)
{
for (int j = nfactors - 1; j > i; j--)
{
p[j + 1] = p[j];
e[j + 1] = e[j];
}
p[i + 1] = prime;
e[i + 1] = m;
factors->nfactors = nfactors + 1;
}
else
{
e[i] += m;
}
}
#define factor_insert(f, p) factor_insert_multiplicity (f, p, 1)
static void
factor_insert_large (struct factors *factors,
uintmax_t p1, uintmax_t p0)
{
if (p1 > 0)
{
assert (factors->plarge[1] == 0);
factors->plarge[0] = p0;
factors->plarge[1] = p1;
}
else
factor_insert (factors, p0);
}
/* Number of bits in an uintmax_t. */
enum { W = sizeof (uintmax_t) * CHAR_BIT };
#define P(a,b,c,d) a,
static const unsigned char primes_diff[] = {
#include "primes.h"
0,0,0,0,0,0,0 /* 7 sentinels for 8-way loop */
};
#undef P
#define PRIMES_PTAB_ENTRIES \
(sizeof (primes_diff) / sizeof (primes_diff[0]) - 8 + 1)
#define P(a,b,c,d) b,
static const unsigned char primes_diff8[] = {
#include "primes.h"
0,0,0,0,0,0,0 /* 7 sentinels for 8-way loop */
};
#undef P
struct primes_dtab
{
uintmax_t binv, lim;
};
#define P(a,b,c,d) {c,d},
static const struct primes_dtab primes_dtab[] = {
#include "primes.h"
{1,0},{1,0},{1,0},{1,0},{1,0},{1,0},{1,0} /* 7 sentinels for 8-way loop */
};
#undef P
/* debugging for developers. Enables devmsg().
This flag is used only in the GMP code. */
static bool dev_debug = false;
/* Prove primality or run probabilistic tests. */
static bool flag_prove_primality = true;
/* Number of Miller-Rabin tests to run when not proving primality. */
#define MR_REPS 25
#ifdef __GNUC__
# define LIKELY(cond) __builtin_expect ((cond), 1)
# define UNLIKELY(cond) __builtin_expect ((cond), 0)
#else
# define LIKELY(cond) (cond)
# define UNLIKELY(cond) (cond)
#endif
static void
factor_insert_refind (struct factors *factors, uintmax_t p, unsigned int i,
unsigned int off)
{
for (unsigned int j = 0; j < off; j++)
p += primes_diff[i + j];
factor_insert (factors, p);
}
/* Trial division with odd primes uses the following trick.
Let p be an odd prime, and B = 2^{W_TYPE_SIZE}. For simplicity,
consider the case t < B (this is the second loop below).
From our tables we get
binv = p^{-1} (mod B)
lim = floor ( (B-1) / p ).
First assume that t is a multiple of p, t = q * p. Then 0 <= q <= lim
(and all quotients in this range occur for some t).
Then t = q * p is true also (mod B), and p is invertible we get
q = t * binv (mod B).
Next, assume that t is *not* divisible by p. Since multiplication
by binv (mod B) is a one-to-one mapping,
t * binv (mod B) > lim,
because all the smaller values are already taken.
This can be summed up by saying that the function
q(t) = binv * t (mod B)
is a permutation of the range 0 <= t < B, with the curious property
that it maps the multiples of p onto the range 0 <= q <= lim, in
order, and the non-multiples of p onto the range lim < q < B.
*/
static uintmax_t
factor_using_division (uintmax_t *t1p, uintmax_t t1, uintmax_t t0,
struct factors *factors)
{
if (t0 % 2 == 0)
{
unsigned int cnt;
if (t0 == 0)
{
count_trailing_zeros (cnt, t1);
t0 = t1 >> cnt;
t1 = 0;
cnt += W_TYPE_SIZE;
}
else
{
count_trailing_zeros (cnt, t0);
rsh2 (t1, t0, t1, t0, cnt);
}
factor_insert_multiplicity (factors, 2, cnt);
}
uintmax_t p = 3;
unsigned int i;
for (i = 0; t1 > 0 && i < PRIMES_PTAB_ENTRIES; i++)
{
for (;;)
{
uintmax_t q1, q0, hi, lo;
q0 = t0 * primes_dtab[i].binv;
umul_ppmm (hi, lo, q0, p);
if (hi > t1)
break;
hi = t1 - hi;
q1 = hi * primes_dtab[i].binv;
if (LIKELY (q1 > primes_dtab[i].lim))
break;
t1 = q1; t0 = q0;
factor_insert (factors, p);
}
p += primes_diff[i + 1];
}
if (t1p)
*t1p = t1;
#define DIVBLOCK(I) \
do { \
for (;;) \
{ \
q = t0 * pd[I].binv; \
if (LIKELY (q > pd[I].lim)) \
break; \
t0 = q; \
factor_insert_refind (factors, p, i + 1, I); \
} \
} while (0)
for (; i < PRIMES_PTAB_ENTRIES; i += 8)
{
uintmax_t q;
const struct primes_dtab *pd = &primes_dtab[i];
DIVBLOCK (0);
DIVBLOCK (1);
DIVBLOCK (2);
DIVBLOCK (3);
DIVBLOCK (4);
DIVBLOCK (5);
DIVBLOCK (6);
DIVBLOCK (7);
p += primes_diff8[i];
if (p * p > t0)
break;
}
return t0;
}
/* Entry i contains (2i+1)^(-1) mod 2^8. */
static const unsigned char binvert_table[128] =
{
0x01, 0xAB, 0xCD, 0xB7, 0x39, 0xA3, 0xC5, 0xEF,
0xF1, 0x1B, 0x3D, 0xA7, 0x29, 0x13, 0x35, 0xDF,
0xE1, 0x8B, 0xAD, 0x97, 0x19, 0x83, 0xA5, 0xCF,
0xD1, 0xFB, 0x1D, 0x87, 0x09, 0xF3, 0x15, 0xBF,
0xC1, 0x6B, 0x8D, 0x77, 0xF9, 0x63, 0x85, 0xAF,
0xB1, 0xDB, 0xFD, 0x67, 0xE9, 0xD3, 0xF5, 0x9F,
0xA1, 0x4B, 0x6D, 0x57, 0xD9, 0x43, 0x65, 0x8F,
0x91, 0xBB, 0xDD, 0x47, 0xC9, 0xB3, 0xD5, 0x7F,
0x81, 0x2B, 0x4D, 0x37, 0xB9, 0x23, 0x45, 0x6F,
0x71, 0x9B, 0xBD, 0x27, 0xA9, 0x93, 0xB5, 0x5F,
0x61, 0x0B, 0x2D, 0x17, 0x99, 0x03, 0x25, 0x4F,
0x51, 0x7B, 0x9D, 0x07, 0x89, 0x73, 0x95, 0x3F,
0x41, 0xEB, 0x0D, 0xF7, 0x79, 0xE3, 0x05, 0x2F,
0x31, 0x5B, 0x7D, 0xE7, 0x69, 0x53, 0x75, 0x1F,
0x21, 0xCB, 0xED, 0xD7, 0x59, 0xC3, 0xE5, 0x0F,
0x11, 0x3B, 0x5D, 0xC7, 0x49, 0x33, 0x55, 0xFF
};
/* Compute n^(-1) mod B, using a Newton iteration. */
#define binv(inv,n) \
do { \
uintmax_t __n = (n); \
uintmax_t __inv; \
\
__inv = binvert_table[(__n / 2) & 0x7F]; /* 8 */ \
if (W_TYPE_SIZE > 8) __inv = 2 * __inv - __inv * __inv * __n; \
if (W_TYPE_SIZE > 16) __inv = 2 * __inv - __inv * __inv * __n; \
if (W_TYPE_SIZE > 32) __inv = 2 * __inv - __inv * __inv * __n; \
\
if (W_TYPE_SIZE > 64) \
{ \
int __invbits = 64; \
do { \
__inv = 2 * __inv - __inv * __inv * __n; \
__invbits *= 2; \
} while (__invbits < W_TYPE_SIZE); \
} \
\
(inv) = __inv; \
} while (0)
/* q = u / d, assuming d|u. */
#define divexact_21(q1, q0, u1, u0, d) \
do { \
uintmax_t _di, _q0; \
binv (_di, (d)); \
_q0 = (u0) * _di; \
if ((u1) >= (d)) \
{ \
uintmax_t _p1, _p0; \
umul_ppmm (_p1, _p0, _q0, d); \
(q1) = ((u1) - _p1) * _di; \
(q0) = _q0; \
} \
else \
{ \
(q0) = _q0; \
(q1) = 0; \
} \
} while (0)
/* x B (mod n). */
#define redcify(r_prim, r, n) \
do { \
uintmax_t _redcify_q; \
udiv_qrnnd (_redcify_q, r_prim, r, 0, n); \
} while (0)
/* x B^2 (mod n). Requires x > 0, n1 < B/2 */
#define redcify2(r1, r0, x, n1, n0) \
do { \
uintmax_t _r1, _r0, _i; \
if ((x) < (n1)) \
{ \
_r1 = (x); _r0 = 0; \
_i = W_TYPE_SIZE; \
} \
else \
{ \
_r1 = 0; _r0 = (x); \
_i = 2*W_TYPE_SIZE; \
} \
while (_i-- > 0) \
{ \
lsh2 (_r1, _r0, _r1, _r0, 1); \
if (ge2 (_r1, _r0, (n1), (n0))) \
sub_ddmmss (_r1, _r0, _r1, _r0, (n1), (n0)); \
} \
(r1) = _r1; \
(r0) = _r0; \
} while (0)
/* Modular two-word multiplication, r = a * b mod m, with mi = m^(-1) mod B.
Both a and b must be in redc form, the result will be in redc form too. */
static inline uintmax_t
mulredc (uintmax_t a, uintmax_t b, uintmax_t m, uintmax_t mi)
{
uintmax_t rh, rl, q, th, tl, xh;
umul_ppmm (rh, rl, a, b);
q = rl * mi;
umul_ppmm (th, tl, q, m);
xh = rh - th;
if (rh < th)
xh += m;
return xh;
}
/* Modular two-word multiplication, r = a * b mod m, with mi = m^(-1) mod B.
Both a and b must be in redc form, the result will be in redc form too.
For performance reasons, the most significant bit of m must be clear. */
static uintmax_t
mulredc2 (uintmax_t *r1p,
uintmax_t a1, uintmax_t a0, uintmax_t b1, uintmax_t b0,
uintmax_t m1, uintmax_t m0, uintmax_t mi)
{
uintmax_t r1, r0, q, p1, p0, t1, t0, s1, s0;
mi = -mi;
assert ( (a1 >> (W_TYPE_SIZE - 1)) == 0);
assert ( (b1 >> (W_TYPE_SIZE - 1)) == 0);
assert ( (m1 >> (W_TYPE_SIZE - 1)) == 0);
/* First compute a0 * <b1, b0> B^{-1}
+-----+
|a0 b0|
+--+--+--+
|a0 b1|
+--+--+--+
|q0 m0|
+--+--+--+
|q0 m1|
-+--+--+--+
|r1|r0| 0|
+--+--+--+
*/
umul_ppmm (t1, t0, a0, b0);
umul_ppmm (r1, r0, a0, b1);
q = mi * t0;
umul_ppmm (p1, p0, q, m0);
umul_ppmm (s1, s0, q, m1);
r0 += (t0 != 0); /* Carry */
add_ssaaaa (r1, r0, r1, r0, 0, p1);
add_ssaaaa (r1, r0, r1, r0, 0, t1);
add_ssaaaa (r1, r0, r1, r0, s1, s0);
/* Next, (a1 * <b1, b0> + <r1, r0> B^{-1}
+-----+
|a1 b0|
+--+--+
|r1|r0|
+--+--+--+
|a1 b1|
+--+--+--+
|q1 m0|
+--+--+--+
|q1 m1|
-+--+--+--+
|r1|r0| 0|
+--+--+--+
*/
umul_ppmm (t1, t0, a1, b0);
umul_ppmm (s1, s0, a1, b1);
add_ssaaaa (t1, t0, t1, t0, 0, r0);
q = mi * t0;
add_ssaaaa (r1, r0, s1, s0, 0, r1);
umul_ppmm (p1, p0, q, m0);
umul_ppmm (s1, s0, q, m1);
r0 += (t0 != 0); /* Carry */
add_ssaaaa (r1, r0, r1, r0, 0, p1);
add_ssaaaa (r1, r0, r1, r0, 0, t1);
add_ssaaaa (r1, r0, r1, r0, s1, s0);
if (ge2 (r1, r0, m1, m0))
sub_ddmmss (r1, r0, r1, r0, m1, m0);
*r1p = r1;
return r0;
}
static uintmax_t
powm (uintmax_t b, uintmax_t e, uintmax_t n, uintmax_t ni, uintmax_t one)
{
uintmax_t y = one;
if (e & 1)
y = b;
while (e != 0)
{
b = mulredc (b, b, n, ni);
e >>= 1;
if (e & 1)
y = mulredc (y, b, n, ni);
}
return y;
}
static uintmax_t
powm2 (uintmax_t *r1m,
const uintmax_t *bp, const uintmax_t *ep, const uintmax_t *np,
uintmax_t ni, const uintmax_t *one)
{
uintmax_t r1, r0, b1, b0, n1, n0;
unsigned int i;
uintmax_t e;
b0 = bp[0];
b1 = bp[1];
n0 = np[0];
n1 = np[1];
r0 = one[0];
r1 = one[1];
for (e = ep[0], i = W_TYPE_SIZE; i > 0; i--, e >>= 1)
{
if (e & 1)
{
r0 = mulredc2 (r1m, r1, r0, b1, b0, n1, n0, ni);
r1 = *r1m;
}
b0 = mulredc2 (r1m, b1, b0, b1, b0, n1, n0, ni);
b1 = *r1m;
}
for (e = ep[1]; e > 0; e >>= 1)
{
if (e & 1)
{
r0 = mulredc2 (r1m, r1, r0, b1, b0, n1, n0, ni);
r1 = *r1m;
}
b0 = mulredc2 (r1m, b1, b0, b1, b0, n1, n0, ni);
b1 = *r1m;
}
*r1m = r1;
return r0;
}
static bool
millerrabin (uintmax_t n, uintmax_t ni, uintmax_t b, uintmax_t q,
unsigned int k, uintmax_t one)
{
uintmax_t y = powm (b, q, n, ni, one);
uintmax_t nm1 = n - one; /* -1, but in redc representation. */
if (y == one || y == nm1)
return true;
for (unsigned int i = 1; i < k; i++)
{
y = mulredc (y, y, n, ni);
if (y == nm1)
return true;
if (y == one)
return false;
}
return false;
}
static bool
millerrabin2 (const uintmax_t *np, uintmax_t ni, const uintmax_t *bp,
const uintmax_t *qp, unsigned int k, const uintmax_t *one)
{
uintmax_t y1, y0, nm1_1, nm1_0, r1m;
y0 = powm2 (&r1m, bp, qp, np, ni, one);
y1 = r1m;
if (y0 == one[0] && y1 == one[1])
return true;
sub_ddmmss (nm1_1, nm1_0, np[1], np[0], one[1], one[0]);
if (y0 == nm1_0 && y1 == nm1_1)
return true;
for (unsigned int i = 1; i < k; i++)
{
y0 = mulredc2 (&r1m, y1, y0, y1, y0, np[1], np[0], ni);
y1 = r1m;
if (y0 == nm1_0 && y1 == nm1_1)
return true;
if (y0 == one[0] && y1 == one[1])
return false;
}
return false;
}
/* Lucas' prime test. The number of iterations vary greatly, up to a few dozen
have been observed. The average seem to be about 2. */
static bool
prime_p (uintmax_t n)
{
int k;
bool is_prime;
uintmax_t a_prim, one, ni;
struct factors factors;
if (n <= 1)
return false;
/* We have already casted out small primes. */
if (n < (uintmax_t) FIRST_OMITTED_PRIME * FIRST_OMITTED_PRIME)
return true;
/* Precomputation for Miller-Rabin. */
uintmax_t q = n - 1;
for (k = 0; (q & 1) == 0; k++)
q >>= 1;
uintmax_t a = 2;
binv (ni, n); /* ni <- 1/n mod B */
redcify (one, 1, n);
addmod (a_prim, one, one, n); /* i.e., redcify a = 2 */
/* Perform a Miller-Rabin test, finds most composites quickly. */
if (!millerrabin (n, ni, a_prim, q, k, one))
return false;
if (flag_prove_primality)
{
/* Factor n-1 for Lucas. */
factor (n - 1, &factors);
}
/* Loop until Lucas proves our number prime, or Miller-Rabin proves our
number composite. */
for (unsigned int r = 0; r < PRIMES_PTAB_ENTRIES; r++)
{
if (flag_prove_primality)
{
is_prime = true;
for (unsigned int i = 0; i < factors.nfactors && is_prime; i++)
{
is_prime
= powm (a_prim, (n - 1) / factors.p[i], n, ni, one) != one;
}
}
else
{
/* After enough Miller-Rabin runs, be content. */
is_prime = (r == MR_REPS - 1);
}
if (is_prime)
return true;
a += primes_diff[r]; /* Establish new base. */
/* The following is equivalent to redcify (a_prim, a, n). It runs faster
on most processors, since it avoids udiv_qrnnd. If we go down the
udiv_qrnnd_preinv path, this code should be replaced. */
{
uintmax_t s1, s0;
umul_ppmm (s1, s0, one, a);
if (LIKELY (s1 == 0))
a_prim = s0 % n;
else
{
uintmax_t dummy;
udiv_qrnnd (dummy, a_prim, s1, s0, n);
}
}
if (!millerrabin (n, ni, a_prim, q, k, one))
return false;
}
error (0, 0, "Lucas prime test failure. This should not happen");
return false;
}
static bool
prime2_p (uintmax_t n1, uintmax_t n0)
{
uintmax_t q[2], nm1[2];
uintmax_t a_prim[2];
uintmax_t one[2];
uintmax_t na[2];
uintmax_t ni;
unsigned int k;
struct factors factors;
if (n1 == 0)
return prime_p (n0);
nm1[1] = n1 - (n0 == 0);
nm1[0] = n0 - 1;
if (nm1[0] == 0)
{
count_trailing_zeros (k, nm1[1]);
q[0] = nm1[1] >> k;
q[1] = 0;
k += W_TYPE_SIZE;
}
else
{
count_trailing_zeros (k, nm1[0]);
rsh2 (q[1], q[0], nm1[1], nm1[0], k);
}
uintmax_t a = 2;
binv (ni, n0);
redcify2 (one[1], one[0], 1, n1, n0);
addmod2 (a_prim[1], a_prim[0], one[1], one[0], one[1], one[0], n1, n0);
/* FIXME: Use scalars or pointers in arguments? Some consistency needed. */
na[0] = n0;
na[1] = n1;
if (!millerrabin2 (na, ni, a_prim, q, k, one))
return false;
if (flag_prove_primality)
{
/* Factor n-1 for Lucas. */
factor (nm1[0], &factors);
}
/* Loop until Lucas proves our number prime, or Miller-Rabin proves our
number composite. */
for (unsigned int r = 0; r < PRIMES_PTAB_ENTRIES; r++)
{
bool is_prime;
uintmax_t e[2], y[2];
if (flag_prove_primality)
{
is_prime = true;
if (factors.plarge[1])
{
uintmax_t pi;
binv (pi, factors.plarge[0]);
e[0] = pi * nm1[0];
e[1] = 0;
y[0] = powm2 (&y[1], a_prim, e, na, ni, one);
is_prime = (y[0] != one[0] || y[1] != one[1]);
}
for (unsigned int i = 0; i < factors.nfactors && is_prime; i++)