368 lines
11 KiB
C
368 lines
11 KiB
C
#include "tommath_private.h"
|
||
#ifdef BN_MP_PRIME_IS_PRIME_C
|
||
/* LibTomMath, multiple-precision integer library -- Tom St Denis
|
||
*
|
||
* LibTomMath is a library that provides multiple-precision
|
||
* integer arithmetic as well as number theoretic functionality.
|
||
*
|
||
* The library was designed directly after the MPI library by
|
||
* Michael Fromberger but has been written from scratch with
|
||
* additional optimizations in place.
|
||
*
|
||
* The library is free for all purposes without any express
|
||
* guarantee it works.
|
||
*/
|
||
|
||
/* portable integer log of two with small footprint */
|
||
static unsigned int floor_ilog2(int value)
|
||
{
|
||
unsigned int r = 0;
|
||
while ((value >>= 1) != 0) {
|
||
r++;
|
||
}
|
||
return r;
|
||
}
|
||
|
||
|
||
int mp_prime_is_prime(const mp_int *a, int t, int *result)
|
||
{
|
||
mp_int b;
|
||
int ix, err, res, p_max = 0, size_a, len;
|
||
unsigned int fips_rand, mask;
|
||
|
||
/* default to no */
|
||
*result = MP_NO;
|
||
|
||
/* valid value of t? */
|
||
if (t > PRIME_SIZE) {
|
||
return MP_VAL;
|
||
}
|
||
|
||
/* Some shortcuts */
|
||
/* N > 3 */
|
||
if (a->used == 1) {
|
||
if (a->dp[0] == 0 || a->dp[0] == 1) {
|
||
*result = 0;
|
||
return MP_OKAY;
|
||
}
|
||
if (a->dp[0] == 2) {
|
||
*result = 1;
|
||
return MP_OKAY;
|
||
}
|
||
}
|
||
|
||
/* N must be odd */
|
||
if (mp_iseven(a) == MP_YES) {
|
||
return MP_OKAY;
|
||
}
|
||
/* N is not a perfect square: floor(sqrt(N))^2 != N */
|
||
if ((err = mp_is_square(a, &res)) != MP_OKAY) {
|
||
return err;
|
||
}
|
||
if (res != 0) {
|
||
return MP_OKAY;
|
||
}
|
||
|
||
/* is the input equal to one of the primes in the table? */
|
||
for (ix = 0; ix < PRIME_SIZE; ix++) {
|
||
if (mp_cmp_d(a, ltm_prime_tab[ix]) == MP_EQ) {
|
||
*result = MP_YES;
|
||
return MP_OKAY;
|
||
}
|
||
}
|
||
#ifdef MP_8BIT
|
||
/* The search in the loop above was exhaustive in this case */
|
||
if (a->used == 1 && PRIME_SIZE >= 31) {
|
||
return MP_OKAY;
|
||
}
|
||
#endif
|
||
|
||
/* first perform trial division */
|
||
if ((err = mp_prime_is_divisible(a, &res)) != MP_OKAY) {
|
||
return err;
|
||
}
|
||
|
||
/* return if it was trivially divisible */
|
||
if (res == MP_YES) {
|
||
return MP_OKAY;
|
||
}
|
||
|
||
/*
|
||
Run the Miller-Rabin test with base 2 for the BPSW test.
|
||
*/
|
||
if ((err = mp_init_set(&b,2)) != MP_OKAY) {
|
||
return err;
|
||
}
|
||
|
||
if ((err = mp_prime_miller_rabin(a, &b, &res)) != MP_OKAY) {
|
||
goto LBL_B;
|
||
}
|
||
if (res == MP_NO) {
|
||
goto LBL_B;
|
||
}
|
||
/*
|
||
Rumours have it that Mathematica does a second M-R test with base 3.
|
||
Other rumours have it that their strong L-S test is slightly different.
|
||
It does not hurt, though, beside a bit of extra runtime.
|
||
*/
|
||
b.dp[0]++;
|
||
if ((err = mp_prime_miller_rabin(a, &b, &res)) != MP_OKAY) {
|
||
goto LBL_B;
|
||
}
|
||
if (res == MP_NO) {
|
||
goto LBL_B;
|
||
}
|
||
|
||
/*
|
||
* Both, the Frobenius-Underwood test and the the Lucas-Selfridge test are quite
|
||
* slow so if speed is an issue, define LTM_USE_FIPS_ONLY to use M-R tests with
|
||
* bases 2, 3 and t random bases.
|
||
*/
|
||
#ifndef LTM_USE_FIPS_ONLY
|
||
if (t >= 0) {
|
||
/*
|
||
* Use a Frobenius-Underwood test instead of the Lucas-Selfridge test for
|
||
* MP_8BIT (It is unknown if the Lucas-Selfridge test works with 16-bit
|
||
* integers but the necesssary analysis is on the todo-list).
|
||
*/
|
||
#if defined (MP_8BIT) || defined (LTM_USE_FROBENIUS_TEST)
|
||
err = mp_prime_frobenius_underwood(a, &res);
|
||
if (err != MP_OKAY && err != MP_ITER) {
|
||
goto LBL_B;
|
||
}
|
||
if (res == MP_NO) {
|
||
goto LBL_B;
|
||
}
|
||
#else
|
||
if ((err = mp_prime_strong_lucas_selfridge(a, &res)) != MP_OKAY) {
|
||
goto LBL_B;
|
||
}
|
||
if (res == MP_NO) {
|
||
goto LBL_B;
|
||
}
|
||
#endif
|
||
}
|
||
#endif
|
||
|
||
/* run at least one Miller-Rabin test with a random base */
|
||
if(t == 0) {
|
||
t = 1;
|
||
}
|
||
|
||
/*
|
||
abs(t) extra rounds of M-R to extend the range of primes it can find if t < 0.
|
||
Only recommended if the input range is known to be < 3317044064679887385961981
|
||
|
||
It uses the bases for a deterministic M-R test if input < 3317044064679887385961981
|
||
The caller has to check the size.
|
||
|
||
Not for cryptographic use because with known bases strong M-R pseudoprimes can
|
||
be constructed. Use at least one M-R test with a random base (t >= 1).
|
||
|
||
The 1119 bit large number
|
||
|
||
80383745745363949125707961434194210813883768828755814583748891752229742737653\
|
||
33652186502336163960045457915042023603208766569966760987284043965408232928738\
|
||
79185086916685732826776177102938969773947016708230428687109997439976544144845\
|
||
34115587245063340927902227529622941498423068816854043264575340183297861112989\
|
||
60644845216191652872597534901
|
||
|
||
has been constructed by F. Arnault (F. Arnault, "Rabin-Miller primality test:
|
||
composite numbers which pass it.", Mathematics of Computation, 1995, 64. Jg.,
|
||
Nr. 209, S. 355-361), is a semiprime with the two factors
|
||
|
||
40095821663949960541830645208454685300518816604113250877450620473800321707011\
|
||
96242716223191597219733582163165085358166969145233813917169287527980445796800\
|
||
452592031836601
|
||
|
||
20047910831974980270915322604227342650259408302056625438725310236900160853505\
|
||
98121358111595798609866791081582542679083484572616906958584643763990222898400\
|
||
226296015918301
|
||
|
||
and it is a strong pseudoprime to all forty-six prime M-R bases up to 200
|
||
|
||
It does not fail the strong Bailley-PSP test as implemented here, it is just
|
||
given as an example, if not the reason to use the BPSW-test instead of M-R-tests
|
||
with a sequence of primes 2...n.
|
||
|
||
*/
|
||
if (t < 0) {
|
||
t = -t;
|
||
/*
|
||
Sorenson, Jonathan; Webster, Jonathan (2015).
|
||
"Strong Pseudoprimes to Twelve Prime Bases".
|
||
*/
|
||
/* 318665857834031151167461 */
|
||
if ((err = mp_read_radix(&b, "437ae92817f9fc85b7e5", 16)) != MP_OKAY) {
|
||
goto LBL_B;
|
||
}
|
||
|
||
if (mp_cmp(a,&b) == MP_LT) {
|
||
p_max = 12;
|
||
}
|
||
else { /* 3317044064679887385961981 */
|
||
if ((err = mp_read_radix(&b, "2be6951adc5b22410a5fd", 16)) != MP_OKAY) {
|
||
goto LBL_B;
|
||
}
|
||
|
||
if (mp_cmp(a,&b) == MP_LT) {
|
||
p_max = 13;
|
||
}
|
||
else {
|
||
err = MP_VAL;
|
||
goto LBL_B;
|
||
}
|
||
}
|
||
|
||
// for compatibility with the current API (well, compatible within a sign's width)
|
||
if (p_max < t) {
|
||
p_max = t;
|
||
}
|
||
|
||
if(p_max > PRIME_SIZE) {
|
||
err = MP_VAL;
|
||
goto LBL_B;
|
||
}
|
||
/* we did bases 2 and 3 already, skip them */
|
||
for (ix = 2; ix < p_max; ix++) {
|
||
mp_set(&b,ltm_prime_tab[ix]);
|
||
if ((err = mp_prime_miller_rabin(a, &b, &res)) != MP_OKAY) {
|
||
goto LBL_B;
|
||
}
|
||
if (res == MP_NO) {
|
||
goto LBL_B;
|
||
}
|
||
}
|
||
}
|
||
/*
|
||
Do "t" M-R tests with random bases between 3 and "a".
|
||
See Fips 186.4 p. 126ff
|
||
*/
|
||
else if (t > 0) {
|
||
/*
|
||
* The mp_digit's have a defined bit-size but the size of the
|
||
* array a.dp is a simple 'int' and this library can not assume full
|
||
* compliance to the current C-standard (ISO/IEC 9899:2011) because
|
||
* it gets used for small embeded processors, too. Some of those MCUs
|
||
* have compilers that one cannot call standard compliant by any means.
|
||
* Hence the ugly type-fiddling in the following code.
|
||
*/
|
||
size_a = mp_count_bits(a);
|
||
mask = (1u << floor_ilog2(size_a)) - 1u;
|
||
/*
|
||
Assuming the General Rieman hypothesis (never thought to write that in a
|
||
comment) the upper bound can be lowered to 2*(log a)^2.
|
||
E. Bach, “Explicit bounds for primality testing and related problems,”
|
||
Math. Comp. 55 (1990), 355–380.
|
||
|
||
size_a = (size_a/10) * 7;
|
||
len = 2 * (size_a * size_a);
|
||
|
||
E.g.: a number of size 2^2048 would be reduced to the upper limit
|
||
|
||
floor(2048/10)*7 = 1428
|
||
2 * 1428^2 = 4078368
|
||
|
||
(would have been ~4030331.9962 with floats and natural log instead)
|
||
That number is smaller than 2^28, the default bit-size of mp_digit.
|
||
*/
|
||
|
||
/*
|
||
How many tests, you might ask? Dana Jacobsen of Math::Prime::Util fame
|
||
does exactly 1. In words: one. Look at the end of _GMP_is_prime() in
|
||
Math-Prime-Util-GMP-0.50/primality.c if you do not believe it.
|
||
|
||
The function mp_rand() goes to some length to use a cryptographically
|
||
good PRNG. That also means that the chance to always get the same base
|
||
in the loop is non-zero, although very low.
|
||
If the BPSW test and/or the addtional Frobenious test have been
|
||
performed instead of just the Miller-Rabin test with the bases 2 and 3,
|
||
a single extra test should suffice, so such a very unlikely event
|
||
will not do much harm.
|
||
|
||
To preemptivly answer the dangling question: no, a witness does not
|
||
need to be prime.
|
||
*/
|
||
for (ix = 0; ix < t; ix++) {
|
||
/* mp_rand() guarantees the first digit to be non-zero */
|
||
if ((err = mp_rand(&b, 1)) != MP_OKAY) {
|
||
goto LBL_B;
|
||
}
|
||
/*
|
||
* Reduce digit before casting because mp_digit might be bigger than
|
||
* an unsigned int and "mask" on the other side is most probably not.
|
||
*/
|
||
fips_rand = (unsigned int) (b.dp[0] & (mp_digit) mask);
|
||
#ifdef MP_8BIT
|
||
/*
|
||
* One 8-bit digit is too small, so concatenate two if the size of
|
||
* unsigned int allows for it.
|
||
*/
|
||
if( (sizeof(unsigned int) * CHAR_BIT)/2 >= (sizeof(mp_digit) * CHAR_BIT) ) {
|
||
if ((err = mp_rand(&b, 1)) != MP_OKAY) {
|
||
goto LBL_B;
|
||
}
|
||
fips_rand <<= sizeof(mp_digit) * CHAR_BIT;
|
||
fips_rand |= (unsigned int) b.dp[0];
|
||
fips_rand &= mask;
|
||
}
|
||
#endif
|
||
/* Ceil, because small numbers have a right to live, too, */
|
||
len = (int) ( (fips_rand + DIGIT_BIT) / DIGIT_BIT);
|
||
/* Unlikely. */
|
||
if(len < 0){
|
||
ix--;
|
||
continue;
|
||
}
|
||
/*
|
||
* As mentioned above, one 8-bit digit is too small and
|
||
* although it can only happen in the unlikely case that
|
||
* an "unsigned int" is smaller than 16 bit a simple test
|
||
* is cheap and the correction even cheaper.
|
||
*/
|
||
#ifdef MP_8BIT
|
||
/* All "a" < 2^8 have been caught before */
|
||
if(len == 1){
|
||
len++;
|
||
}
|
||
#endif
|
||
if ((err = mp_rand(&b, len)) != MP_OKAY) {
|
||
goto LBL_B;
|
||
}
|
||
/*
|
||
* That number might got too big and the witness has to be
|
||
* smaller than or equal to "a"
|
||
*/
|
||
len = mp_count_bits(&b);
|
||
if (len > size_a) {
|
||
len = len - size_a;
|
||
mp_div_2d(&b, len, &b, NULL);
|
||
}
|
||
|
||
/* Although the chance for b <= 3 is miniscule, try again. */
|
||
if (mp_cmp_d(&b,3) != MP_GT) {
|
||
ix--;
|
||
continue;
|
||
}
|
||
if ((err = mp_prime_miller_rabin(a, &b, &res)) != MP_OKAY) {
|
||
goto LBL_B;
|
||
}
|
||
if (res == MP_NO) {
|
||
goto LBL_B;
|
||
}
|
||
}
|
||
}
|
||
|
||
/* passed the test */
|
||
*result = MP_YES;
|
||
LBL_B:
|
||
mp_clear(&b);
|
||
return err;
|
||
}
|
||
|
||
#endif
|
||
|
||
/* ref: $Format:%D$ */
|
||
/* git commit: $Format:%H$ */
|
||
/* commit time: $Format:%ai$ */
|