#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.
*
* SPDX-License-Identifier: Unlicense
*/
/* portable integer log of two with small footprint */
static unsigned int s_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] == 0u) || (a->dp[0] == 1u)) {
*result = 0;
return MP_OKAY;
}
if (a->dp[0] == 2u) {
*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, 2uL)) != 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".
*/
/* 0x437ae92817f9fc85b7e5 = 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 {
/* 0x2be6951adc5b22410a5fd = 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 << s_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;
if ((err = mp_div_2d(&b, len, &b, NULL)) != MP_OKAY) {
goto LBL_B;
}
}
/* Although the chance for b <= 3 is miniscule, try again. */
if (mp_cmp_d(&b, 3uL) != 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: $Format:%ai$ */