Edit
kc3-lang/libtommath/bn_mp_prime_is_prime.c
Steffen Jaeckel
- bn_mp_prime_is_prime.c
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#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 if (fips_rand > ((unsigned int) INT_MAX - DIGIT_BIT)) { len = INT_MAX / DIGIT_BIT; } else { 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: $Format:%ai$ */