Add support for random numbers, and prime generation and testing.
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145
src/unsigned/primes.rs
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145
src/unsigned/primes.rs
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use rand::RngCore;
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/// Functions related to the generation of random numbers and primes.
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pub trait PrimeGen: Sized + PartialOrd {
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/// Generate a random prime number, using the given RNG and running
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/// the primality check for the given number of iterations. This is
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/// equivalent to calling `random_primef` with the identity function
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/// as the modifier.
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fn random_prime<R: RngCore>(rng: &mut R, iters: usize) -> Self {
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Self::random_primef(rng, iters, |x| Some(x))
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}
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/// Generate a random prime number, using a modification function
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/// and running the primality check for the given number of iterations.
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/// The modifier function is run after the routine generates a random
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/// number, but before the primality check, and can be used to force
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/// the return value to have certain properties: the low bit set, the
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/// high bit set, and/or the number is above a certain value.
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fn random_primef<F,R>(rng: &mut R, iters: usize, prep: F) -> Self
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where F: Fn(Self) -> Option<Self>, R: RngCore;
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/// Determine if the given number is probably prime. This should be
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/// an implementation of Miller-Rabin, with some quick sanity checks,
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/// over the given number of iterations.
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fn probably_prime<R: RngCore>(&self, rng: &mut R, iters: usize) -> bool;
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}
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pub static SMALL_PRIMES: [u64; 310] = [
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2, 3, 5, 7, 11, 13, 17, 19, 23, 29,
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31, 37, 41, 43, 47, 53, 59, 61, 67, 71,
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73, 79, 83, 89, 97, 101, 103, 107, 109, 113,
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127, 131, 137, 139, 149, 151, 157, 163, 167, 173,
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179, 181, 191, 193, 197, 199, 211, 223, 227, 229,
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233, 239, 241, 251, 257, 263, 269, 271, 277, 281,
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283, 293, 307, 311, 313, 317, 331, 337, 347, 349,
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353, 359, 367, 373, 379, 383, 389, 397, 401, 409,
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419, 421, 431, 433, 439, 443, 449, 457, 461, 463,
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467, 479, 487, 491, 499, 503, 509, 521, 523, 541,
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547, 557, 563, 569, 571, 577, 587, 593, 599, 601,
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607, 613, 617, 619, 631, 641, 643, 647, 653, 659,
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661, 673, 677, 683, 691, 701, 709, 719, 727, 733,
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739, 743, 751, 757, 761, 769, 773, 787, 797, 809,
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811, 821, 823, 827, 829, 839, 853, 857, 859, 863,
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877, 881, 883, 887, 907, 911, 919, 929, 937, 941,
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947, 953, 967, 971, 977, 983, 991, 997, 1009, 1013,
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1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069,
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1087, 1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151,
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1153, 1163, 1171, 1181, 1187, 1193, 1201, 1213, 1217, 1223,
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1229, 1231, 1237, 1249, 1259, 1277, 1279, 1283, 1289, 1291,
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1297, 1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367, 1373,
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1381, 1399, 1409, 1423, 1427, 1429, 1433, 1439, 1447, 1451,
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1453, 1459, 1471, 1481, 1483, 1487, 1489, 1493, 1499, 1511,
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1523, 1531, 1543, 1549, 1553, 1559, 1567, 1571, 1579, 1583,
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1597, 1601, 1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657,
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1663, 1667, 1669, 1693, 1697, 1699, 1709, 1721, 1723, 1733,
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1741, 1747, 1753, 1759, 1777, 1783, 1787, 1789, 1801, 1811,
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1823, 1831, 1847, 1861, 1867, 1871, 1873, 1877, 1879, 1889,
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1901, 1907, 1913, 1931, 1933, 1949, 1951, 1973, 1979, 1987,
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1993, 1997, 1999, 2003, 2011, 2017, 2027, 2029, 2039, 2053];
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macro_rules! prime_gen_impls {
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($name: ident) => {
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impl PrimeGen for $name {
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fn random_primef<F,R>(rng: &mut R, iters: usize, modifier: F) -> Self
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where
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F: Fn($name) -> Option<$name>,
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R: RngCore
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{
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loop {
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let base = rng.gen();
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if let Some(candidate) = modifier(base) {
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let good = candidate.probably_prime(rng, iters);
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if good {
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return candidate;
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}
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}
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}
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}
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fn probably_prime<R: RngCore>(&self, rng: &mut R, iters: usize) -> bool
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{
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for tester in SMALL_PRIMES.iter() {
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if self.is_multiple_of(*tester) {
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return false;
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}
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}
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self.miller_rabin(rng, iters)
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}
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}
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impl $name {
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fn miller_rabin<R: RngCore>(&self, rng: &mut R, iters: usize) -> bool
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{
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let one = $name::from(1u64);
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let two = $name::from(2u64);
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let nm1 = self - $name::from(1u64);
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// Quoth Wikipedia:
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// write n - 1 as 2^r*d with d odd by factoring powers of 2 from n - 1
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let mut d = nm1.clone();
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let mut r = 0;
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while d.is_even() {
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d >>= 1;
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r += 1;
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assert!(r < $name::bit_length());
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}
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// WitnessLoop: repeat k times
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'WitnessLoop: for _k in 0..iters {
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// pick a random integer a in the range [2, n - 2]
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let a = rng.gen_range(&two, &nm1);
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// x <- a^d mod n
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let mut x = a.modexp(&d, self);
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// if x = 1 or x = n - 1 then
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if (&x == &one) || (&x == &nm1) {
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// continue WitnessLoop
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continue 'WitnessLoop;
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}
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// repeat r - 1 times:
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for _i in 0..r {
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// x <- x^2 mod n
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x = x.modexp(&two, self);
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// if x = 1 then
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if &x == &one {
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// return composite
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return false;
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}
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// if x = n - 1 then
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if &x == &nm1 {
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// continue WitnessLoop
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continue 'WitnessLoop;
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}
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}
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// return composite
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return false;
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}
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// return probably prime
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true
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}
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fn is_multiple_of(&self, x: u64) -> bool
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{
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(self % $name::from(x)).is_zero()
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}
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}
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};
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}
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