slow_primes 0.1.14

Deprecated in favour of `primal`. A library to generate, identify and handle prime numbers and related properties. This library includes slow enumeration of primes up to a bound, slow factorisation of arbitrary numbers, fast primality tests and state-of-the-art estimation of upper and lower bounds for π(n) (the number of primes below n) and p_k (the k-th prime).
Documentation 
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91

fn mod_exp(mut x: u64, mut d: u64, n: u64) -> u64 {
    let mut ret = 1;
    while d != 0 {
        if d % 2 == 1 {
            ret *= x;
            if ret >= n {
                ret %= n;
            }
        }
        d /= 2;
        x *= x;
        if x >= n {
            x %= n;
        }
    }
    ret
}

/// Test if `n` is prime, using the deterministic version of the
/// Miller-Rabin test.
///
/// Doing a lot of primality tests with numbers strictly below some
/// upper bound will be faster using the `is_prime` method of a
/// `Primes` instance.
pub fn is_prime_miller_rabin(n: u64) -> bool {
    const HINT: &'static [u64] = &[2];

    // we have a strict upper bound, so we can just use the witness
    // table of Pomerance, Selfridge & Wagstaff and Jeaschke to be as
    // efficient as possible, without having to fall back to
    // randomness.
    const WITNESSES: &'static [(u64, &'static [u64])] =
        &[(2_046, HINT),
          (1_373_652, &[2, 3]),
          (9_080_190, &[31, 73]),
          (25_326_000, &[2, 3, 5]),
          (4_759_123_140, &[2, 7, 61]),
          (1_112_004_669_632, &[2, 13, 23, 1662803]),
          (2_152_302_898_746, &[2, 3, 5, 7, 11]),
          (3_474_749_660_382, &[2, 3, 5, 7, 11, 13]),
          (341_550_071_728_320, &[2, 3, 5, 7, 11, 13, 17]),
          (0xFFFF_FFFF_FFFF_FFFF, &[2, 3, 5, 7, 11, 13, 17, 19, 23])
         ];

    if n % 2 == 0 { return n == 2 }
    if n == 1 { return false }

    let mut d = n - 1;
    let mut s = 0;
    while d % 2 == 0 { d /= 2; s += 1 }

    let witnesses =
        WITNESSES.iter().find(|&&(hi, _)| hi >= n)
            .map(|&(_, wtnss)| wtnss).unwrap();
    'next_witness: for &a in witnesses.iter() {
        let mut power = mod_exp(a, d, n);
        if power == 1 { continue 'next_witness }

        for _r in 0..s {
            if power == n - 1 {
                continue 'next_witness
            }
            power *= power;
            if power >= n {
                power %= n;
            }
        }
        return false
    }

    true
}

#[cfg(test)]
mod tests {
    use Primes;
    use super::is_prime_miller_rabin;

    #[test]
    fn miller_rabin() {
        const LIMIT: usize = 1_000_000;
        let sieve = Primes::sieve(LIMIT);
        for x in 0..LIMIT {
            let s = sieve.is_prime(x);
            let mr = is_prime_miller_rabin(x as u64);

            assert!(s == mr, "miller_rabin {} mismatches sieve {} for {}",
                    mr, s, x)
        }
    }
}