num_bigint_dig/

prime.rs

// https://github.com/RustCrypto/RSA/blob/master/src/prime.rs
//! Implements probabilistic prime checkers.

use integer::Integer;
use num_traits::{FromPrimitive, One, ToPrimitive, Zero};
use rand::rngs::StdRng;
use rand::SeedableRng;

use crate::algorithms::jacobi;
use crate::big_digit;
use crate::bigrand::RandBigInt;
use crate::Sign::Plus;
use crate::{BigInt, BigUint, IntoBigUint};

lazy_static! {
    pub(crate) static ref BIG_1: BigUint = BigUint::one();
    pub(crate) static ref BIG_2: BigUint = BigUint::from_u64(2).unwrap();
    pub(crate) static ref BIG_3: BigUint = BigUint::from_u64(3).unwrap();
    pub(crate) static ref BIG_64: BigUint = BigUint::from_u64(64).unwrap();
}

const PRIMES_A: u64 = 3 * 5 * 7 * 11 * 13 * 17 * 19 * 23 * 37;
const PRIMES_B: u64 = 29 * 31 * 41 * 43 * 47 * 53;

/// Records the primes < 64.
const PRIME_BIT_MASK: u64 = 1 << 2
    | 1 << 3
    | 1 << 5
    | 1 << 7
    | 1 << 11
    | 1 << 13
    | 1 << 17
    | 1 << 19
    | 1 << 23
    | 1 << 29
    | 1 << 31
    | 1 << 37
    | 1 << 41
    | 1 << 43
    | 1 << 47
    | 1 << 53
    | 1 << 59
    | 1 << 61;

/// ProbablyPrime reports whether x is probably prime,
/// applying the Miller-Rabin test with n pseudorandomly chosen bases
/// as well as a Baillie-PSW test.
///
/// If x is prime, ProbablyPrime returns true.
/// If x is chosen randomly and not prime, ProbablyPrime probably returns false.
/// The probability of returning true for a randomly chosen non-prime is at most ¼ⁿ.
///
/// ProbablyPrime is 100% accurate for inputs less than 2⁶⁴.
/// See Menezes et al., Handbook of Applied Cryptography, 1997, pp. 145-149,
/// and FIPS 186-4 Appendix F for further discussion of the error probabilities.
///
/// ProbablyPrime is not suitable for judging primes that an adversary may
/// have crafted to fool the test.
///
/// This is a port of `ProbablyPrime` from the go std lib.
pub fn probably_prime(x: &BigUint, n: usize) -> bool {
    if x.is_zero() {
        return false;
    }

    if x < &*BIG_64 {
        return (PRIME_BIT_MASK & (1 << x.to_u64().unwrap())) != 0;
    }

    if x.is_even() {
        return false;
    }

    let r_a = &(x % PRIMES_A);
    let r_b = &(x % PRIMES_B);

    if (r_a % 3u32).is_zero()
        || (r_a % 5u32).is_zero()
        || (r_a % 7u32).is_zero()
        || (r_a % 11u32).is_zero()
        || (r_a % 13u32).is_zero()
        || (r_a % 17u32).is_zero()
        || (r_a % 19u32).is_zero()
        || (r_a % 23u32).is_zero()
        || (r_a % 37u32).is_zero()
        || (r_b % 29u32).is_zero()
        || (r_b % 31u32).is_zero()
        || (r_b % 41u32).is_zero()
        || (r_b % 43u32).is_zero()
        || (r_b % 47u32).is_zero()
        || (r_b % 53u32).is_zero()
    {
        return false;
    }

    probably_prime_miller_rabin(x, n + 1, true) && probably_prime_lucas(x)
}

const NUMBER_OF_PRIMES: usize = 127;
const PRIME_GAP: [u64; 167] = [
    2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, 4, 6,
    2, 10, 2, 6, 6, 4, 6, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4, 6, 2, 10, 6, 6, 6, 2, 6, 4, 2, 10,
    14, 4, 2, 4, 14, 6, 10, 2, 4, 6, 8, 6, 6, 4, 6, 8, 4, 8, 10, 2, 10, 2, 6, 4, 6, 8, 4, 2, 4, 12,
    8, 4, 8, 4, 6, 12, 2, 18, 6, 10, 6, 6, 2, 6, 10, 6, 6, 2, 6, 6, 4, 2, 12, 10, 2, 4, 6, 6, 2,
    12, 4, 6, 8, 10, 8, 10, 8, 6, 6, 4, 8, 6, 4, 8, 4, 14, 10, 12, 2, 10, 2, 4, 2, 10, 14, 4, 2, 4,
    14, 4, 2, 4, 20, 4, 8, 10, 8, 4, 6, 6, 14, 4, 6, 6, 8, 6, 12,
];

const INCR_LIMIT: usize = 0x10000;

/// Calculate the next larger prime, given a starting number `n`.
pub fn next_prime(n: &BigUint) -> BigUint {
    if n < &*BIG_2 {
        return 2u32.into_biguint().unwrap();
    }

    // We want something larger than our current number.
    let mut res = n + &*BIG_1;

    // Ensure we are odd.
    res |= &*BIG_1;

    // Handle values up to 7.
    if let Some(val) = res.to_u64() {
        if val < 7 {
            return res;
        }
    }

    let nbits = res.bits();
    let prime_limit = if nbits / 2 >= NUMBER_OF_PRIMES {
        NUMBER_OF_PRIMES - 1
    } else {
        nbits / 2
    };

    // Compute the residues modulo small odd primes
    let mut moduli = vec![BigUint::zero(); prime_limit];

    'outer: loop {
        let mut prime = 3;
        for i in 0..prime_limit {
            moduli[i] = &res % prime;
            prime += PRIME_GAP[i];
        }

        // Check residues
        let mut difference: usize = 0;
        for incr in (0..INCR_LIMIT as u64).step_by(2) {
            let mut prime: u64 = 3;

            let mut cancel = false;
            for i in 0..prime_limit {
                let r = (&moduli[i] + incr) % prime;
                prime += PRIME_GAP[i];

                if r.is_zero() {
                    cancel = true;
                    break;
                }
            }

            if !cancel {
                res += difference;
                difference = 0;
                if probably_prime(&res, 20) {
                    break 'outer;
                }
            }

            difference += 2;
        }

        res += difference;
    }

    res
}

/// Reports whether n passes reps rounds of the Miller-Rabin primality test, using pseudo-randomly chosen bases.
/// If `force2` is true, one of the rounds is forced to use base 2.
///
/// See Handbook of Applied Cryptography, p. 139, Algorithm 4.24.
pub fn probably_prime_miller_rabin(n: &BigUint, reps: usize, force2: bool) -> bool {
    // println!("miller-rabin: {}", n);
    let nm1 = n - &*BIG_1;
    // determine q, k such that nm1 = q << k
    let k = nm1.trailing_zeros().unwrap() as usize;
    let q = &nm1 >> k;

    let nm3 = n - &*BIG_2;

    let mut rng = StdRng::seed_from_u64(n.get_limb(0) as u64);

    'nextrandom: for i in 0..reps {
        let x = if i == reps - 1 && force2 {
            BIG_2.clone()
        } else {
            rng.gen_biguint_below(&nm3) + &*BIG_2
        };

        let mut y = x.modpow(&q, n);
        if y.is_one() || y == nm1 {
            continue;
        }

        for _ in 1..k {
            y = y.modpow(&*BIG_2, n);
            if y == nm1 {
                continue 'nextrandom;
            }
            if y.is_one() {
                return false;
            }
        }
        return false;
    }

    true
}

/// Reports whether n passes the "almost extra strong" Lucas probable prime test,
/// using Baillie-OEIS parameter selection. This corresponds to "AESLPSP" on Jacobsen's tables (link below).
/// The combination of this test and a Miller-Rabin/Fermat test with base 2 gives a Baillie-PSW test.
///
///
/// References:
///
/// Baillie and Wagstaff, "Lucas Pseudoprimes", Mathematics of Computation 35(152),
/// October 1980, pp. 1391-1417, especially page 1401.
/// http://www.ams.org/journals/mcom/1980-35-152/S0025-5718-1980-0583518-6/S0025-5718-1980-0583518-6.pdf
///
/// Grantham, "Frobenius Pseudoprimes", Mathematics of Computation 70(234),
/// March 2000, pp. 873-891.
/// http://www.ams.org/journals/mcom/2001-70-234/S0025-5718-00-01197-2/S0025-5718-00-01197-2.pdf
///
/// Baillie, "Extra strong Lucas pseudoprimes", OEIS A217719, https://oeis.org/A217719.
///
/// Jacobsen, "Pseudoprime Statistics, Tables, and Data", http://ntheory.org/pseudoprimes.html.
///
/// Nicely, "The Baillie-PSW Primality Test", http://www.trnicely.net/misc/bpsw.html.
/// (Note that Nicely's definition of the "extra strong" test gives the wrong Jacobi condition,
/// as pointed out by Jacobsen.)
///
/// Crandall and Pomerance, Prime Numbers: A Computational Perspective, 2nd ed.
/// Springer, 2005.
pub fn probably_prime_lucas(n: &BigUint) -> bool {
    // println!("lucas: {}", n);
    // Discard 0, 1.
    if n.is_zero() || n.is_one() {
        return false;
    }

    // Two is the only even prime.
    if n.to_u64() == Some(2) {
        return false;
    }

    // Baillie-OEIS "method C" for choosing D, P, Q,
    // as in https://oeis.org/A217719/a217719.txt:
    // try increasing P ≥ 3 such that D = P² - 4 (so Q = 1)
    // until Jacobi(D, n) = -1.
    // The search is expected to succeed for non-square n after just a few trials.
    // After more than expected failures, check whether n is square
    // (which would cause Jacobi(D, n) = 1 for all D not dividing n).
    let mut p = 3u64;
    let n_int = BigInt::from_biguint(Plus, n.clone());

    loop {
        if p > 10000 {
            // This is widely believed to be impossible.
            // If we get a report, we'll want the exact number n.
            panic!("internal error: cannot find (D/n) = -1 for {:?}", n)
        }

        let d_int = BigInt::from_u64(p * p - 4).unwrap();
        let j = jacobi(&d_int, &n_int);

        if j == -1 {
            break;
        }
        if j == 0 {
            // d = p²-4 = (p-2)(p+2).
            // If (d/n) == 0 then d shares a prime factor with n.
            // Since the loop proceeds in increasing p and starts with p-2==1,
            // the shared prime factor must be p+2.
            // If p+2 == n, then n is prime; otherwise p+2 is a proper factor of n.
            return n_int.to_i64() == Some(p as i64 + 2);
        }
        if p == 40 {
            // We'll never find (d/n) = -1 if n is a square.
            // If n is a non-square we expect to find a d in just a few attempts on average.
            // After 40 attempts, take a moment to check if n is indeed a square.
            let t1 = n.sqrt();
            let t1 = &t1 * &t1;
            if &t1 == n {
                return false;
            }
        }

        p += 1;
    }

    // Grantham definition of "extra strong Lucas pseudoprime", after Thm 2.3 on p. 876
    // (D, P, Q above have become Δ, b, 1):
    //
    // Let U_n = U_n(b, 1), V_n = V_n(b, 1), and Δ = b²-4.
    // An extra strong Lucas pseudoprime to base b is a composite n = 2^r s + Jacobi(Δ, n),
    // where s is odd and gcd(n, 2*Δ) = 1, such that either (i) U_s ≡ 0 mod n and V_s ≡ ±2 mod n,
    // or (ii) V_{2^t s} ≡ 0 mod n for some 0 ≤ t < r-1.
    //
    // We know gcd(n, Δ) = 1 or else we'd have found Jacobi(d, n) == 0 above.
    // We know gcd(n, 2) = 1 because n is odd.
    //
    // Arrange s = (n - Jacobi(Δ, n)) / 2^r = (n+1) / 2^r.
    let mut s = n + &*BIG_1;
    let r = s.trailing_zeros().unwrap() as usize;
    s = &s >> r;
    let nm2 = n - &*BIG_2; // n - 2

    // We apply the "almost extra strong" test, which checks the above conditions
    // except for U_s ≡ 0 mod n, which allows us to avoid computing any U_k values.
    // Jacobsen points out that maybe we should just do the full extra strong test:
    // "It is also possible to recover U_n using Crandall and Pomerance equation 3.13:
    // U_n = D^-1 (2V_{n+1} - PV_n) allowing us to run the full extra-strong test
    // at the cost of a single modular inversion. This computation is easy and fast in GMP,
    // so we can get the full extra-strong test at essentially the same performance as the
    // almost extra strong test."

    // Compute Lucas sequence V_s(b, 1), where:
    //
    //	V(0) = 2
    //	V(1) = P
    //	V(k) = P V(k-1) - Q V(k-2).
    //
    // (Remember that due to method C above, P = b, Q = 1.)
    //
    // In general V(k) = α^k + β^k, where α and β are roots of x² - Px + Q.
    // Crandall and Pomerance (p.147) observe that for 0 ≤ j ≤ k,
    //
    //	V(j+k) = V(j)V(k) - V(k-j).
    //
    // So in particular, to quickly double the subscript:
    //
    //	V(2k) = V(k)² - 2
    //	V(2k+1) = V(k) V(k+1) - P
    //
    // We can therefore start with k=0 and build up to k=s in log₂(s) steps.
    let mut vk = BIG_2.clone();
    let mut vk1 = BigUint::from_u64(p).unwrap();

    for i in (0..s.bits()).rev() {
        if is_bit_set(&s, i) {
            // k' = 2k+1
            // V(k') = V(2k+1) = V(k) V(k+1) - P
            let t1 = (&vk * &vk1) + n - p;
            vk = &t1 % n;
            // V(k'+1) = V(2k+2) = V(k+1)² - 2
            let t1 = (&vk1 * &vk1) + &nm2;
            vk1 = &t1 % n;
        } else {
            // k' = 2k
            // V(k'+1) = V(2k+1) = V(k) V(k+1) - P
            let t1 = (&vk * &vk1) + n - p;
            vk1 = &t1 % n;
            // V(k') = V(2k) = V(k)² - 2
            let t1 = (&vk * &vk) + &nm2;
            vk = &t1 % n;
        }
    }

    // Now k=s, so vk = V(s). Check V(s) ≡ ±2 (mod n).
    if vk.to_u64() == Some(2) || vk == nm2 {
        // Check U(s) ≡ 0.
        // As suggested by Jacobsen, apply Crandall and Pomerance equation 3.13:
        //
        //	U(k) = D⁻¹ (2 V(k+1) - P V(k))
        //
        // Since we are checking for U(k) == 0 it suffices to check 2 V(k+1) == P V(k) mod n,
        // or P V(k) - 2 V(k+1) == 0 mod n.
        let mut t1 = &vk * p;
        let mut t2 = &vk1 << 1;

        if t1 < t2 {
            core::mem::swap(&mut t1, &mut t2);
        }

        t1 -= t2;

        if (t1 % n).is_zero() {
            return true;
        }
    }

    // Check V(2^t s) ≡ 0 mod n for some 0 ≤ t < r-1.
    for _ in 0..r - 1 {
        if vk.is_zero() {
            return true;
        }

        // Optimization: V(k) = 2 is a fixed point for V(k') = V(k)² - 2,
        // so if V(k) = 2, we can stop: we will never find a future V(k) == 0.
        if vk.to_u64() == Some(2) {
            return false;
        }

        // k' = 2k
        // V(k') = V(2k) = V(k)² - 2
        let t1 = (&vk * &vk) - &*BIG_2;
        vk = &t1 % n;
    }

    false
}

/// Checks if the i-th bit is set
#[inline]
fn is_bit_set(x: &BigUint, i: usize) -> bool {
    get_bit(x, i) == 1
}

/// Returns the i-th bit.
#[inline]
fn get_bit(x: &BigUint, i: usize) -> u8 {
    let j = i / big_digit::BITS;
    // if is out of range of the set words, it is always false.
    if i >= x.bits() {
        return 0;
    }

    (x.get_limb(j) >> (i % big_digit::BITS) & 1) as u8
}

#[cfg(test)]
mod tests {
    use super::*;
    use alloc::vec::Vec;
    // use RandBigInt;

    use crate::biguint::ToBigUint;

    lazy_static! {
        static ref PRIMES: Vec<&'static str> = vec![
        "2",
        "3",
        "5",
        "7",
        "11",

        "13756265695458089029",
        "13496181268022124907",
        "10953742525620032441",
        "17908251027575790097",

        // https://golang.org/issue/638
        "18699199384836356663",

        "98920366548084643601728869055592650835572950932266967461790948584315647051443",
        "94560208308847015747498523884063394671606671904944666360068158221458669711639",

        // http://primes.utm.edu/lists/small/small3.html
        "449417999055441493994709297093108513015373787049558499205492347871729927573118262811508386655998299074566974373711472560655026288668094291699357843464363003144674940345912431129144354948751003607115263071543163",
        "230975859993204150666423538988557839555560243929065415434980904258310530753006723857139742334640122533598517597674807096648905501653461687601339782814316124971547968912893214002992086353183070342498989426570593",
        "5521712099665906221540423207019333379125265462121169655563495403888449493493629943498064604536961775110765377745550377067893607246020694972959780839151452457728855382113555867743022746090187341871655890805971735385789993",
        "203956878356401977405765866929034577280193993314348263094772646453283062722701277632936616063144088173312372882677123879538709400158306567338328279154499698366071906766440037074217117805690872792848149112022286332144876183376326512083574821647933992961249917319836219304274280243803104015000563790123",
        // ECC primes: http://tools.ietf.org/html/draft-ladd-safecurves-02
        "3618502788666131106986593281521497120414687020801267626233049500247285301239",                                                                                  // Curve1174: 2^251-9
        "57896044618658097711785492504343953926634992332820282019728792003956564819949",                                                                                 // Curve25519: 2^255-19
        "9850501549098619803069760025035903451269934817616361666987073351061430442874302652853566563721228910201656997576599",                                           // E-382: 2^382-105
        "42307582002575910332922579714097346549017899709713998034217522897561970639123926132812109468141778230245837569601494931472367",                                 // Curve41417: 2^414-17
        "6864797660130609714981900799081393217269435300143305409394463459185543183397656052122559640661454554977296311391480858037121987999716643812574028291115057151", // E-521: 2^521-1
        ];

        static ref COMPOSITES: Vec<&'static str> = vec![
            "0",
            "1",

            "21284175091214687912771199898307297748211672914763848041968395774954376176754",
            "6084766654921918907427900243509372380954290099172559290432744450051395395951",
            "84594350493221918389213352992032324280367711247940675652888030554255915464401",
            "82793403787388584738507275144194252681",

            // Arnault, "Rabin-Miller Primality Test: Composite Numbers Which Pass It",
            // Mathematics of Computation, 64(209) (January 1995), pp. 335-361.
            "1195068768795265792518361315725116351898245581", // strong pseudoprime to prime bases 2 through 29
            // strong pseudoprime to all prime bases up to 200
            "8038374574536394912570796143419421081388376882875581458374889175222974273765333652186502336163960045457915042023603208766569966760987284043965408232928738791850869166857328267761771029389697739470167082304286871099974399765441448453411558724506334092790222752962294149842306881685404326457534018329786111298960644845216191652872597534901",

            // Extra-strong Lucas pseudoprimes. https://oeis.org/A217719
            "989",
            "3239",
            "5777",
            "10877",
            "27971",
            "29681",
            "30739",
            "31631",
            "39059",
            "72389",
            "73919",
            "75077",
            "100127",
            "113573",
            "125249",
            "137549",
            "137801",
            "153931",
            "155819",
            "161027",
            "162133",
            "189419",
            "218321",
            "231703",
            "249331",
            "370229",
            "429479",
            "430127",
            "459191",
            "473891",
            "480689",
            "600059",
            "621781",
            "632249",
            "635627",

            "3673744903",
            "3281593591",
            "2385076987",
            "2738053141",
            "2009621503",
            "1502682721",
            "255866131",
            "117987841",
            "587861",

            "6368689",
            "8725753",
            "80579735209",
            "105919633",
        ];

        // Test Cases from #51
        static ref ISSUE_51: Vec<&'static str> = vec![
            "1579751",
            "1884791",
            "3818929",
            "4080359",
            "4145951",
        ];
    }

    #[test]
    fn test_primes() {
        for prime in PRIMES.iter() {
            let p = BigUint::parse_bytes(prime.as_bytes(), 10).unwrap();
            for i in [0, 1, 20].iter() {
                assert!(
                    probably_prime(&p, *i as usize),
                    "{} is a prime ({})",
                    prime,
                    i,
                );
            }
        }
    }

    #[test]
    fn test_composites() {
        for comp in COMPOSITES.iter() {
            let p = BigUint::parse_bytes(comp.as_bytes(), 10).unwrap();
            for i in [0, 1, 20].iter() {
                assert!(
                    !probably_prime(&p, *i as usize),
                    "{} is a composite ({})",
                    comp,
                    i,
                );
            }
        }
    }

    #[test]
    fn test_issue_51() {
        for num in ISSUE_51.iter() {
            let p = BigUint::parse_bytes(num.as_bytes(), 10).unwrap();
            assert!(probably_prime(&p, 20), "{} is a prime number", num);
        }
    }

    macro_rules! test_pseudo_primes {
        ($name:ident, $cond:expr, $want:expr) => {
            #[test]
            fn $name() {
                let mut i = 3;
                let mut want = $want;
                while i < 100000 {
                    let n = BigUint::from_u64(i).unwrap();
                    let pseudo = $cond(&n);
                    if pseudo && (want.is_empty() || i != want[0]) {
                        panic!("cond({}) = true, want false", i);
                    } else if !pseudo && !want.is_empty() && i == want[0] {
                        panic!("cond({}) = false, want true", i);
                    }
                    if !want.is_empty() && i == want[0] {
                        want = want[1..].to_vec();
                    }
                    i += 2;
                }

                if !want.is_empty() {
                    panic!("forgot to test: {:?}", want);
                }
            }
        };
    }

    test_pseudo_primes!(
        test_probably_prime_miller_rabin,
        |n| probably_prime_miller_rabin(n, 1, true) && !probably_prime_lucas(n),
        vec![
            2047, 3277, 4033, 4681, 8321, 15841, 29341, 42799, 49141, 52633, 65281, 74665, 80581,
            85489, 88357, 90751,
        ]
    );

    test_pseudo_primes!(
        test_probably_prime_lucas,
        |n| probably_prime_lucas(n) && !probably_prime_miller_rabin(n, 1, true),
        vec![989, 3239, 5777, 10877, 27971, 29681, 30739, 31631, 39059, 72389, 73919, 75077,]
    );

    #[test]
    fn test_bit_set() {
        let v = &vec![0b10101001];
        let num = BigUint::from_slice(&v);
        assert!(is_bit_set(&num, 0));
        assert!(!is_bit_set(&num, 1));
        assert!(!is_bit_set(&num, 2));
        assert!(is_bit_set(&num, 3));
        assert!(!is_bit_set(&num, 4));
        assert!(is_bit_set(&num, 5));
        assert!(!is_bit_set(&num, 6));
        assert!(is_bit_set(&num, 7));
    }

    #[test]
    fn test_next_prime_basics() {
        let primes1 = (0..2048u32)
            .map(|i| next_prime(&i.to_biguint().unwrap()))
            .collect::<Vec<_>>();
        let primes2 = (0..2048u32)
            .map(|i| {
                let i = i.to_biguint().unwrap();
                let p = next_prime(&i);
                assert!(&p > &i);
                p
            })
            .collect::<Vec<_>>();

        for (p1, p2) in primes1.iter().zip(&primes2) {
            assert_eq!(p1, p2);
            assert!(probably_prime(p1, 25));
        }
    }

    #[test]
    fn test_next_prime_bug_44() {
        let i = 1032989.to_biguint().unwrap();
        let next = next_prime(&i);
        assert_eq!(1033001.to_biguint().unwrap(), next);
    }
}