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//! number theory algorithms on primality
pub mod test {
//! primality test algorithms (not unit tests)
use crate::rng_static_xorshift64::static_xorshift64;
/// pre-generate random bases for some tests.
pub fn rng_bases(k: usize) -> Vec<u64> {
(0..k).map(|_| static_xorshift64()).collect()
}
pub(crate) mod test_cases {
#[allow(dead_code)]
pub(crate) const P: &'static [u64] = &[2, 998_244_353, 1_000_000_007];
#[allow(dead_code)]
pub(crate) const NP: &'static [u64] = &[0, 1, 561, 512_461];
}
pub(crate) fn trivial_primality(n: u64) -> Option<bool> {
if n == 2 {
return Some(true);
}
if n < 2 || n & 1 == 0 {
return Some(false);
}
None
}
/// n is composite but b^(n-1) = 1 (mod n) for all b such gcd(b, n) = 1
pub const CARMICHAEL_NUMS: &'static [u64] = &[
561, 1105, 1729, 2465, 2821, 6601, 8911, 10585, 15841, 29341, 41041,
46657, 52633, 62745, 63973, 75361, 101101, 115921, 126217, 162401,
172081, 188461, 252601, 278545, 294409, 314821, 334153, 340561, 399001,
410041, 449065, 488881, 512461,
];
pub mod mr {
//! Miller Rabin's Test
pub mod bases {
//! deterministic Miller Rabin bases
pub const B32: [u32; 3] = [2, 7, 61];
pub const B64_12: [u64; 12] =
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37];
pub const B64_7: [u64; 7] =
[2, 325, 9_375, 28_178, 450_775, 9_780_504, 1_795_265_022];
}
pub fn is_p(n: u64) -> bool {
use crate::{
montgomery_modular_multiplication_64::*,
power::pow_semigroup,
};
if let Some(bl) = super::trivial_primality(n) {
return bl;
}
let s = (n - 1).trailing_zeros();
let d = (n - 1) >> s;
// n - 1 = 2^s*d
let m = MontgomeryMultiplication64::new(n);
let mul = |x, y| m.mul(x, y);
let is_c = |b| -> bool {
let mut x = pow_semigroup(&mul, b, d);
if x == 1 {
return false;
}
for _ in 0..s {
if x == n - 1 {
return false;
}
x = mul(x, x);
}
true
};
bases::B64_7
.iter()
.map(|&b| b % n)
.filter(|&b| 2 <= b && b < n - 1)
.all(|b| !is_c(b))
}
#[test]
fn test_mr() {
use super::test_cases::*;
for x in P.into_iter() {
assert!(is_p(*x));
}
for x in NP.into_iter() {
assert!(!is_p(*x));
}
}
}
pub mod solovay_strassen {
//! solovay strassen's test
use crate::{
euler_criterion::try_euler_criterion,
jacobi_symbol::jacobi_symbol,
};
pub fn is_p(
b: &[u64],
n: u64,
) -> bool {
if let Some(bl) = super::trivial_primality(n) {
return bl;
}
// compare jcobi symbol and euler's criterion.
let is_c = |b| -> bool {
let j = jacobi_symbol(n, b);
if j == 0 {
return true;
}
if let Ok(e) = try_euler_criterion(n, b) {
let j = if j == 1 { 1 } else { n - 1 };
e != j
} else {
true
}
};
b.iter()
.map(|&b| b % n)
.filter(|&b| 2 <= b && b < n)
.all(|b| !is_c(b))
}
#[test]
fn test() {
use super::{
rng_bases,
test_cases::*,
};
let bases = rng_bases(20);
for x in P.into_iter() {
assert!(is_p(&bases, *x));
}
for x in NP.into_iter() {
assert!(!is_p(&bases, *x));
}
}
}
pub mod miller_rabin_solovay_strassen {
//! Miller Rabin - Solovay Strassen's Test
}
pub mod fermat {
//! Fermat's Test
use crate::{
greatest_common_divisor_euclidean::gcd,
montgomery_modular_multiplication_64::MontgomeryMultiplication64,
power::pow_semigroup,
};
pub fn is_p(
b: &[u64],
n: u64,
) -> bool {
if let Some(bl) = super::trivial_primality(n) {
return bl;
}
let m = MontgomeryMultiplication64::new(n);
let mul = |x, y| m.mul(x, y);
b.iter()
.map(|&b| b % n)
.filter(|&b| 2 <= b && b < n - 1)
.all(|b| gcd(b, n) == 1 && pow_semigroup(&mul, b, n - 1) == 1)
}
#[test]
fn test() {
use super::{
rng_bases,
test_cases::*,
};
let bases = rng_bases(30);
for x in P.into_iter() {
assert!(is_p(&bases, *x));
}
for x in NP.into_iter() {
assert!(!is_p(&bases, *x));
}
for x in super::CARMICHAEL_NUMS.iter() {
assert!(!is_p(&bases, *x));
}
}
}
pub mod adleman_pomerance_rumely {}
pub mod aks {
//! Agrawal, Kayal, Sexena test
// TODO:
}
pub mod atkin_morain_elliptic_curve {}
pub mod baillie_psw {}
pub mod elliptic_curve {}
pub mod frobenius {}
pub mod goldwasser_kilian {}
pub mod lucas {}
pub mod lucas_lehmer {}
pub mod lucas_lehmer_reisel {}
pub mod miller {}
pub mod pepin {}
pub mod pocklington {}
pub mod proth_theorem {}
pub mod trial_division {
// TODO
// use prime factorize trial division.
// kind of factors is just one.
}
pub mod wilson_theorem {}
}