dashu-int 0.6.2

Arbitrary-precision integer math library for Rust, balancing ergonomics and efficiency. Provides UBig and IBig (unsigned and signed), with small values inlined on the stack. Efficient arithmetic, modular arithmetic, and number theory (pow, ilog, gcd, gcd_ext); two's-complement bit operations; parsing and formatting in base 2-36; optional serde, rand, num-traits, rkyv, and zeroize.
Documentation 
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//! Implement num-modular traits.
//!
//! This module implements [ModularSymbols] (the Jacobi/Legendre/Kronecker
//! symbols) for [UBig]. Together with the `Reducer` implementations for the
//! rings of the [modular](crate::modular) and [monty](crate::monty) modules,
//! it makes `UBig` usable as the base integer type for the carried-modulus
//! integers of the `num-modular` ecosystem (e.g. `num_prime::mint::Mint`).

use num_modular::ModularSymbols;

use crate::{fast_div::ConstDivisor, ubig::UBig};
use dashu_base::BitTest;

impl ModularSymbols<&UBig> for UBig {
    #[inline]
    fn checked_legendre(&self, n: &UBig) -> Option<i8> {
        let ring = ConstDivisor::new(n.clone());
        let r = ring
            .reduce(self.clone())
            .pow(&((n - UBig::ONE) >> 1))
            .residue();
        if r.is_zero() {
            Some(0)
        } else if r == UBig::ONE {
            Some(1)
        } else if r + UBig::ONE == *n {
            Some(-1)
        } else {
            None
        }
    }

    fn checked_jacobi(&self, n: &UBig) -> Option<i8> {
        // the Jacobi symbol is only defined for positive odd integers
        if !n.bit(0) {
            return None;
        }
        if self.is_zero() {
            return Some(if n == &UBig::ONE { 1 } else { 0 });
        }
        if self == &UBig::ONE {
            return Some(1);
        }

        let mut a = self % n;
        let mut n = n.clone();
        let mut t: i8 = 1;
        while !a.is_zero() {
            // strip factors of two from a, tracking their effect on the symbol:
            // (2|n) = -1 iff n mod 8 in {3,5}, i.e. bit 0 set and bit 1 != bit 2
            let s = a.trailing_zeros().unwrap_or(0);
            if s > 0 {
                if s & 1 == 1 && n.bit(1) != n.bit(2) {
                    t = -t;
                }
                a = &a >> s;
            }

            // (a|n) = (n|a) * (-1)^((a-1)/2 * (n-1)/2), the extra sign is -1
            // only if both a and n are 3 mod 4
            core::mem::swap(&mut a, &mut n);
            if a.bit(0) && a.bit(1) && n.bit(0) && n.bit(1) {
                t = -t;
            }
            a %= &n;
        }
        Some(if n == UBig::ONE { t } else { 0 })
    }

    fn kronecker(&self, n: &UBig) -> i8 {
        if n.is_zero() {
            return if self == &UBig::ONE { 1 } else { 0 };
        }
        if n == &UBig::ONE {
            return 1;
        }
        if self.is_zero() {
            return 0;
        }
        if *n == UBig::from(2u8) {
            // (a|2) = 1 if a mod 8 in {1,7}, -1 if in {3,5}, 0 if a is even
            return if !self.bit(0) {
                0
            } else if self.bit(1) == self.bit(2) {
                1
            } else {
                -1
            };
        }

        let s = n.trailing_zeros().unwrap_or(0);
        if s == 0 {
            // n is odd, the Kronecker symbol equals the Jacobi symbol
            return self.checked_jacobi(n).unwrap_or(0);
        }
        let n_odd = n >> s;
        let t1 = self.kronecker(&UBig::from(2u8));
        if t1 == 0 {
            return 0;
        }
        let t2 = self.checked_jacobi(&n_odd).unwrap_or(0);
        if s & 1 == 0 {
            t2
        } else {
            t1 * t2
        }
    }
}

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

    fn u(v: u8) -> UBig {
        UBig::from(v)
    }

    #[test]
    fn jacobi_symbol() {
        // hand-computed cases, cross-checked by Euler's criterion
        assert_eq!(u(2).checked_jacobi(&u(5)), Some(-1));
        assert_eq!(u(2).checked_jacobi(&u(7)), Some(1));
        assert_eq!(u(2).checked_jacobi(&u(9)), Some(1)); // (2|3)^2
        assert_eq!(u(3).checked_jacobi(&u(5)), Some(-1));
        assert_eq!(u(4).checked_jacobi(&u(5)), Some(1));
        assert_eq!(u(5).checked_jacobi(&u(5)), Some(0));
        assert_eq!(u(6).checked_jacobi(&u(7)), Some(-1)); // 6 = -1 (mod 7)
        assert_eq!(u(5).checked_jacobi(&u(7)), Some(-1));
        assert_eq!(u(2).checked_jacobi(&u(15)), Some(1)); // (2|3)(2|5)
        assert_eq!(u(14).checked_jacobi(&u(15)), Some(-1));
        assert_eq!(u(0).checked_jacobi(&u(9)), Some(0));
        assert_eq!(u(0).checked_jacobi(&u(1)), Some(1));
        assert_eq!(u(1).checked_jacobi(&u(9)), Some(1));

        // not defined for even modulus
        assert_eq!(u(3).checked_jacobi(&u(8)), None);

        // reciprocity: (a|n)(n|a) = -1 when both are 3 mod 4, +1 otherwise
        assert_eq!(u(5).checked_jacobi(&u(21)), u(21).checked_jacobi(&u(5)));
        assert_eq!(u(11).checked_jacobi(&u(13)), u(13).checked_jacobi(&u(11)));
        assert_eq!(u(13).checked_jacobi(&u(17)), u(17).checked_jacobi(&u(13)));
        assert_eq!(u(7).checked_jacobi(&u(15)), Some(-1)); // (7|3)(7|5)
        assert_eq!(u(15).checked_jacobi(&u(7)), Some(1));

        // multiplicativity in the first argument
        assert_eq!(
            u(6).checked_jacobi(&u(35)),
            Some(u(2).checked_jacobi(&u(35)).unwrap() * u(3).checked_jacobi(&u(35)).unwrap())
        );
    }

    #[test]
    fn legendre_symbol() {
        // 7 is prime
        assert_eq!(u(2).checked_legendre(&u(7)), Some(1));
        assert_eq!(u(3).checked_legendre(&u(7)), Some(-1));
        assert_eq!(u(7).checked_legendre(&u(7)), Some(0));
        // quadratic residues mod 11: 1^2..5^2 = 1,4,9,5,3
        for a in [1u8, 3, 4, 5, 9] {
            assert_eq!(u(a).checked_legendre(&u(11)), Some(1), "a={a}");
        }
        for a in [2u8, 6, 7, 8, 10] {
            assert_eq!(u(a).checked_legendre(&u(11)), Some(-1), "a={a}");
        }
    }

    #[test]
    fn kronecker_symbol() {
        assert_eq!(u(1).kronecker(&u(0)), 1);
        assert_eq!(u(5).kronecker(&u(0)), 0);
        assert_eq!(u(5).kronecker(&u(1)), 1);
        assert_eq!(u(3).kronecker(&u(2)), -1);
        assert_eq!(u(7).kronecker(&u(2)), 1);
        assert_eq!(u(1).kronecker(&u(2)), 1);
        assert_eq!(u(5).kronecker(&u(2)), -1);
        assert_eq!(u(4).kronecker(&u(2)), 0); // even self
        assert_eq!(u(3).kronecker(&u(8)), -1); // (3|2)^3
        assert_eq!(u(5).kronecker(&u(8)), -1); // (5|2)^3
        assert_eq!(u(4).kronecker(&u(8)), 0);

        // consistency with the Jacobi symbol for odd n
        assert_eq!(u(6).kronecker(&u(35)), u(6).checked_jacobi(&u(35)).unwrap());
    }
}