qfall_math/integer_mod_q/polynomial_ring_zq/

norm.rs

// Copyright © 2025 Marcel Luca Schmidt
//
// This file is part of qFALL-math.
//
// qFALL-math is free software: you can redistribute it and/or modify it under
// the terms of the Mozilla Public License Version 2.0 as published by the
// Mozilla Foundation. See <https://mozilla.org/en-US/MPL/2.0/>.

//! This module includes functionality to compute several norms
//! defined on polynomials.

use super::PolynomialRingZq;
use crate::{
    integer::Z,
    integer_mod_q::fmpz_mod_helpers::length,
    traits::{GetCoefficient, Pow},
};
use std::cmp::max;

impl PolynomialRingZq {
    /// Returns the squared Euclidean norm or squared 2-norm of the given polynomial.
    /// The squared Euclidean norm for a polynomial is obtained by treating the coefficients
    /// of the polynomial as a vector and then applying the standard squared Euclidean norm.
    ///
    /// Each length of an entry in this vector is defined as the shortest distance
    /// to the next zero representative modulo q.
    ///
    /// # Examples
    /// ```
    /// use qfall_math::{integer::Z, integer_mod_q::PolynomialRingZq};
    /// use std::str::FromStr;
    ///
    /// let poly = PolynomialRingZq::from_str("3  1 2 3 / 4  1 2 3 4 mod 11").unwrap();
    ///
    /// let sqrd_2_norm = poly.norm_eucl_sqrd();
    ///
    /// // 1*1 + 2*2 + 3*3 = 14
    /// assert_eq!(Z::from(14), sqrd_2_norm);
    /// ```
    pub fn norm_eucl_sqrd(&self) -> Z {
        let mut res = Z::ZERO;
        for i in 0..=self.get_degree() {
            let coeff: Z = unsafe { self.get_coeff_unchecked(i) };
            res += length(&coeff.value, &self.modulus.get_fq_ctx().ctxp[0].n[0])
                .pow(2)
                .unwrap();
        }
        res
    }

    /// Returns the infinity norm or the maximal absolute value of a
    /// coefficient of the given polynomial.
    /// The infinity norm for a polynomial is obtained by treating the coefficients
    /// of the polynomial as a vector and then applying the standard infinity norm.
    ///
    /// Each length of an entry in this vector is defined as the shortest distance
    /// to the next zero representative modulo q.
    ///
    /// # Examples
    /// ```
    /// use qfall_math::{integer::Z, integer_mod_q::PolynomialRingZq};
    /// use std::str::FromStr;
    ///
    /// let poly = PolynomialRingZq::from_str("3  1 2 4 / 4  1 2 3 4 mod 7").unwrap();
    ///
    /// let infty_norm = poly.norm_infty();
    ///
    /// // max coefficient is 4 = -3
    /// assert_eq!(Z::from(3), infty_norm);
    /// ```
    pub fn norm_infty(&self) -> Z {
        let mut res = Z::ZERO;

        for i in 0..=self.get_degree() {
            let coeff: Z = unsafe { self.get_coeff_unchecked(i) };
            let len = length(&coeff.value, &self.modulus.get_fq_ctx().ctxp[0].n[0]);
            res = max(res, len);
        }
        res
    }
}

#[cfg(test)]
mod test_norm_eucl_sqrd {
    use super::Z;
    use crate::integer_mod_q::PolynomialRingZq;
    use std::str::FromStr;

    /// Check whether the squared euclidean norm for polynomials
    /// with small coefficients is calculated correctly
    #[test]
    fn poly_small_coefficient() {
        let poly_1 = PolynomialRingZq::from_str("0 / 2  1 2 mod 11").unwrap();
        let poly_2 = PolynomialRingZq::from_str("3  1 2 3 / 4  1 2 3 4 mod 11").unwrap();
        let poly_3 = PolynomialRingZq::from_str("3  1 20 194 / 4  1 2 3 4 mod 195").unwrap();

        assert_eq!(poly_1.norm_eucl_sqrd(), Z::ZERO);
        assert_eq!(poly_2.norm_eucl_sqrd(), Z::from(14));
        assert_eq!(poly_3.norm_eucl_sqrd(), Z::from(402));
    }

    /// Check whether the squared euclidean norm for polynomials
    /// with small coefficients is calculated correctly
    #[test]
    fn poly_large_coefficient() {
        let poly_1 =
            PolynomialRingZq::from_str(&format!("1  {} / 2  1 2 mod {}", u64::MAX, u128::MAX))
                .unwrap();
        let poly_2 = PolynomialRingZq::from_str(&format!(
            "3  {} {} {} / 4  1 2 3 4 mod {}",
            u64::MAX,
            i64::MIN,
            i64::MAX,
            u64::MAX - 58
        ))
        .unwrap();

        assert_eq!(
            poly_1.norm_eucl_sqrd(),
            Z::from(u64::MAX) * Z::from(u64::MAX)
        );
        assert_eq!(
            poly_2.norm_eucl_sqrd(),
            Z::from(58) * Z::from(58)
                + Z::from((u64::MAX - 1) / 2 - 57) * Z::from((u64::MAX - 1) / 2 - 57)
                + Z::from((u64::MAX - 1) / 2 - 58) * Z::from((u64::MAX - 1) / 2 - 58)
        );
    }
}

#[cfg(test)]
mod test_norm_infty {
    use super::Z;
    use crate::integer_mod_q::PolynomialRingZq;
    use std::str::FromStr;

    /// Check whether the infinity norm for polynomials
    /// with small coefficients is calculated correctly
    #[test]
    fn poly_small_coefficient() {
        let poly_1 = PolynomialRingZq::from_str("0 / 2  1 2 mod 11").unwrap();
        let poly_2 = PolynomialRingZq::from_str("3  1 2 3 / 4  1 2 3 4 mod 5").unwrap();
        let poly_3 = PolynomialRingZq::from_str("3  1 20 194 / 4  1 2 3 4 mod 195").unwrap();

        assert_eq!(poly_1.norm_infty(), Z::ZERO);
        assert_eq!(poly_2.norm_infty(), Z::from(2));
        assert_eq!(poly_3.norm_infty(), Z::from(20));
    }

    /// Check whether the infinity norm for polynomials
    /// with small coefficients is calculated correctly
    #[test]
    fn poly_large_coefficient() {
        let poly_1 =
            PolynomialRingZq::from_str(&format!("1  {} / 2  1 2 mod {}", u64::MAX, u128::MAX))
                .unwrap();
        let poly_2 = PolynomialRingZq::from_str(&format!(
            "3  {} {} {} / 4  1 2 3 4 mod {}",
            u64::MAX,
            i64::MIN,
            i64::MAX,
            u64::MAX - 58
        ))
        .unwrap();

        assert_eq!(poly_1.norm_infty(), Z::from(u64::MAX));
        assert_eq!(poly_2.norm_infty(), Z::from((u64::MAX - 1) / 2 - 57));
    }
}