Module error

Binary univariate polynomial algebra

This repo is highly inspired from https://gist.github.com/mildsunrise/e21ae2b1649532813f2594932f9e9371, and https://github.com/uranix/factormod

---

Installation

On your project’s directory, run the following command:

cargo add binary_polynomial_mod_algebra

Usage

Polynomial creation

You can create a BinaryPolynomial from BigUint, where the least significant bit is the lowest degree coefficient.

use num_bigint::BigUint;
use binary_polynomial_mod_algebra::BinaryPolynomial;

let polynomial = BinaryPolynomial::from(BigUint::from(13u32)); // or 0b1101u32
assert_eq!(polynomial.to_string(), "x^3 + x^2 + 1");

You can also create a BinaryPolynomial from a Vec<bool>, where the first element is the highest degree coefficient.

use binary_polynomial_mod_algebra::BinaryPolynomial;

let polynomial = BinaryPolynomial::from(vec![true, true, false, true]);
assert_eq!(polynomial.to_string(), "x^3 + x^2 + 1");

Or from a &str:

use binary_polynomial_mod_algebra::BinaryPolynomial;
let polynomial = BinaryPolynomial::try_from("x^4 + x + 1").unwrap();
assert_eq!(polynomial.to_string(), "x^4 + x + 1");

Finally, you can create the null or unitary BinaryPolynomial:

use binary_polynomial_mod_algebra::BinaryPolynomial;
use num_traits::identities::One;
use num_traits::Zero;

let polynomial = BinaryPolynomial::zero();
assert_eq!(polynomial.to_string(), "0");

let polynomial = BinaryPolynomial::one();
assert_eq!(polynomial.to_string(), "1");

Non-zero polynomial

Some functions require a NonZeroBinaryPolynomial argument, you can create it from a BinaryPolynomial:

use binary_polynomial_mod_algebra::BinaryPolynomial;
use binary_polynomial_mod_algebra::NonZeroBinaryPolynomial;
use num_traits::identities::Zero;

let zero_polynomial = BinaryPolynomial::zero();
let try_non_zero_polynomial = NonZeroBinaryPolynomial::new(zero_polynomial);
assert!(try_non_zero_polynomial.is_none());

let non_zero_polynomial = BinaryPolynomial::from(vec![true, true, false, true]);
let non_zero_polynomial = NonZeroBinaryPolynomial::new(non_zero_polynomial).unwrap();
assert!(non_zero_polynomial.get().to_string() == "x^3 + x^2 + 1");

Example

Polynomial inversion modulo a non-zero polynomial:

use binary_polynomial_mod_algebra::BinaryPolynomial;
use binary_polynomial_mod_algebra::NonZeroBinaryPolynomial;

let polynomial = NonZeroBinaryPolynomial::new(
        BinaryPolynomial::from(vec![true, false, true, true]) // x^3 + x + 1
    ).unwrap();
let modulo = NonZeroBinaryPolynomial::new(
        BinaryPolynomial::from(vec![true, false, false, false, true]) // x^4 + 1
    ).unwrap();
let inverse = polynomial.inv_mod(&modulo);
assert!(inverse.is_some()); // Inverse exists
assert_eq!(inverse.unwrap().to_string(), "x^3 + x + 1");

Operators

You can use the following operators:

Operator	operand 1	operand 2	Description
+	BinaryPolynomial	BinaryPolynomial	Polynomial addition (equivalent to XOR)
*	BinaryPolynomial	BinaryPolynomial	Polynomial multiplication (without modulo, use mul_mod for modular multiplication)
*	NonZeroBinaryPolynomial	NonZeroBinaryPolynomial	Polynomial multiplication (without modulo, use mul_mod for modular multiplication)
%	BinaryPolynomial	NonZeroBinaryPolynomial	Polynomial modulo
/	BinaryPolynomial	NonZeroBinaryPolynomial	Polynomial division (returns quotient)
pow	BinaryPolynomial	usize	Polynomial exponentiation (without modulo, use pow_mod for modular exponentiation)

Available operations

div_mod(BinaryPolynomial, modulo: NonZeroBinaryPolynomial)

Returns quotient and modulo

mul_mod(BinaryPolynomial, BinaryPolynomial, modulo: NonZeroBinaryPolynomial)

Multiplication modulo

pow_mod(BinaryPolynomial, BinaryPolynomial, modulo: NonZeroBinaryPolynomial)

Exponentiation modulo

congruent_mod(BinaryPolynomial, BinaryPolynomial, modulo: NonZeroBinaryPolynomial)

Check if polynomials are congruent modulo

derivative(BinaryPolynomial)

Returns polynomial derivative

coprime(NonZeroBinaryPolynomial, NonZeroBinaryPolynomial)

Check if two polynomials are coprime

gcd(NonZeroBinaryPolynomial, NonZeroBinaryPolynomial)

Returns greatest common divisor of two polynomials

egcd(NonZeroBinaryPolynomial, NonZeroBinaryPolynomial)

Returns extended greatest common divisor of two polynomials

inv_mod(NonZeroBinaryPolynomial, modulo: NonZeroBinaryPolynomial)

Returns modular inverse of a polynomial

is_irreducible(NonZeroBinaryPolynomial)

Check if polynomial is irreducible using Rabin’s irreducibility test

is_primitive(NonZeroBinaryPolynomial)

Check if polynomial is primitive

irreducible_factors(NonZeroBinaryPolynomial)

Compute irreducible factors of polynomial using Berlekamp’s algorithm

square_free_irreducible_factors(NonZeroBinaryPolynomial)

Compute irreducible factors of square free polynomial

is_square_free(NonZeroBinaryPolynomial)

Check if polynomial is square free Errors in polynomial operations

Error module

Enums

BinaryPolynomialError

Error type for binary polynomial operations