Struct Cyclo 

pub struct Cyclo {
    pub n: usize,
    pub coeffs: Vec<BigInt>,
}

An element of R = ℤ[X]/(Xⁿ + 1): the coefficient vector [a₀, …, a_{n−1}] of Σ aᵢ Xⁱ, with n a power of two.

Fields

n: usize

coeffs: Vec<BigInt>

Implementations

impl Cyclo

pub fn new(n: usize, coeffs: Vec<BigInt>) -> Cyclo

An element from a coefficient vector (padded/verified to length n). n must be a power of two.

pub fn from_ints(n: usize, v: &[i64]) -> Cyclo

Convenience constructor from small integers.

pub fn zero(n: usize) -> Cyclo

pub fn one(n: usize) -> Cyclo

pub fn monomial(n: usize, i: usize, coeff: BigInt) -> Cyclo

coeff · Xⁱ reduced into R (any i ≥ 0): Xⁱ = (−1)^{⌊i/n⌋} · X^{i mod n}.

pub fn add(&self, o: &Cyclo) -> Cyclo

pub fn sub(&self, o: &Cyclo) -> Cyclo

pub fn neg(&self) -> Cyclo

pub fn mul(&self, o: &Cyclo) -> Cyclo

Negacyclic product: multiply as polynomials, then reduce Xⁿ ≡ −1.

pub fn galois(&self, t: u64) -> Cyclo

The Galois automorphism σ_t : X ↦ X^t (t odd, i.e. a unit of ℤ/2n). On monomials σ_t(Xⁱ) = X^{it}, reduced by Xⁿ ≡ −1. A ring automorphism of R.

pub fn conjugate(&self) -> Cyclo

Complex conjugation σ_{−1} : X ↦ X^{−1} = X^{2n−1} — the order-two element of the Galois group.

pub fn norm(&self) -> BigInt

The field norm N(a) = ∏_{σ ∈ Gal} σ(a) ∈ ℤ — multiplicative, and ±1 exactly on the units.

pub fn trace(&self) -> BigInt

The field trace Tr(a) = Σ_{σ ∈ Gal} σ(a) ∈ ℤ — additive; Tr(1) = n.

pub fn is_unit(&self) -> bool

Whether a is a unit of R — equivalently N(a) = ±1.

pub fn unit_inverse(&self) -> Option<Cyclo>

The exact ring inverse of a unit, with no field division: since N(u) = u·∏_{t≠1}σ_t(u) = ±1, we have u⁻¹ = N(u)·∏_{t≠1}σ_t(u). None for a non-unit.

pub fn coeff_norm(&self) -> i64

The sum of absolute values of the coefficients — a coarse length used to recognize a short generator.

Trait Implementations

impl Clone for Cyclo

fn clone(&self) -> Cyclo

Returns a duplicate of the value.

fn clone_from(&mut self, source: &Self)

Performs copy-assignment from source.

impl Debug for Cyclo

fn fmt(&self, f: &mut Formatter<'_>) -> Result

Formats the value using the given formatter.

impl PartialEq for Cyclo

fn eq(&self, other: &Cyclo) -> bool

Tests for self and other values to be equal, and is used by ==.

fn ne(&self, other: &Rhs) -> bool

Tests for !=. The default implementation is almost always sufficient, and should not be overridden without very good reason.

impl Eq for Cyclo

impl StructuralPartialEq for Cyclo