Enum galois_2p8::field::IrreducablePolynomial

pub enum IrreducablePolynomial {
    Poly84310,
    Poly84320,
    Poly85310,
    Poly85320,
    Poly85430,
    Poly8543210,
    Poly86320,
    Poly8643210,
    Poly86510,
    Poly86520,
    Poly86530,
    Poly86540,
    Poly8654210,
    Poly8654310,
    Poly87210,
    Poly87310,
    Poly87320,
    Poly8743210,
    Poly87510,
    Poly87530,
    Poly87540,
    Poly8754320,
    Poly87610,
    Poly8763210,
    Poly8764210,
    Poly8764320,
    Poly8765210,
    Poly8765410,
    Poly8765420,
    Poly8765430,
}

Represents an irreducable polynomial of GF(2^8).

Each polynomial is named according to the nonzero positions of the coefficients. Each digit after the Poly prefix corresponds to the exponent of the variable for which a nonzero coefficient is present. Recall that in GF(2^x), the only possible coefficients are either 0 or 1.

For example, Poly84310 represents x^8 + x^4 + x^3 + x + 1.

Variants

Poly84310

Poly84320

Poly85310

Poly85320

Poly85430

Poly8543210

Poly86320

Poly8643210

Poly86510

Poly86520

Poly86530

Poly86540

Poly8654210

Poly8654310

Poly87210

Poly87310

Poly87320

Poly8743210

Poly87510

Poly87530

Poly87540

Poly8754320

Poly87610

Poly8763210

Poly8764210

Poly8764320

Poly8765210

Poly8765410

Poly8765420

Poly8765430

Implementations

impl IrreducablePolynomial

pub fn to_u16(&self) -> u16

Converts the IrreducablePolynomial to its binary representation.

In GF(2^x), coefficients in the extension polynomial may only take values of 0 or 1. As a result, there is a natural mapping of values in GF(2^x) to bits. 0b1011 maps to x^3 + x^1 + 1, for example. The greatest degree that can be encoded by an eight-bit byte is 7, as a consequence of the least significant bit being treated as x^0 == 1. In order to represent a degree 8 polynomial, there must be at least 9 bits available, so it can be encoded as a 16-bit value.

pub fn is_primitive(&self) -> bool

Determines whether the IrreducablePolynomial is a primitive polynomial.

In GF(2^x) wherein arithmetic operations are performed modulo a polynomial, the polynomial is said to be primitive if for some alpha, each nonzero member value of the field can be uniquely represented by alpha ^ p for p < 2^x, where alpha is a root of the polynomial. That is, for some root of the polynomial, every member of the field can be represented as the exponentiation of said root.

In GF(2^x), the only nontrivial prime root of any given IrreducablePolynomial is two. We say “is primitive” as a shorthand for meaning that the IrreducablePolynomial is primitive assuming a root of two.

The use of primitive polynomials confers an immediate performance benefit for single values: we can represent multiplication and division as addition and subtraction within logarithms and exponents. Additionally, some usages of Galois fields, e.g. Reed-Solomon syndrome coding, require primitive polynomials to function properly.

This method consults the PRIMITIVES table to determine if the IrreducablePolynomial is actually primitive. That is to say,

poly.is_primitive() == PRIMITIVES.binary_search(&poly).is_ok()

Trait Implementations

impl Clone for IrreducablePolynomial

fn clone(&self) -> IrreducablePolynomial

Returns a copy of the value.

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

Performs copy-assignment from source.

impl Debug for IrreducablePolynomial

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

Formats the value using the given formatter.

impl Hash for IrreducablePolynomial

fn hash<__H: Hasher>(&self, state: &mut __H)

Feeds this value into the given Hasher.

fn hash_slice<H>(data: &[Self], state: &mut H)where
    H: Hasher,
    Self: Sized,

Feeds a slice of this type into the given Hasher.

impl Ord for IrreducablePolynomial

fn cmp(&self, other: &IrreducablePolynomial) -> Ordering

This method returns an Ordering between self and other.

fn max(self, other: Self) -> Selfwhere
    Self: Sized,

Compares and returns the maximum of two values.

fn min(self, other: Self) -> Selfwhere
    Self: Sized,

Compares and returns the minimum of two values.

fn clamp(self, min: Self, max: Self) -> Selfwhere
    Self: Sized + PartialOrd<Self>,

Restrict a value to a certain interval.

impl PartialEq<IrreducablePolynomial> for IrreducablePolynomial

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

This method tests for self and other values to be equal, and is used by ==.

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

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

impl PartialOrd<IrreducablePolynomial> for IrreducablePolynomial

fn partial_cmp(&self, other: &IrreducablePolynomial) -> Option<Ordering>

This method returns an ordering between self and other values if one exists.

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

This method tests less than (for self and other) and is used by the < operator.

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

This method tests less than or equal to (for self and other) and is used by the <= operator.

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

This method tests greater than (for self and other) and is used by the > operator.

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

This method tests greater than or equal to (for self and other) and is used by the >= operator.

impl Copy for IrreducablePolynomial

impl Eq for IrreducablePolynomial

impl StructuralEq for IrreducablePolynomial

impl StructuralPartialEq for IrreducablePolynomial