Documentation

Binary Polynomial represented by an integer

Addition over F₂m. This is just a synonym of xor.

Multiplication over F₂m.

This function is undefined for negative arguments, because their bit representation is platform-dependent. Zero modulus is also prohibited.

Squaring over F₂m without reduction by modulo.

The implementation utilizes the fact that for binary polynomial S(x) we have S(x)^2 = S(x^2). In other words, insert a zero bit between every bits of argument: 1101 -> 1010001.

This function is undefined for negative arguments, because their bit representation is platform-dependent.

Squaring over F₂m.

This function is undefined for negative arguments, because their bit representation is platform-dependent. Zero modulus is also prohibited.

Exponentiation in F₂m by computing a^b mod fx.

This implements an exponentiation by squaring based solution. It inherits the same restrictions as squareF2m. Negative exponents are disallowed.

Reduction by modulo over F₂m.

This function is undefined for negative arguments, because their bit representation is platform-dependent. Zero modulus is also prohibited.

Square rooot in F₂m.

We exploit the fact that a^(2^m) = a, or in particular, a^(2^m - 1) = 1 from a classical result by Lagrange. Thus the square root is simply a^(2^(m - 1)).

Modular inversion over F₂m. If n doesn't have an inverse, Nothing is returned.

This function is undefined for negative arguments, because their bit representation is platform-dependent. Zero modulus is also prohibited.

Division over F₂m. If the dividend doesn't have an inverse it returns Nothing.

This function is undefined for negative arguments, because their bit representation is platform-dependent. Zero modulus is also prohibited.