Polynomial Basis - Squaring

Squaring

Squaring is an important operation because it can be used for general exponentiation as well as inversion of an element. The most basic way to square an element in the polynomial basis would be to apply a chosen multiplication algorithm on an element twice. In general case, there are minor optimizations that can be made, specifically related to the fact that when multiplying an element by itself, all the bits will be the same. In practice, however, the irreducible polynomial for the field is chosen with very few nonzero coefficients which makes squaring in polynomial basis of GF(2m) much simpler than multiplication.