$schmi{GF(2^n)}$ Shifted Polynomial Basis Multipliers Based on Subquadratic Toeplitz Matrix-Vector Product App

Besides Karatsuba´s algorithm, optimal Toeplitz matrix-vector product (TMVP) formulae is another approach to design GF(2ⁿ) subquadratic multipliers. However, when GF(2ⁿ) elements are represented using a polynomial basis or its generalization-shifted polynomial basis-this approach is currently appliable only to fields GF(2ⁿ) generated by an irreducible trinomial or a special type of irreducible pentanomials, but not to a general irreducible pentanomial. The reason is that no transformation matrix, which transforms the Mastrovito matrix into a Toeplitz matrix, has been found. In this article, we propose such a transformation matrix and its inverse matrix for an arbitrary irreducible pentanomial.