$\schmi{GF(2^n)}$ Shifted Polynomial Basis Multipliers Based on Subquadratic Toeplitz Matrix-Vector Product Approach for All Irreducible Pentanomials

Besides Karatsuba’s algorithm, optimal Toeplitz matrix-vector product (TMVP) formulae is another approach to design
$GF(2^n)$
subquadratic multipliers. However, when
$GF(2^n)$
elements are represented using a polynomial basis or its generalization—shifted polynomial basis—this approach is currently appliable only to fields
$GF(2^n)$
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.