Relationship between GF(2^m) Montgomery and Shifted Polynomial Basis Multiplication Algorithms

Abstract

Applying the matrix-vector product idea of the Mastrovito multiplier to the GF(2^{m}) Montgomery multiplication algorithm, we present a new parallel multiplier for irreducible trinomials. This multiplier and the corresponding shifted polynomial basis (SPB) multiplier have the same circuit structure for the same set of parameters. Furthermore, by establishing isomorphisms between the Montgomery and the SPB constructions of GF(2^{m}), we show that the Montgomery algorithm can be used to perform the SPB multiplication without any changes and vice versa.