GF(2n) Redundant Representation Using Matrix Embedding for Irreducible Trinomials

By embedding a Toeplitz matrix-vector product (MVP) of dimension n into a circulant MVP of dimension N = 2n+ δ− 1, where δ can be any nonnegative integer, we present a GF(2) multiplication algorithm. This algorithm leads to a new redundant representation, and it has two merits: 1. The flexible choices of δ make it possible to select a proper N such that the multiplication operation in ring GF(2)[x]/(x + 1) can be performed using some asymptotically faster algorithms, e.g. the Fast Fourier Transformation (FFT)-based multiplication algorithm; 2. The redundant degrees, which are defined as N/n, are smaller than those of most previous GF(2) redundant representations, and in fact they are approximately equal to 2 for all applicable cases.