Subquadratic Space Complexity Multiplier for a Class of Binary Fields Using Toeplitz Matrix Approach

In the recent past, subquadratic space complexity multipliers have been proposed for binary fields defined by irreducible trinomials and some specific pentanomials. For such multipliers, alternative irreducible polynomials can also be used, in particular, nearly all one polynomials (NAOPs) seem to be better than pentanomials (see [7]). For improved efficiency, multiplication modulo an NAOP is performed via modulo a quadrinomial whose degree is one more than that of the original NAOP. In this paper, we present a Toeplitz matrix-vector product based approach for multiplication modulo a quadrinomial. We obtain a fully parallel (nonsequential) multiplier with a subquadratic space complexity, which has the same order of space complexity as that of Fan and Hasan. The Toeplitz matrix-vector product based approach is also interesting in the design of sequential multipliers. In this paper, we present two such multipliers: one with bit serial output and the other bit parallel output.