A new class of irreducible pentanomials for polynomial based multipliers in binary fields

TL;DR

This paper introduces a new class of irreducible pentanomials for binary field multipliers, providing an efficient reduction process and a multiplier design with improved XOR and AND complexity, suitable for hardware implementation.

Contribution

The paper presents a novel class of irreducible pentanomials and a corresponding multiplier design that enhances efficiency in binary field arithmetic.

Findings

Reduced XOR and AND complexity in the proposed multiplier

Comparable time delay to existing Karatsuba-based multipliers

Exact operation count for polynomial reduction

Abstract

We introduce a new class of irreducible pentanomials over $\mathbb{F}_2$ of the form $f(x) = x^{2b+c} + x^{b+c} + x^b + x^c + 1$. Let $m=2b+c$ and use $f$ to define the finite field extension of degree $m$. We give the exact number of operations required for computing the reduction modulo $f$. We also provide a multiplier based on Karatsuba algorithm in $\mathbb{F}_2[x]$ combined with our reduction process. We give the total cost of the multiplier and found that the bit-parallel multiplier defined by this new class of polynomials has improved XOR and AND complexity. Our multiplier has comparable time delay when compared to other multipliers based on Karatsuba algorithm.