Paper 2018/091

Polynomial multiplication over binary finite fields: new upper bounds

Alessandro De Piccoli, Andrea Visconti, and Ottavio Giulio Rizzo

Abstract

When implementing a cryptographic algorithm, efficient operations have high relevance both in hardware and software. Since a number of operations can be performed via polynomial multiplication, the arithmetic of polynomials over finite fields plays a key role in real-life implementations. One of the most interesting paper that addressed the problem has been published in 2009. In [5], Bernstein suggests to split polynomials into parts and presents a new recursive multiplication technique which is faster than those commonly used. In order to further reduce the number of bit operations [6] required to multiply n-bit polynomials, researchers adopt different approaches. In [18] a greedy heuristic has been applied to linear straight-line sequences listed in [6]. In 2013, D'angella, Schiavo and Visconti [20] skip some redundant operations of the multiplication algorithms described in [5]. In 2015, Cenk, Negre and Hasan [12] suggest new multiplication algorithms. In this paper, (a) we present a "k-1"-level Recursion algorithm that can be used to reduce the effective number of bit operations required to multiply n-bit polynomials; and (b) we use algebraic extensions of F_2 combined with Lagrange interpolation to improve the asymptotic complexity.