**Polynomial multiplication over binary finite fields: new upper bounds**

*Alessandro De Piccoli and 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.

**Category / Keywords: **foundations / Polynomial multiplication, Karatsuba, Two-level Seven-way Recursion algorithm, binary fields, fast software implementations.

**Date: **received 25 Jan 2018

**Contact author: **andrea visconti at unimi it

**Available format(s): **PDF | BibTeX Citation

**Version: **20180128:213224 (All versions of this report)

**Short URL: **ia.cr/2018/091

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