Fast multiplication of binary polynomials with the forthcoming vectorized VPCLMULQDQ instruction

Abstract

Polynomial multiplication over binary fields $\mathbb{F}_{2^{n}}$ is a common primitive, used for example by current cryptosystems such as AES-GCM (with $n=128)$. It also turns out to be a primitive for other cryptosystems, that are being designed for the Post Quantum era, with values $n\gg 128$. Examples from the recent submissions to the NIST Post-Quantum Cryptography project, are BIKE, LEDAKem, and GeMSS, where the performance of the polynomial multiplications, is significant. Therefore, efficient polynomial multiplication over $\mathbb{F}_{2^{n}}$, with large $n$, is a significant emerging optimization target. Anticipating future applications, Intel has recently announced that its future architecture (codename “Ice Lake”) will introduce a new vectorized way to use the current VPCLMULQDQ instruction. In this paper, we demonstrate how to use this instruction for accelerating polynomial multiplication. Our analysis shows a prediction for at least 2x speedup for multiplications with polynomials of degree 512 or more.