Finite Field Multiplication Combining AMNS and DFT Approach for Pairing Cryptography

Abstract

Pairings over elliptic curves use fields $\mathbb{F}_{p^k}$ with p ≥ 2¹⁶⁰ and 6 < k ≤ 32. In this paper we propose to represent elements in $\mathbb{F}_p$ with AMNS sytem of [1]. For well chosen AMNS we get roots of unity with sparse representation. The multiplication by these roots are thus really efficient in $\mathbb{F}_p$. The DFT/FFT approach for multiplication in extension field $F_{p^k}$ is thus optimized. The resulting complexity of a multiplication in $\mathbb{F}_{p^k}$ combining AMNS and DFT is about 50% less than the previously recommended approach [2].