Finite Field Multiplication Combining AMNS and DFT Approach for Pairing Cryptography

Pairings over ellitpic curve use fields GF(p^k) with p >= 2^{160} and 6 < k <=32. In this paper we propose to represent elements in GF(p) with AMNS (Bajard et al. SAC04). For well chosen AMNS we get roots of unity with sparse representation. The multiplication by these roots are thus really efficient in GF(p). The DFT/FFT approach for multiplication in extension field GF(p^k) is thus optimized. The resulting complexity of a multiplication in GF(p^k) combining AMNS and DFT is about 50\% less than the previously recommended approach