Paper 2026/885 Original: eprint.iacr.org/2026/885 ↗

Optimized Final Exponentiation for Optimal Ate Pairings Using Cyclotomic Cubing

Abstract

Pairing-based cryptography relies heavily on the efficiency of bilinear pairings, the computation of which is dominated by the final exponentiation step. This paper describes an efficient cubing operation in the cyclotomic subgroup of $\mathbb{F}_{q^6}$ for $q\equiv1\mod{6}$. As an application, we use existing results for computing Frobenius maps to optimize the cost of the optimal Ate pairing final exponentiation over the SG54 curve. Furthermore, we introduce a novel decomposition for the hard part of the final exponentiation for this curve. Additionally, we apply established methods for cyclotomic cubing to accelerate the final exponentiation for the BLS15 and BLS27 curves. Compared to previous works, our approach achieves efficiency gains of $24\%$ for SG54 and $22\%$ for the BLS15 and BLS27 curves.