Paper 2022/367

Efficient Algorithms for Large Prime Characteristic Fields and Their Application to Bilinear Pairings

Patrick Longa, Microsoft Research

We propose a novel approach that generalizes interleaved modular multiplication algorithms to the computation of sums of products over large prime fields. This operation has widespread use and is at the core of many cryptographic applications. The method reformulates the widely used lazy reduction technique, crucially avoiding the need for storage and computation of "double-precision" operations. Moreover, it can be easily adapted to the different methods that exist to compute modular multiplication, producing algorithms that are significantly more efficient and memory-friendly. We showcase the performance of the proposed approach in the computation of multiplication over an extension field $\mathbb{F}_{p^k}$, and demonstrate its impact with a record-breaking implementation for bilinear pairings: a full optimal ate pairing over the popular BLS12-381 curve is computed in under half a millisecond on a 3.2GHz Intel Coffee Lake processor, which is about $1.40\times$ faster than the state-of-the-art.

Note: The main case study has been retargeted to focus on pairings, after taking into account the recent break of SIDH/SIKE.

Available format(s)
Public-key cryptography
Publication info
Prime fields extension fields bilinear pairings BLS12-381 efficient computation
Contact author(s)
plonga @ microsoft com
2022-10-06: revised
2022-03-22: received
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      author = {Patrick Longa},
      title = {Efficient Algorithms for Large Prime Characteristic Fields and Their Application to Bilinear Pairings},
      howpublished = {Cryptology ePrint Archive, Paper 2022/367},
      year = {2022},
      note = {\url{}},
      url = {}
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