International Association
for Cryptologic Research

The bottleneck in the proving algorithm of most of elliptic-curve-based SNARK proof systems is the Multi-Scalar-Multiplication (MSM) algorithm. In this paper we give an overview of a variant of the Pippenger MSM algorithm together with a set of optimizations tailored for curves that admit a twisted Edwards form. We prove that this is the case for SNARK-friendly chains and cycles of elliptic curves, which are useful for recursive constructions. Our contribution is twofold: first, we optimize the arithmetic of finite fields by improving on the well-known Coarsely Integrated Operand Scanning (CIOS) modular multiplication. This is a contribution of independent interest that applies to many different contexts. Second, we propose a new coordinate system for twisted Edwards curves tailored for the Pippenger MSM algorithm.Accelerating the MSM over these curves is critical for deployment of recursive proof< systems applications such as proof-carrying-data, blockchain rollups and blockchain light clients. We implement our work in Go and benchmark it on two different CPU architectures (x86 and arm64). We show that our implementation achieves a 40-47% speedup over the state-of-the-art implementation (which was implemented in Rust). This MSM implementation won the first place in the ZPrize competition in the open division “Accelerating MSM on Mobile” and will be deployed in two real-world applications: Linea zkEVM by ConsenSys and probably Celo network.

At CANS'20, El Housni and Guillevic introduced a new 2-chain of pairing-friendly elliptic curves for recursive zero-knowledge Succinct Non-interactive ARguments of Knowledge (zk-SNARKs) made of the former BLS12-377 curve (a Barreto--Lynn--Scott curve over a 377-bit prime field) and the new BW6-761 curve (a Brezing--Weng curve of embedding degree 6 over a 761-bit prime field). First we generalise the curve construction, the pairing formulas ($e \colon \G_1 \times \G_2 \to \G_T$) and the group operations to any BW6 curve defined on top of any BLS12 curve, forming a family of 2-chain pairing-friendly curves. Second, we investigate other possible 2-chain families made on top of the BLS12 and BLS24 curves. We compare BW6 to Cocks--Pinch curves of higher embedding degrees 8 and 12 (CP8, CP12) at the 128-bit security level. We derive formulas for efficient optimal ate and optimal Tate pairings on our new CP curves. We show that for both BLS12 and BLS24, the BW6 construction always gives the fastest pairing and curve arithmetic compared to Cocks-Pinch curves. Finally, we suggest a short list of curves suitable for Groth16 and KZG-based universal SNARKs and present an optimized implementation of these curves. Based on Groth16 and PlonK (a KZG-based SNARK) implementations in the \texttt{gnark} ecosystem, we obtain that the BLS12-377/BW6-761 pair is optimized for the former while the BLS24-315/BW6-672 pair is optimized for the latter.