A survey of elliptic curves for proof systems

Diego F. Aranha, Youssef El Housni, Aurore Guillevic

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Elliptic curves have become key ingredients for instantiating zero-knowledge proofs and more generally proof systems. Recently, there have been many tailored constructions of these curves that aim at efficiently implementing different kinds of proof systems. In this survey we provide the reader with a comprehensive overview on existing work and revisit the contributions in terms of efficiency and security. We present an overview at three stages of the process: curves to instantiate a SNARK, curves to instantiate a recursive SNARK, and also curves to express an elliptic-curve related statement. We provide new constructions of curves for SNARKs and generalize the state-of-the-art constructions for recursive SNARKs. We also exhaustively document the existing work and open-source implementations.

Original languageEnglish
JournalDesigns, Codes and Cryptography
Pages (from-to)3333–3378
Number of pages46
Publication statusPublished - Oct 2023


  • Elliptic curves
  • Pairings
  • Proof systems
  • SNARKs


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