Families of SNARK-Friendly 2-Chains of Elliptic Curves (original) (raw)

References

  1. Aranha, D.F., Barreto, P.S.L.M., Longa, P., Ricardini, J.E.: The realm of the pairings. In: Lange, T., Lauter, K., Lisoněk, P. (eds.) SAC 2013. LNCS, vol. 8282, pp. 3–25. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-662-43414-7_1
    Chapter Google Scholar
  2. Aranha, D.F., Karabina, K., Longa, P., Gebotys, C.H., López, J.: Faster explicit formulas for computing pairings over ordinary curves. In: Paterson, K.G. (ed.) EUROCRYPT 2011. LNCS, vol. 6632, pp. 48–68. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-20465-4_5
    Chapter Google Scholar
  3. Barbulescu, R., Duquesne, S.: Updating key size estimations for pairings. J. Cryptol. 32(4), 1298–1336 (2019). https://doi.org/10.1007/s00145-018-9280-5
    Article MathSciNet MATH Google Scholar
  4. Barreto, P.S.L.M., Lynn, B., Scott, M.: Constructing elliptic curves with prescribed embedding degrees. In: Cimato, S., Persiano, G., Galdi, C. (eds.) SCN 2002. LNCS, vol. 2576, pp. 257–267. Springer, Heidelberg (2003). https://doi.org/10.1007/3-540-36413-7_19
    Chapter Google Scholar
  5. Ben-Sasson, E., Chiesa, A., Tromer, E., Virza, M.: Scalable zero knowledge via cycles of elliptic curves. In: Garay, J.A., Gennaro, R. (eds.) CRYPTO 2014, Part II. LNCS, vol. 8617, pp. 276–294. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-662-44381-1_16
    Chapter Google Scholar
  6. Bernstein, D.J., Doumen, J., Lange, T., Oosterwijk, J.-J.: Faster batch forgery identification. In: Galbraith, S., Nandi, M. (eds.) INDOCRYPT 2012. LNCS, vol. 7668, pp. 454–473. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-34931-7_26
    Chapter Google Scholar
  7. Bitansky, N., Canetti, R., Chiesa, A., Tromer, E.: From extractable collision resistance to succinct non-interactive arguments of knowledge, and back again. In: Goldwasser, S. (ed.) ITCS 2012, pp. 326–349. ACM, January 2012. https://doi.org/10.1145/2090236.2090263
  8. Bitansky, N., Canetti, R., Chiesa, A., Tromer, E.: Recursive composition and bootstrapping for SNARKS and proof-carrying data. In: Boneh, D., Roughgarden, T., Feigenbaum, J. (eds.) 45th ACM STOC, pp. 111–120. ACM Press, June 2013. https://doi.org/10.1145/2488608.2488623
  9. Botrel, G., Piellard, T., Housni, Y.E., Kubjas, I., Tabaie, A.: Consensys/gnark: v0.6.0, January 2022. https://doi.org/10.5281/zenodo.5819105
  10. Bowe, S., Chiesa, A., Green, M., Miers, I., Mishra, P., Wu, H.: ZEXE: enabling decentralized private computation. In: 2020 IEEE Symposium on Security and Privacy, pp. 947–964. IEEE Computer Society Press, May 2020. https://doi.org/10.1109/SP40000.2020.00050
  11. Chiesa, A., Hu, Y., Maller, M., Mishra, P., Vesely, N., Ward, N.: Marlin: preprocessing zkSNARKs with universal and updatable SRS. In: Canteaut, A., Ishai, Y. (eds.) EUROCRYPT 2020, Part I. LNCS, vol. 12105, pp. 738–768. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-45721-1_26
    Chapter Google Scholar
  12. Costello, C., Lange, T., Naehrig, M.: Faster pairing computations on curves with high-degree twists. In: Nguyen, P.Q., Pointcheval, D. (eds.) PKC 2010. LNCS, vol. 6056, pp. 224–242. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-13013-7_14
    Chapter Google Scholar
  13. De Micheli, G., Gaudry, P., Pierrot, C.: Lattice enumeration for tower NFS: a 521-bit discrete logarithm computation. In: Tibouchi, M., Wang, H. (eds.) ASIACRYPT 2021, Part I. LNCS, vol. 13090, pp. 67–96. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-92062-3_3
    Chapter Google Scholar
  14. El Housni, Y.: A fork of gnark-crypto: Golang library for finite fields, fft, and elliptic curves (2021). https://github.com/yelhousni/gnark-crypto
  15. El Housni, Y., Guillevic, A.: Optimized and secure pairing-friendly elliptic curves suitable for one layer proof composition. In: Krenn, S., Shulman, H., Vaudenay, S. (eds.) CANS 2020. LNCS, vol. 12579, pp. 259–279. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-65411-5_13
    Chapter Google Scholar
  16. El Housni, Y., Guillevic, A.: Families of SNARK-friendly 2-chains of elliptic curves. ePrint 2021/1359 (2021)
    Google Scholar
  17. El Housni, Y., Guillevic, A.: Families of SNARK-friendly 2-chains of elliptic curves (2021). MIT License. https://gitlab.inria.fr/zk-curves/snark-2-chains
  18. Fotiadis, G., Konstantinou, E.: TNFS resistant families of pairing-friendly elliptic curves. Theor. Comput. Sci. 800, 73–89 (2019). https://doi.org/10.1016/j.tcs.2019.10.017
    Article MathSciNet MATH Google Scholar
  19. Freeman, D., Scott, M., Teske, E.: A taxonomy of pairing-friendly elliptic curves. J. Cryptol. 23(2), 224–280 (2010). https://doi.org/10.1007/s00145-009-9048-z
    Article MathSciNet MATH Google Scholar
  20. Gabizon, A., Williamson, Z.J.: plookup: a simplified polynomial protocol for lookup tables. ePrint 2020/315 (2020)
    Google Scholar
  21. Gabizon, A., Williamson, Z.J., Ciobotaru, O.: PLONK: permutations over lagrange-bases for oecumenical noninteractive arguments of knowledge. ePrint 2019/953 (2019)
    Google Scholar
  22. Galbraith, S.D., Lin, X., Scott, M.: Endomorphisms for faster elliptic curve cryptography on a large class of curves. In: Joux, A. (ed.) EUROCRYPT 2009. LNCS, vol. 5479, pp. 518–535. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-01001-9_30
    Chapter Google Scholar
  23. Granger, R., Scott, M.: Faster squaring in the cyclotomic subgroup of sixth degree extensions. In: Nguyen, P.Q., Pointcheval, D. (eds.) PKC 2010. LNCS, vol. 6056, pp. 209–223. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-13013-7_13
    Chapter Google Scholar
  24. Grassi, L., Khovratovich, D., Rechberger, C., Roy, A., Schofnegger, M.: Poseidon: a new hash function for zero-knowledge proof systems. In: USENIX Security Symposium (2021)
    Google Scholar
  25. Groth, J.: On the size of pairing-based non-interactive arguments. In: Fischlin, M., Coron, J.-S. (eds.) EUROCRYPT 2016, Part II. LNCS, vol. 9666, pp. 305–326. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-49896-5_11
    Chapter Google Scholar
  26. Guillevic, A.: A short-list of pairing-friendly curves resistant to special TNFS at the 128-bit security level. In: Kiayias, A., Kohlweiss, M., Wallden, P., Zikas, V. (eds.) PKC 2020, Part II. LNCS, vol. 12111, pp. 535–564. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-45388-6_19
    Chapter Google Scholar
  27. Guillevic, A., Masson, S., Thomé, E.: Cocks-Pinch curves of embedding degrees five to eight and optimal ate pairing computation. Des. Codes Cryptogr. 88, 1047–1081 (2020). https://doi.org/10.1007/s10623-020-00727-w
    Article MathSciNet MATH Google Scholar
  28. Guillevic, A., Singh, S.: On the alpha value of polynomials in the tower number field sieve algorithm. Math. Cryptol. 1(1), 1–39 (2021). https://journals.flvc.org/mathcryptology/article/view/125142
  29. Hayashida, D., Hayasaka, K., Teruya, T.: Efficient final exponentiation via cyclotomic structure for pairings over families of elliptic curves. ePrint 2020/875 (2020)
    Google Scholar
  30. Hess, F., Smart, N.P., Vercauteren, F.: The eta pairing revisited. IEEE Trans. Inf. Theory 52(10), 4595–4602 (2006). https://doi.org/10.1109/TIT.2006.881709
    Article MathSciNet MATH Google Scholar
  31. Karabina, K.: Squaring in cyclotomic subgroups. Math. Comput. 82(281), 555–579 (2013). https://doi.org/10.1090/S0025-5718-2012-02625-1
    Article MathSciNet MATH Google Scholar
  32. Kate, A., Zaverucha, G.M., Goldberg, I.: Constant-size commitments to polynomials and their applications. In: Abe, M. (ed.) ASIACRYPT 2010. LNCS, vol. 6477, pp. 177–194. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-17373-8_11
    Chapter Google Scholar
  33. Kilian, J.: A note on efficient zero-knowledge proofs and arguments (extended abstract). In: 24th ACM STOC, pp. 723–732. ACM Press, May 1992. https://doi.org/10.1145/129712.129782
  34. Kim, T., Barbulescu, R.: Extended tower number field sieve: a new complexity for the medium prime case. In: Robshaw, M., Katz, J. (eds.) CRYPTO 2016, Part I. LNCS, vol. 9814, pp. 543–571. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53018-4_20
    Chapter Google Scholar
  35. Micali, S.: CS proofs (extended abstracts). In: 35th FOCS, pp. 436–453. IEEE Computer Society Press, November 1994. https://doi.org/10.1109/SFCS.1994.365746
  36. Schoof, R.: Nonsingular plane cubic curves over finite fields. J. Comb. Theory Ser. A 46(2), 183–211 (1987). https://doi.org/10.1016/0097-3165(87)90003-3
    Article MathSciNet MATH Google Scholar
  37. Scott, M.: A note on group membership tests for \(\mathbb{G}_1\), \(\mathbb{G}_2\) and \(\mathbb{G}_{T}\) on BLS pairing-friendly curves. ePrint 2021/1130 (2021)
    Google Scholar
  38. Scott, M.: Pairing implementation revisited. ePrint 2019/077 (2019)
    Google Scholar
  39. Scott, M.: Unbalancing pairing-based key exchange protocols. ePrint 2013/688 (2013)
    Google Scholar
  40. Silverman, J.H.: The Arithmetic of Elliptic Curves. Graduate Texts in Mathematics, Springer, Dordrecht (2009). https://doi.org/10.1007/978-0-387-09494-6
    Book MATH Google Scholar
  41. Valiant, P.: Incrementally verifiable computation or proofs of knowledge imply time/space efficiency. In: Canetti, R. (ed.) TCC 2008. LNCS, vol. 4948, pp. 1–18. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-78524-8_1
    Chapter MATH Google Scholar
  42. Vercauteren, F.: Optimal pairings. IEEE Trans. Inf. Theory 56(1), 455–461 (2010). https://doi.org/10.1109/TIT.2009.2034881
    Article MathSciNet MATH Google Scholar
  43. Wahby, R.S., Boneh, D.: Fast and simple constant-time hashing to the BLS12-381 elliptic curve. IACR TCHES 2019(4), 154–179 (2019). https://doi.org/10.13154/tches.v2019.i4.154-179

Download references