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A New Isogeny Representation and Applications to Cryptography

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Advances in Cryptology – ASIACRYPT 2022 (ASIACRYPT 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13792))

Abstract

This paper focuses on isogeny representations, defined as ways to evaluate isogenies and verify membership to the language of isogenous supersingular curves (the set of triples \(D,E_1,E_2\) with a cyclic isogeny of degree D between \(E_1\) and \(E_2\)). The tasks of evaluating and verifying isogenies are fundamental for isogeny-based cryptography.

Our main contribution is the design of the suborder representation, a new isogeny representation targetted at the case of (big) prime degree. The core of our new method is the revelation of endomorphisms of smooth norm inside a well-chosen suborder of the codomain’s endomorphism ring. This new representation appears to be opening interesting prospects for isogeny-based cryptography under the hardness of a new computational problem: the SubOrder to Ideal Problem (SOIP). As an application, we introduce pSIDH, a new NIKE based on the suborder representation. Studying new assumption appears to be particularly crucial in the light of the recent attacks against isogeny-based cryptography.

In order to manipulate efficiently the suborder representation, we develop several heuristic algorithmic tools to solve norm equations inside a new family of quaternion orders. These new algorithms may be of independent interest.

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References

  1. Arpin, S., Chen, M., Lauter, K.E., Scheidler, R., Stange, K.E., Tran, H.T.: Orienteering with one endomorphism. arXiv preprint arXiv:2201.11079 (2022)

  2. Burdges, J., De Feo, L.: Delay encryption. In: Canteaut, A., Standaert, F.-X. (eds.) EUROCRYPT 2021. LNCS, vol. 12696, pp. 302–326. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-77870-5_11

    Chapter  Google Scholar 

  3. Bernstein, D. J., De Feo, L., Leroux, A., Smith, B.: Faster computation of isogenies of large prime degree. In: Galbraith, S., editor, ANTS-XIV - 14th Algorithmic Number Theory Symposium, pp. 39–55, Auckland, New Zealand (2020)

    Google Scholar 

  4. Biasse, J.-F., Jao, D., Sankar, A.: A quantum algorithm for computing isogenies between supersingular elliptic curves. In: Meier, W., Mukhopadhyay, D. (eds.) INDOCRYPT 2014. LNCS, vol. 8885, pp. 428–442. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-13039-2_25

    Chapter  Google Scholar 

  5. Beullens, W., Kleinjung, T., Vercauteren, F.: CSI-FiSh: efficient isogeny based signatures through class group computations. In: Galbraith, S.D., Moriai, S. (eds.) ASIACRYPT 2019. LNCS, vol. 11921, pp. 227–247. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-34578-5_9

    Chapter  Google Scholar 

  6. Boneh, D., Kogan, D., Woo, K.: Oblivious pseudorandom functions from isogenies. In: Moriai, S., Wang, H. (eds.) ASIACRYPT 2020. LNCS, vol. 12492, pp. 520–550. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-64834-3_18

    Chapter  Google Scholar 

  7. Castryck, W., Decru, T.: An efficient key recovery attack on SIDH (preliminary version). Cryptology ePrint Archive (2022)

    Google Scholar 

  8. Childs, A., Jao, D., Soukharev, V.: Constructing elliptic curve isogenies in quantum subexponential time. J. Math. Cryptol. 8(1), 1–29 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. Colò, L., Kohel, D.: Orienting supersingular isogeny graphs. Number-Theoretic Methods in Cryptology (2019)

    Google Scholar 

  10. Chavez-Saab, J., Rodríguez-Henríquez, F., Tibouchi, M.: Verifiable isogeny walks: towards an isogeny-based postquantum VDF. In: AlTawy, R., Hülsing, A. (eds.) SAC 2021. LNCS, vol. 13203, pp. 441–460. Springer, Cham (2022). https://doi.org/10.1007/978-3-030-99277-4_21

    Chapter  Google Scholar 

  11. De Feo, L., et al.: Séta: supersingular encryption from torsion attacks. In: Tibouchi, M., Wang, H. (eds.) ASIACRYPT 2021. LNCS, vol. 13093, pp. 249–278. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-92068-5_9

    Chapter  Google Scholar 

  12. De Feo, L., Galbraith, S.D.: SeaSign: compact isogeny signatures from class group actions. In: Ishai, Y., Rijmen, V. (eds.) EUROCRYPT 2019. LNCS, vol. 11478, pp. 759–789. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-17659-4_26

    Chapter  Google Scholar 

  13. De Feo, L., Kohel, D., Leroux, A., Petit, C., Wesolowski, B.: SQISign: compact post-quantum signatures from quaternions and isogenies. In: Moriai, S., Wang, H. (eds.) ASIACRYPT 2020. LNCS, vol. 12491, pp. 64–93. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-64837-4_3

    Chapter  Google Scholar 

  14. De Feo, L., Leroux, A., Wesolowski, B.: SQISign twice as fast. Cryptology ePrint Archive, New algorithms for the deuring correspondence (2022)

    Google Scholar 

  15. De Feo, L., Masson, S., Petit, C., Sanso, A.: Verifiable delay functions from supersingular isogenies and pairings. In: Galbraith, S.D., Moriai, S. (eds.) ASIACRYPT 2019. LNCS, vol. 11921, pp. 248–277. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-34578-5_10

    Chapter  Google Scholar 

  16. Eisenträger, K., Hallgren, S., Lauter, K., Morrison, T., Petit, C.: Supersingular isogeny graphs and endomorphism rings: reductions and solutions. In: Nielsen, J.B., Rijmen, V. (eds.) EUROCRYPT 2018. LNCS, vol. 10822, pp. 329–368. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-78372-7_11

    Chapter  Google Scholar 

  17. Eisenträger, K., Hallgren, S., Leonardi, C., Morrison, T., Park, J.: Computing endomorphism rings of supersingular elliptic curves and connections to path-finding in isogeny graphs. Open Book Ser. 4(1), 215–232 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  18. Fouotsa, T.B., Kutas, P., Merz, S.P., Ti, Y.B.: On the isogeny problem with torsion point information. In: Hanaoka, G., Shikata, J., Watanabe, Y. (eds.) Public-Key Cryptography PKC 2022. Lecture Notes in Computer Science, vol. 13177, pp. 142–161. Springer, Cham (2022)

    Chapter  Google Scholar 

  19. Fouotsa, T.B., Petit, C.: A new adaptive attack on SIDH. In: Galbraith, S.D. (ed.) CT-RSA 2022. LNCS, vol. 13161, pp. 322–344. Springer, Cham (2022). https://doi.org/10.1007/978-3-030-95312-6_14

    Chapter  Google Scholar 

  20. Galbraith, S.D., Petit, C., Silva, J.: Identification protocols and signature schemes based on supersingular isogeny problems. In: Takagi, T., Peyrin, T. (eds.) ASIACRYPT 2017. LNCS, vol. 10624, pp. 3–33. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-70694-8_1

    Chapter  Google Scholar 

  21. Galbraith, S.D., Petit, C., Shani, B., Ti, Y.B.: On the security of supersingular isogeny cryptosystems. In: Cheon, J.H., Takagi, T. (eds.) ASIACRYPT 2016. LNCS, vol. 10031, pp. 63–91. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53887-6_3

    Chapter  Google Scholar 

  22. Jao, D., De Feo, L.: Towards quantum-resistant cryptosystems from supersingular elliptic curve isogenies. In: Yang, B.-Y. (ed.) PQCrypto 2011. LNCS, vol. 7071, pp. 19–34. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-25405-5_2

    Chapter  MATH  Google Scholar 

  23. Jao, D., Soukharev, V.: Isogeny-based quantum-resistant undeniable signatures. In: Mosca, M. (ed.) PQCrypto 2014. LNCS, vol. 8772, pp. 160–179. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-11659-4_10

    Chapter  MATH  Google Scholar 

  24. Kohel, D., Lauter, K., Petit, C., Tignol, J.P.: On the quaternion-isogeny path problem. LMS J. Comput. Math. 17(A), 418–432 (2014)

    Google Scholar 

  25. Kutas, P., Martindale, C., Panny, L., Petit, C., Stange, K.E. : Weak instances of SIDH variants under improved torsion-point attacks. Cryptology ePrint Archive, Report 2020/633 (2020). https://eprint.iacr.org/2020/633

  26. Kutas, P., Merz, S.-P., Petit, C., Weitkämper, C.: One-way functions and malleability oracles: hidden shift attacks on isogeny-based protocols. In: Canteaut, A., Standaert, F.-X. (eds.) EUROCRYPT 2021. LNCS, vol. 12696, pp. 242–271. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-77870-5_9

    Chapter  Google Scholar 

  27. Kohel, D.: Endomorphism rings of elliptic curves over finite fields. PhD thesis, University of California at Berkeley (1996)

    Google Scholar 

  28. Maino, L., Martindale, C.: An attack on SIDH with arbitrary starting curve. Cryptology ePrint Archive (2022)

    Google Scholar 

  29. Petit, C.: Faster algorithms for isogeny problems using torsion point images. In: Takagi, T., Peyrin, T. (eds.) ASIACRYPT 2017. LNCS, vol. 10625, pp. 330–353. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-70697-9_12

    Chapter  Google Scholar 

  30. Robert, D.: Breaking SIDH in polynomial time. Cryptology ePrint Archive (2022)

    Google Scholar 

  31. Schoof, R.: Counting points on elliptic curves over finite fields. J. de théorie des nombres de Bordeaux 7(1), 219–254 (1995)

    MathSciNet  MATH  Google Scholar 

  32. Ueno, R., Xagawa, K., Tanaka, Y., Ito, A., Takahashi, J., Homma, N.: Curse of re-encryption: a generic power/em analysis on post-quantum kems. IACR Trans. Cryptographic Hardw. Embed. Syst., 296–322 (2022)

    Google Scholar 

  33. Vélu, J.: Isogénies entre courbes elliptiques. Comptes-Rendus de l’Académie des Sciences, Série I, 273:238–241, juillet (1971)

    Google Scholar 

  34. Voight, J.: Quaternion Algebras. Springer Cham (2018)

    Google Scholar 

  35. Waterhouse, W.C.: Abelian varieties over finite fields. Annales Scientifiques de l’E.N.S, (1969)

    Google Scholar 

  36. Wesolowski, B.: The supersingular isogeny path and endomorphism ring problems are equivalent. In: FOCS 2021–62nd Annual IEEE Symposium on Foundations of Computer Science (2022)

    Google Scholar 

  37. Yoo, Y., Azarderakhsh, R., Jalali, A., Jao, D., Soukharev, V.: A post-quantum digital signature scheme based on supersingular isogenies. In: Kiayias, A. (ed.) FC 2017. LNCS, vol. 10322, pp. 163–181. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-70972-7_9

    Chapter  Google Scholar 

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Acknowledgements

We are very grateful to Steven Galbraith for a very thorough review of the paper and numerous comments to help improve the current write-up. We would also like to thank anonymous reviewers for their insight on our work. Finally, we thank Luca De Feo for useful remarks regarding the best way to define an isogeny representation.

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Authors and Affiliations

  1. DGA, Paris, France

    Antonin Leroux

  2. LIX, CNRS, Ecole Polytechnique, Institut Polytechnique de Paris, Palaiseau, France

    Antonin Leroux

  3. INRIA, Le Chesnay, France

    Antonin Leroux

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  1. Antonin Leroux
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Correspondence to Antonin Leroux .

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Editors and Affiliations

  1. Indian Institute of Technology Madras, Chennai, India

    Shweta Agrawal

  2. Chinese Academy of Sciences, Beijing, China

    Dongdai Lin

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Leroux, A. (2022). A New Isogeny Representation and Applications to Cryptography. In: Agrawal, S., Lin, D. (eds) Advances in Cryptology – ASIACRYPT 2022. ASIACRYPT 2022. Lecture Notes in Computer Science, vol 13792. Springer, Cham. https://doi.org/10.1007/978-3-031-22966-4_1

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  • DOI: https://doi.org/10.1007/978-3-031-22966-4_1

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