Abstract
SIDH/SIKE-style protocols benefit from key compression to minimize their bandwidth requirements, but proposed key compression mechanisms rely on computing bilinear pairings. Pairing computation is a notoriously expensive operation, and, unsurprisingly, it is typically one of the main efficiency bottlenecks in SIDH key compression, incurring processing time penalties that are only mitigated at the cost of trade-offs with precomputed tables. We address this issue by describing how to compress isogeny-based keys without pairings. As a bonus, we also substantially reduce the storage requirements of other operations involved in key compression.
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- 1.
The actual SIKE setting matches this convention.
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For the binary torsion, our methods require an extra table, \(\mathsf {L}_6\) or \(\mathsf {L}_6^*\), containing just 9 points over \(\mathbb {F}_{p^2}\), a small fraction of the space required for the other tables.
References
Azarderakhsh, R., et al.: Supersingular Isogeny Key Encapsulation. SIKE Team (2020). https://sike.org/
Azarderakhsh, R., Jao, D., Kalach, K., Koziel, B., Leonardi, C.: Key compression for isogeny-based cryptosystems. In: Proceedings of the 3rd ACM International Workshop on ASIA Public-Key Cryptography, pp. 1–10. ACM (2016)
Bernstein, D.J., Birkner, P., Joye, M., Lange, T., Peters, C.: Twisted Edwards Curves. In: Vaudenay, S. (ed.) AFRICACRYPT 2008. LNCS, vol. 5023, pp. 389–405. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-68164-9_26
Chuengsatiansup, C.: Optimizing curve-based cryptography. PhD thesis, Technische Universiteit Eindhoven (2017)
Costello, C., Jao, D., Longa, P., Naehrig, M., Renes, J., Urbanik, D.: Efficient Compression of SIDH Public Keys. In: Coron, J.-S., Nielsen, J.B. (eds.) EUROCRYPT 2017. LNCS, vol. 10210, pp. 679–706. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-56620-7_24
Costello, C., Longa, P., Naehrig, M.: Efficient Algorithms for Supersingular Isogeny Diffie-Hellman. In: Robshaw, M., Katz, J. (eds.) CRYPTO 2016. LNCS, vol. 9814, pp. 572–601. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53018-4_21
De Feo, L., Jao, D., Plût, J.: Towards quantum-resistant cryptosystems from supersingular elliptic curve isogenies. J. Math. Crypto. 8(3), 209–247 (2014)
Galbraith, S.D., Rotger, V.: Easy decision-Diffie-Hellman groups. LMS J. Comput. Math. 7, 201–218 (2004)
Hutchinson, A., Karabina, K., Pereira, G.: Memory Optimization Techniques for Computing Discrete Logarithms in Compressed SIKE. Cryptology ePrint Archive, Report 2021/368 (2020). http://eprint.iacr.org/2021/368
Menezes, A.J., van Oorschot, P.C., Vanstone, S.A.: Handbook of Applied Cryptography. CRC Press, Boca Raton, USA (1999)
Naehrig, M., Renes, J.: Dual Isogenies and Their Application to Public-Key Compression for Isogeny-Based Cryptography. In: Galbraith, S.D., Moriai, S. (eds.) ASIACRYPT 2019. LNCS, vol. 11922, pp. 243–272. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-34621-8_9
Pereira, G., Doliskani, J., Jao, D.: x-only point addition formula and faster compressed SIKE. J. Cryptographic Eng. 11(1), 1–13, (2020)
Pohlig, S.C., Hellman, M.E.: An improved algorithm for computing logarithms over \(GF(p)\) and its cryptographic significance. IEEE Trans. Inf. Theory 24(1), 106–110 (1978)
Shoup, V.: A Computational Introduction to Number Theory and Algebra. Cambridge University Press, Cambridge (2005)
Sutherland, A.V.: Structure computation and discrete logarithms in finite Abelian \(p\)-groups. Math. Comput. 80, 477–500 (2011)
Teske, E.: The Pohlig-Hellman method generalized for group structure computation. J. Symbolic Comput. 27(6), 521–534 (1999)
Zanon, G.H.M., Simplicio, M.A., Pereira, G.C.C.F., Doliskani, J., Barreto, P.S.L.M.: Faster Isogeny-Based Compressed Key Agreement. In: Lange, T., Steinwandt, R. (eds.) PQCrypto 2018. LNCS, vol. 10786, pp. 248–268. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-79063-3_12
Zanon, G.H.M., Simplicio Jr., M.A., Pereira, G.C.C.F., Doliskani, J., Barreto, P.S.L.M.: Faster key compression for isogeny-based cryptosystems. IEEE Trans. Comput. 68(5), 688–701 (2018)
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This work is supported in part by NSERC, CryptoWorks21, Canada First Research Excellence Fund, Public Works and Government Services Canada.
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Pereira, G.C.C.F., Barreto, P.S.L.M. (2021). Isogeny-Based Key Compression Without Pairings. In: Garay, J.A. (eds) Public-Key Cryptography – PKC 2021. PKC 2021. Lecture Notes in Computer Science(), vol 12710. Springer, Cham. https://doi.org/10.1007/978-3-030-75245-3_6
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