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Abstract
For primes p≡3mod4, we show that setting up CSIDH on the surface, i.e., using supersingular elliptic curves with endomorphism ring Z[(1+−p−−−√)/2], amounts to just a few sign switches in the underlying arithmetic. If p≡7mod8 then horizontal 2-isogenies can be used to help compute the class group action. The formulas we derive for these 2-isogenies are very efficient (they basically amount to a single exponentiation in Fp) and allow for a noticeable speed-up, e.g., our resulting CSURF-512 protocol runs about 5.68% faster than CSIDH-512. This improvement is completely orthogonal to all previous speed-ups, constant-time measures and construction of cryptographic primitives that have appeared in the literature so far. At the same time, moving to the surface gets rid of the redundant factor Z3 of the acting ideal-class group, which is present in the case of CSIDH and offers no extra security.
Keywords
Isogeny-based cryptography, Hard homogeneous spaces, CSIDH, Montgomery curves

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Citation

Please use this url to cite or link to this publication:

MLA
Castryck, Wouter, and Thomas Decru. “CSIDH on the Surface.” Post-Quantum Cryptography, 11th International Conference, PQCrypto 2020, edited by J. Ding and J. P. Tillich, vol. 12100, Springer, 2020, pp. 111–29, doi:10.1007/978-3-030-44223-1_7.
APA
Castryck, W., & Decru, T. (2020). CSIDH on the surface. In J. Ding & J. P. Tillich (Eds.), Post-quantum cryptography, 11th international conference, PQCrypto 2020 (Vol. 12100, pp. 111–129). Cham: Springer. https://doi.org/10.1007/978-3-030-44223-1_7
Chicago author-date
Castryck, Wouter, and Thomas Decru. 2020. “CSIDH on the Surface.” In Post-Quantum Cryptography, 11th International Conference, PQCrypto 2020, edited by J. Ding and J. P. Tillich, 12100:111–29. Cham: Springer. https://doi.org/10.1007/978-3-030-44223-1_7.
Chicago author-date (all authors)
Castryck, Wouter, and Thomas Decru. 2020. “CSIDH on the Surface.” In Post-Quantum Cryptography, 11th International Conference, PQCrypto 2020, ed by. J. Ding and J. P. Tillich, 12100:111–129. Cham: Springer. doi:10.1007/978-3-030-44223-1_7.
Vancouver
1.
Castryck W, Decru T. CSIDH on the surface. In: Ding J, Tillich JP, editors. Post-quantum cryptography, 11th international conference, PQCrypto 2020. Cham: Springer; 2020. p. 111–29.
IEEE
[1]
W. Castryck and T. Decru, “CSIDH on the surface,” in Post-quantum cryptography, 11th international conference, PQCrypto 2020, Paris, FRANCE, 2020, vol. 12100, pp. 111–129.
@inproceedings{8665496,
  abstract     = {For primes p≡3mod4, we show that setting up CSIDH on the surface, i.e., using supersingular elliptic curves with endomorphism ring Z[(1+−p−−−√)/2], amounts to just a few sign switches in the underlying arithmetic. If p≡7mod8 then horizontal 2-isogenies can be used to help compute the class group action. The formulas we derive for these 2-isogenies are very efficient (they basically amount to a single exponentiation in Fp) and allow for a noticeable speed-up, e.g., our resulting CSURF-512 protocol runs about 5.68% faster than CSIDH-512. This improvement is completely orthogonal to all previous speed-ups, constant-time measures and construction of cryptographic primitives that have appeared in the literature so far. At the same time, moving to the surface gets rid of the redundant factor Z3 of the acting ideal-class group, which is present in the case of CSIDH and offers no extra security.},
  author       = {Castryck, Wouter and Decru, Thomas},
  booktitle    = {Post-quantum cryptography, 11th international conference, PQCrypto 2020},
  editor       = {Ding, J. and Tillich, J. P.},
  isbn         = {9783030442224},
  issn         = {0302-9743},
  keywords     = {Isogeny-based cryptography,Hard homogeneous spaces,CSIDH,Montgomery curves},
  language     = {eng},
  location     = {Paris, FRANCE},
  pages        = {111--129},
  publisher    = {Springer},
  title        = {CSIDH on the surface},
  url          = {http://dx.doi.org/10.1007/978-3-030-44223-1_7},
  volume       = {12100},
  year         = {2020},
}

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