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Computing cardinalities of $\mathbb{Q}$ -curve reductions over finite fields

  • François Morain (a1), Charlotte Scribot (a2) and Benjamin Smith (a3)

Abstract

We present a specialized point-counting algorithm for a class of elliptic curves over $\mathbb{F}_{p^{2}}$ that includes reductions of quadratic $\mathbb{Q}$ -curves modulo inert primes and, more generally, any elliptic curve over $\mathbb{F}_{p^{2}}$ with a low-degree isogeny to its Galois conjugate curve. These curves have interesting cryptographic applications. Our algorithm is a variant of the Schoof–Elkies–Atkin (SEA) algorithm, but with a new, lower-degree endomorphism in place of Frobenius. While it has the same asymptotic asymptotic complexity as SEA, our algorithm is much faster in practice.

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References

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1. Bostan, A., Morain, F., Salvy, B. and Schost, É., ‘Fast algorithms for computing isogenies between elliptic curves’, Math. Comp. 77 (2008) no. 263, 17551778.
2. Cheng, Q., ‘Straight-line programs and torsion points on elliptic curves’, Comput. Complexity 12 (2003) 150161.
3. Costello, C., Hisil, H. and Smith, B., ‘Faster compact Diffie–Hellman: endomorphisms on the x-line’, Advances in cryptology – EUROCRYPT 2014 – Proceedings of the 33rd Annual International Conference on the Theory and Applications of Cryptographic Techniques, Copenhagen, Denmark, May 11–15, 2014 , Lecture Notes in Computer Science 8441 (eds Nguyen, P. Q. and Oswald, E.; Springer, Berlin, 2014) 183200.
4. Costello, C. and Longa, P., ‘Fourℚ: four-dimensional decompositions on a ℚ-curve over the Mersenne prime’, Advances in cryptology – ASIACRYPT 2015 – 21st International Conference on the Theory and Application of Cryptology and Information Security, Auckland, New Zealand, November 29–December 3, 2015, Proceedings, Part I , Lecture Notes in Computer Science 9452 (eds Iwata, T. and Cheon, J. H.; Springer, Berlin, 2015) 214235.
5. Couveignes, J.-M. and Morain, F., ‘Schoof’s algorithm and isogeny cycles’, Algorithmic number theory, 1st Algorithmic Number Theory Symposium, Cornell University, May 6–9, 1994 , Lecture Notes in Computer Science 877 (eds Adleman, L. and Huang, M.-D.; Springer, Berlin, 1994) 4358.
6. Fouquet, M. and Morain, F., ‘Isogeny volcanoes and the SEA algorithm’, Algorithmic number theory, Proceedings of the 5th International Symposium, ANTS-V, Sydney, Australia, July 2002 , Lecture Notes in Computer Science 2369 (eds Fieker, C. and Kohel, D. R.; Springer, Berlin, 2002) 276291.
7. von zur Gathen, J. and Gerhard, J., Modern computer algebra (Cambridge University Press, Cambridge, 1999).
8. Gaudry, P. and Morain, F., ‘Fast algorithms for computing the eigenvalue in the Schoof–Elkies–Atkin algorithm’, ISSAC ’06: Proceedings of the 2006 International Symposium on Symbolic and Algebraic Computation (ACM Press, New York, NY, 2006) 109115.
9. González, J., ‘Isogenies of polyquadratic ℚ-curves to their Galois conjugates’, Arch. Math. 77 (2001) 383390.
10. Guillevic, A. and Ionica, S., ‘Four-dimensional GLV via the Weil restriction’, Advances in cryptology – ASIACRYPT 2013 , Lecture Notes in Computer Science 8269 (eds Sako, K. and Sarkar, P.; Springer, 2013) 7996.
11. Hasegawa, Y., ‘Q-curves over quadratic fields’, Manuscripta Math. 94 (1997) no. 1, 347364.
12. Joux, A. and Lercier, R., ‘Chinese & match, an alternative to Atkin’s match and sort method used in the SEA algorithm’, Math. Comp. 70 (2001) no. 234, 827836.
13. Kaltofen, E. and Shoup, V., ‘Subquadratic-time factoring of polynomials over finite fields’, Math. Comp. 67 (1998) no. 223, 11791197.
14. Lercier, R., ‘Finding good random elliptic curves for cryptosystems defined over F2 n ’, Advances in cryptology – EUROCRYPT ’97 , Lecture Notes in Computer Science 1233 (ed. Fumy, W.; Springer, Berlin, 1997) 379392.
15. Mihăilescu, P., Morain, F. and Schost, É., ‘Computing the eigenvalue in the Schoof–Elkies–Atkin algorithm using Abelian lifts’, ISSAC ’07: Proceedings of the 2007 International Symposium on Symbolic and Algebraic Computation (ACM Press, New York, NY, 2007) 285292.
16. Sako, K. and Sarkar, P. (eds), Advances in cryptology – ASIACRYPT 2013 , Lecture Notes in Computer Science 8269 (Springer, 2013).
17. Satoh, T., ‘On p-adic point counting algorithms for elliptic curves over finite fields’, Algorithmic number theory, Proceedings of the 5th International Symposium, ANTS-V, Sydney, Australia, July 7–12, 2002 , Lecture Notes in Computer Science 2369 (eds Fieker, C. and Kohel, D. R.; Springer, 2002) 4366.
18. Schoof, R., ‘Elliptic curves over finite fields and the computation of square roots mod p ’, Math. Comp. 44 (1985) 483494.
19. Schoof, R., ‘Nonsingular plane cubic curves over finite fields’, J. Combin. Theory Ser. A 46 (1987) no. 2, 183211.
20. Schoof, R., ‘Counting points on elliptic curves over finite fields’, J. Théor. Nombres Bordeaux 7 (1995) 219254.
21. Shparlinski, I. E. and Sutherland, A. V., ‘On the distribution of Atkin and Elkies primes’, Found. Comput. Math. 14 (2014) no. 2, 285297.
22. Shparlinski, I. E. and Sutherland, A. V., ‘On the distribution of Atkin and Elkies primes for reductions of elliptic curves on average’, LMS J. Comput. Math. 18 (2015) no. 1, 308322.
23. Smith, B., ‘Families of fast elliptic curves from ℚ-curves’, Advances in cryptology – ASIACRYPT 2013 , Lecture Notes in Computer Science 8269 (eds Sako, K. and Sarkar, P.; Springer, 2013) 6178.
24. Smith, B., ‘The ℚ-curve construction for endomorphism-accelerated elliptic curves’, J. Cryptology (2015).
25. Sutherland, A. V., ‘Identifying supersingular elliptic curves’, LMS J. Comput. Math. 15 (2012) 317325.
26. Sutherland, A. V., ‘On the evaluation of modular polynomials’, ANTS X—Proceedings of the Tenth Algorithmic Number Theory Symposium, Open Book Series 1 (Mathematical Science Publishers, Berkeley, CA, 2013) 531555.
27. Zhu, H. J., ‘Group structures of elementary supersingular abelian varieties over finite fields’, J. Number Theory 81 (2000) 292309.
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LMS Journal of Computation and Mathematics
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