Abstract
A prohibitive barrier faced by elliptic curve users is the difficulty of computing the curves’ cardinalities. Despite recent theoretical breakthroughs, point counting still remains very cumbersome and intensively time consuming.
In this paper we show that point counting can be avoided at the cost of a protocol slow-down. This slow-down factor is quite important (typically ≅) 500) but proves that the existence of secure elliptic-curve signatures is not necessarily conditioned by point counting.
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de Bruijn, N.: On the number of positive integers ≤ x and free of prime factos ≥ y. Indagationes Mathematicae 13, 50–60 (1951)
de Bruijn, N.: On the number of positive integers ≤ x and free of prime factos ≥ y, II. Indagationes Mathematicae 28, 236–247 (1966)
Couveignes, J.-M., Dewaghe, L., Morain, F.: Isogeny cycles and the Schoof-Elkies-Atkin algorithm, Rapport de recherche LIX/RR/96/03, Laboratoire d’informatique de l’École Polytechnique (1996)
Couveignes, J.-M., Morain, F.: Schoof’s algorithm and isogeny cycles. In: Huang, M.-D.A., Adleman, L.M. (eds.) ANTS 1994. LNCS, vol. 877, pp. 43–58. Springer, Heidelberg (1994)
Dickman, K.: On the frequency of numbers containing prime factors of a certain relative magnitude. Arkiv för matematik, astronomi och fysik 22A(10), 1–14 (1930)
Dixon, J.: Asymptotically fast factorization of integers. Mathematics of computation 36(153), 255–260 (1981)
Halberstam, H.: On integers whose prime factors are small. Proceedings of the London mathematical society 3(21), 102–107 (1970)
Howe, E.: On the group orders of elliptic curves over finite fields. Compositio mathematica 85, 229–247 (1993)
Koblitz, N.: Primality of the number of points on an elliptic curve over a finite field. Pacific Journal of Mathematics 131, 157–165 (1988)
Kunihiro, N., Koyama, K.: Equivalence of counting the number of points on elliptic curve over the ring Zn and factoring n. In: Nyberg, K. (ed.) EUROCRYPT 1998. LNCS, vol. 1403, pp. 47–58. Springer, Heidelberg (1998)
Lay, G., Zimmer, H.: Constructing elliptic curves with given group order over large finite fields. In: Huang, M.-D.A., Adleman, L.M. (eds.) ANTS 1994. LNCS, vol. 877, Springer, Heidelberg (1994)
Lenstra Jr., H.: Factoring integers with elliptic curves. Ann. math. 126, 649–673 (1987)
Lercier, R.: Computing isogenies in GF(2n). In: Cohen, H. (ed.) ANTS 1996. LNCS, vol. 1122, pp. 197–212. Springer, Heidelberg (1996)
Lercier, R., Morain, F.: Counting the number of points on elliptic curves over finite fields: strategies and performances. In: Guillou, L.C., Quisquater, J.-J. (eds.) EUROCRYPT 1995. LNCS, vol. 921, pp. 79–94. Springer, Heidelberg (1995)
Lercier, R., Morain, F.: Counting the number of points on elliptic curves over F\(_{p^n}\) using Couveigne’s algorithm, Rapport de recherche LIX/RR/95/09, Laboratoire d’informatique de l’École Polytechnique (1995)
Menezes, A.: Elliptic curve public key cryptosystems, p. 25. Kluwer academic publishers, Dordrecht (1983)
Menezes, A., Vanstone, S., Zuccharato, R.: Counting points on elliptic curves over F\(_{2^m}\). Mathematics of computation 60(201), 407–420 (1993)
Pohlig, S., Hellman, M.: An improved algorithm for computing logarithms over GF(p) and its cryptographic significance. IEEE Transactions on Information Theory 24, 106–110 (1978)
Poupard, G., Stern, J.: A practical and provably secure design for on the fly authentication and signature generation. In: Nyberg, K. (ed.) EUROCRYPT 1998. LNCS, vol. 1403, pp. 422–436. Springer, Heidelberg (1998)
Schoof, R.: Elliptic curves over finite fields and the computation of square roots mod p. Mathematics of computation 44, 483–494 (1985)
Schoof, R.: Counting points on elliptic curves over finite fields. CACM 21(2), 120–126 (1978)
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Coron, JS., Handschuh, H., Naccache, D. (1999). ECC: Do We Need to Count?. In: Lam, KY., Okamoto, E., Xing, C. (eds) Advances in Cryptology - ASIACRYPT’99. ASIACRYPT 1999. Lecture Notes in Computer Science, vol 1716. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-48000-6_11
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DOI: https://doi.org/10.1007/978-3-540-48000-6_11
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