N. Elkies, Elliptic and modular curves over nite elds and related computational issues, in Computational perspectives on number theory, (ed. D. Buell, J. Teitelbaum), AMS/IP Studies in Adv. Math., 7, 1998, 21-76.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... assistant, sponsored by the Fund for Scienti c Research Flanders (Belgium) The problem of counting the number of points on elliptic curves over nite elds of any characteristic can be solved in polynomial time using Schoof s algorithm [26] and its improvements due to Atkin [2] and Elkies [6]. An excellent account of the resulting SEA algorithm can be found in [3] and [17] For nite elds of small characteristic, Satoh [25] described an algorithm based on p adic methods which is asymptotically faster than the SEA algorithm. Skjernaa [27] and Fouquet, Gaudry and Harley [8] extended the ....

N. Elkies. Elliptic and modular curves over nite elds and related computational issues. Computational Perspectives on Number Theory, pages 21-76, 1998.

....general algorithm there is not very practical. As previously mentioned, our algorithm can be extended to arbitrary hyperelliptic curves in characteristic 2, but we focus on the simplest cases in this paper. We refer to the references in [2] for the large literature on point counting, including [7, 19], and the more recent work [8, 9, 10, 11, 12, 17, 18, 22, 23, 25] Sections 2, 3, 4 and 5 lay the mathematical foundation of our algorithm: it is based mainly upon an extension of the work of Dwork [6] due to Adolphson and Sperber [1] Section 6 contains a statement of the algorithm for what we ....

N. Elkies, Elliptic and modular curves over nite elds and related computational issues, in \Computational perspectives in number theory: Proceedings of a conference in honour of A.O.L. Atkin" , (D.A. Buell and J.T. Teitelbaum), American Mathematical Society International Press 7, 1998, 21-76.

....) if n is odd; n 1 f n 1 if n is even. Once we have a root of (X; J) V elu s formulas [30] enable us to compute a factor g (X) X) This property is at the heart of the improvements of Elkies and Atkin to Schoof s algorithm for computing the cardinality of curves over nite elds [1, 25, 10] (see also [19] for technicalities related to the actual computations) In the table below, for prime , we suppose v is a root of (X; j) and we give the factor of that can be obtained in Table 2. factor 2 (v 2 8) X v 2 16; X v 3 30 v 3 ....

....r 1 and r j 1. If k denotes the number of factors of , then ( 1) We can make precise the rst part of the theorem as follows: Theorem 3.2. Let p = U ) 4. If j V , then (X) has 1 roots modulo p. Proof: See Kohel s thesis [14] 2 We brie y summarize Elkies s idea [10]. Let be the Frobenius of the curve, sending any point P = x; y) of E(F p ) to (x ; y ) Theorem 3.3. Let (X) X UX p denote the characteristic polynomial of the Frobenius of the elliptic E of cardinality p 1 U . When 6= 1, the restriction of to E[ denoted by j E[ ....

N. D. Elkies. Elliptic and modular curves over nite elds and related computational issues. In D. A. Buell and J. T. Teitelbaum, editors, Computational Perspectives on Number Theory: Proceedings of a Conference in Honor of A. O. L. Atkin, volume 7 of AMS/IP Studies in Advanced Mathematics, pages 21-76. American Mathematical Society, International Press, 1998.

.... systems work in the Jacobian of superelliptic curves [12] and of C ab curves [1] The problem of counting the number of points on elliptic curves over nite elds of any characteristic can be solved in polynomial time using Schoof s algorithm [33] and its improvements due to Atkin [2] and Elkies [7]. An excellent F.W.O. research assistant, sponsored by the Fund for Scienti c Research Flanders (Belgium) account of the resulting SEA algorithm can be found in [3] and [20] For nite elds of small characteristic, Satoh [30] described an algorithm based on p adic methods which is ....

N. Elkies. Elliptic and modular curves over nite elds and related computational issues. Computational Perspectives on Number Theory, pages 21-76, 1998.

....and only if they have the same number of F q points. Schoof [30] proposed a polynomial time algorithm for counting the number of points on an elliptic curve over a nite eld. There has been a considerable amount of research building on Schoof s idea (for instance, Atkin [1] 2] Elkies [10] [11], Couveignes [8] Couveignes and Morain [9] Lercier [20] Lercier and Morain [21] This means that there is an ecient solution to the problem of determining whether two elliptic curves over F q are isogenous; namely, compute the number of points on each curve and see if the total is the same. ....

....nd the path connecting j 1 to j 2 . Add this chain of l isogenies to those already found in Stage 1 and use the theory of Elkies and V elu to construct the explicit polynomial form of each l isogeny. We now give further details about the stages. Stage 0 involves standard methods, see Elkies [11] for details. Of course, rather than computing all the equations in advance, one would just compute and store them as they are required. Stage 1: We know that End(E i ) is an order in OK with conductor dividing c. Note that, due to equation (2) it is more likely that End(E 1 ) and End(E 2 ) have ....

N. Elkies, Elliptic and modular curves over nite elds and related computational issues, in Computational perspectives on number theory: proceedings of a conference in honor of A.O.L. Atkin, September 1995.

....one should avoid curves with special properties and instead choose a random curve whose number of points is divisible by a large prime, over a prime eld or an extension of prime degree. This ideal procedure was made possible in practice by the SEA algorithm due to Schoof [Sch85] Sch95] Elkies [Elk98], Atkin [Atk92] and others [Cou94] Cou96] Mor95] Ler97a] M ul95] Dew98] etc. With this method, counting points on one given curve is reasonably fast. However nding a cryptographically suitable curve requires testing many curves and this takes much more time. For instance, Johnson and ....

....over F q with this algorithm is O(log 5 q) using asymptotically fast methods for arithmetic (or O(log 8 q) using na ve arithmetic) The degree of the division polynomial is O( 2 ) which grows quickly and causes this algorithm to be slow in practice. In large characteristic, Elkies [Elk98] and Atkin [Atk92] improved Schoof s method yielding the so called SEA algorithm (see [Sch95] with run time reduced to O(log 4 q) or O(log 6 q) under reasonable hypotheses. Their idea is to construct a factor of degree O( of the division polynomial and work with it instead. Such a ....

N. Elkies. Elliptic and modular curves over nite elds and related computational issues. In D.A. Buell and eds. J.T. Teitelbaum, editors, Computational Perspectives on Number Theory, pages 21-76. AMS/International Press, 1998. Proceedings of a Conference in Honor of A.O.L. Atkin.

....the data. Point Discriminant 1 Cusp 1 1 D = 12 (0; 2) D = 27 (0; 2) D = 27 (1; 1) Cusp [1 : 7] 1; 1) D = 12 (3; 7) Exceptional (3; 7) D = 3 (3=2; 7=8) Exceptional (3=2; 7=8) D = 3 The exceptional points correspond to quadratic Q curves. The j invariants can be computed using the method of Elkies [7]. The point (3; 7) corresponds to the elliptic curve having j invariant equal to j 1 = 27048390693611915236875=2 14 6098504215856136863625=2 14 p 3 29: The point (3=2; 7=8) corresponds to the elliptic curve having j invariant equal to j 2 = 8366877442964720618049886816125=2 92 ....

N. Elkies, Elliptic and modular curves over nite elds and related computational issues, in D. A. Buell and J. T. Teitelbaum (eds.), Computational Perspectives on Number Theory, AMS Studies in Advanced Math., p. 21-76 (1998)

....it is applied to groups whose order is very dicult to compute. This is not the case for groups of elliptic curves over nite elds GF(q) q = 2 k or q a large prime) because of the polynomial time algorithm of Schoof [32] and recent improvements by Atkins, Couveignes, Elkies, Lercier and Morain [1, 4, 5, 8, 18, 19, 23]. The expected run time of the best of these algorithms is O( log q) 6 ) Our notation and assumptions are described in x2, and the new algorithms for determining a cryptosystem, signing and verifying a message are given in x3. The new algorithms are compared with the standard ElGamal ....

N. D. Elkies, Elliptic and modular curves over nite elds and related computational issues, in Advances in Cryptology, Proc. ASIACRYPT'98, Lecture Notes in Computer Science 1514, 1998, 21-76. 9

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N. Elkies, Elliptic and modular curves over nite elds and related computational issues, in Computational perspectives on number theory, (ed. D. Buell, J. Teitelbaum), AMS/IP Studies in Adv. Math., 7, 1998, 21-76.

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N. D. Elkies. Elliptic and modular curves over nite elds and related computational issues. In D. A. Buell and J. T. Teitelbaum, editors, Computational perspectives on number theory, volume 7 of AMS/IP Stud. Adv. Math., pages 21-76. Amer. Math. Soc., Providence, RI, 1998.

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N. Elkies, Elliptic and modular curves over nite elds and related computational issues, in D. A. Buell and J. T. Teitelbaum (eds.) Computational perspectives on number theory, AMS (1997) 21-76.

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Noam D. Elkies. Elliptic and modular curves over nite elds and related computational issues. In Computational perspectives on number theory, in honor of A.O.L Atkin, volume 7 of AMS/IP Studies in Advanced Mathematics, pages 21-76. AMS/IP, 1998.

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N. Elkies. Elliptic and modular curves over nite elds and related computational issues. Computational Perspectives on Number Theory, pages 21-76, 1998.

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N. Elkies, Elliptic and modular curves over nite elds and related computational issues, in \Computational perspectives in number theory: Proceedings of a conference in honour of A.O.L. Atkin" , (D.A. Buell and J.T. Teitelbaum), American Mathematical Society International Press 7, 1998, 21-76.

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