R. Schoof. Counting points on elliptic curves over finite fields. Journal de Theorie des nombres de Bordeaux, 7:219--254, 1995. Available at http://www.cirm.univ-mrs.fr/EMIS/journals/JTNB/.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....The existence of a subgroup of E with the desired properties can be guaranteed by exploiting the fact that the order of any subgroup (which is defined by the order of a point that generates the subgroup) will divide the order of the curve E, denoted #E. #E can be computed using Schoof s Algorithm [115] and then factored using conventional factoring algorithm (e.g. 76] 99] to determine if it has a subgroup with a sufficiently large prime factor. If not, or if the factoring operation does not complete within a reasonable amount of time, another curve E is chosen at random and the process is ....

R. Schoof, "Counting points on elliptic curves over finite fields," Journal Theorie des Nombres de Bordeaux, vol. 7, 1995, pp. 219-254.

....the run time, in large characteristic, Elkies [Elk98] and Atkin [Atk92] proposed replacing the division polynomials with lower degree ones found using the factorization of modular polynomials. In this way they reduced the degree from O( 2 ) to O( yielding the so called SEA algorithm (see [Sch95]) The improved run time, under reasonable hypotheses, is O(log 4 q) or O(log 6 q) Further work by Morain [Mor95a] Muller [Mul95] Dewaghe [Dew98] and others led to more efficient implementations of SEA. Morain carried out actual computations for q as large as 10 499 153 [Mor95b] ....

R. Schoof. Counting points on elliptic curves over finite fields. J. Th'eor. Nombres Bordeaux, 7:219--254, 1995.

....with coordinates in some finite field. When the finite field is large this is generally difficult to do. Ren e Schoof gave a polynomial time algorithm for counting points on elliptic curves i.e. those of genus 1, in his ground breaking paper [Sch85] Subsequent improvements by Elkies and Atkin ([Sch95], Mor95] Elk98] lowered the exponent to the point where efficient implementations became possible. After further improvements ( Cou96] Ler97] several implementations of the Schoof ElkiesAtkin algorithm were actually written and very large finite fields can now be handled in practice ....

R. Schoof. Counting points on elliptic curves over finite fields. J. Th'eor. Nombres Bordeaux, 7:219--254, 1995.

....curve of the form y 2 = x 3 ax b mod p where p is a prime number, x 3 ax b = 0 mod p does not have multiple roots, and 4a 3 27b 2 6= 0 mod p. #E p (a; b) denotes the order of E p (a; b) and can be calculated in polynomial time using algorithms such as Schoof s algorithm [22, 8, 10], and combination of Schoof s algorithm with Shanks baby step giant step algorithm [12] Definition 1. 12] An elliptic curve discrete logarithm problem (ECDL) is defined as follows. Let ff 2 E p (a; b) be a point of order q, and let fi = dff. Given ff and fi, determine the unique integer d, ....

Ren'e Schoof. Counting Points on Elliptic Curves Over Finite Fields. Journal de Th'eeorie des Nombres, Bordeaux, 7:219-254, 1995.

....the trace of an elliptic curve over a finite field has attracted considerable interest in recent years . Thanks to the work of many people in this field of research, we have now adequate tools to perform this task. From a theoretical point of view, Schoof s deterministic polynomial time algorithm [Sch85, Sch95] which enables to perform this task in O(log 8 q) elementary operations was first largely improved by Atkin [Atk88, Atk91] and Elkies [Elk91, Elk97] to yield a probabilistic algorithm whose complexity is O(log 6 q) for finite fields F q of large characteristic (using usual finite field ....

R. Schoof. Counting points on elliptic curves over finite fields. J. Th'eor. Nombres Bordeaux, 7:219--254, 1995. Available at http://almira.ceremab.u-bordeaux.fr/jntb/jtnb1995-1.html. 14

....the cost of computing the number of points on a randomly chosen curve is no longer prohibitive. For finite fields of characteristic two (specially attractive for industrial applications) the improvements of Schoof s algorithm due to Atkin, Elkies, Morain, Couveignes, Muller, Dewaghe, [Sch85, CM94, Mul95, Sch95, CDM96] were significantly speeded up by replacing the isogeny computation algorithm of Couveignes [Cou94] with a recent heuristic algorithm of the author [Ler96] In this article, once briefly recalled some basic facts about elliptic curves in Section 2, we describe in Section 3 our current ....

....and Atkin gave a construction based on modular equations to obtain E b for Elkies primes . This works in any finite field. Unfortunately, the nice analytical method that they proposed for computing explicitly the isogeny between E a and E b is only valid in finite fields of large characteristic [Sch95]. 3.2 Isogenies between Elliptic Curves in IF 2 n . Since the original method by Atkin and Elkies for computing isogenies between two elliptic curves E a and E b does not work in finite fields of small characteristic p [Sch95] only Schoof s algorithm was available during a while to count points ....

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R. Schoof. Counting points on elliptic curves over finite fields. J. Th'eor. Nombres Bordeaux, 7:219--254, 1995. Available at http://www.emath.fr/Maths/Jtnb/jtnb1995-1.html.

.... is an elliptic curve E : y 2 = x 3 ax b; a; b 2 F q ; c fl1997 American Mathematical Society 1 2 DAQING WAN a polynomial time algorithm was obtained by Schoof [Sc1] More practical but probabilistic versions were obtained later by Atkin, Elkies, Couveignes and a few other authors, see [Sc2] for an updated exposition. Schoof s algorithm was generalized to abelian varieties and curves by Pila [Pi] with some improvements by Adleman Huang [AH] Curves and abelian varieties are the cases which have been studied most extensively in the literature. For more general varieties, no ....

R. Schoof, Counting points on elliptic curves over finite fields, Journal de Th'eorie des Nombres de Bordeaux, 7(1995), 219-254.

....problems for which the general noisy Chinese remaindering problem (in which one allows different sizes for the sets S i s) arises. The first problem is point counting on elliptic curves over finite fields. The best general algorithm for this problem is the Schoof Elkies Atkin (SEA) algorithm [33, 12, 4, 5] (see [22] for implementation issues) Let E be an elliptic curve over a finite field of cardinality q. Hasse s theorem states that the cardinality of E is of the form q 1 Gamma t where jtj 2 p q. The SEA algorithm tries to determine this t, using Chinese remainders. However, in practice, it ....

R. Schoof. Counting points on elliptic curves over finite fields. J. Th'eor. Nombres Bordeaux, 7:219--254, 1995.

....less work than the previous ones thus bringing new people to this kind of calculation. 1 Introduction The first polynomial time algorithm for the enumeration of points on elliptic curves defined over finite fields was given by Schoof in [11] This algorithm was made efficient by Atkin and Elkies [12, 10, 4, 1, 2]. Elkies improvement requires the computation of some explicit l isogeny between two given elliptic curves known to be l isogenous. Elkies proposed nice modular equations for this problem. This is quite efficient as long as l p where p is the characteritic. For the case p l we proposed in [3] ....

Ren'e Schoof. Counting points on elliptic curves over finite fields. to appear in the Journal de Th'eorie des nombres de Bordeaux.

....of the torsion points. As we compute (2) in the ring GF (p) x; y] y 2 0x 3 0Ax0B; f (x) the dominant steps is the computation of x p and y p in that ring. From this, the complexity of this algorithm will be O(log 8 (p) 2. 2 Atkin Elkies Improvement (SEA) Elkies idea [7, 23] is to make use of a degree ( 0 1) 2 factor g of f when it is possible to compute g in GF (p) x] In this case, is called an Elkies prime. Otherwise is called an Atkin prime) The factor g represents an eigenspace of the Frobenius map OE, which can be computed as a kernel of an ....

....map OE, which can be computed as a kernel of an isogeny map. Thus, t mod is calculated by the eigenvalue of the eigenspace. As the ratio of Elkies primes is expected 1=2, this method will reduce the complexity to O(log 6 (p) Rather than determining the unique value of t mod , Atkin [1, 23] obtained certain restrictions on the value. Then the real value of t is found among a lot of candidates by, for example, the match and sort technique [12] The Schoof Elkies Atkin(SEA) method is obtained by combining the above two and so it is consisting of two stages, namely, I) collecting ....

[Article contains additional citation context not shown here]

Schoof, R., Counting points on elliptic curves over finite fields, J. Th'eor. Nombres Bordeaux 7 (1995) 219--254.

.... X X S2F XP S ; Y X S2F YP S : 5) 3. 2 Properties The improvements of Schoof s original algorithm [9] to count the number of points on an elliptic curve can be seen as computing isogenies between this curve and other elliptic curves easily found by solving modular equations [10]. Theorem 3 shows examples of such isogenies. Theorem 3. Let be an odd integer and d = Gamma 1) 2. Let E a be an elliptic curve defined over F 2 n such that isogenies of degree defined from it can be found. There exists a factor Q(X) of degree d of f (X) on E a such that one of these ....

Schoof, R. Counting points on elliptic curves over finite fields. To appear in Proc. Journ'ees Arithm'etiques 93, Jan. 1995.

....the main algorithmic problems to be solved is the efficient computation of the cardinality of elliptic curves over finite fields. It was not until recently that Schoof s polynomial time algorithm for solving this problem could be efficiently used, due to the work of Atkin [1, 2] and [15] see also [48, 44] and the results of the implementation given in [44, 30, 46] It gave satisfactory results in the large characteristic case, and only very recently was it possible to make it work as well in the small characteristic case, using Couveignes s thesis [10] The aim of this paper is to explain how ....

....his own solution to the problem of computing g (X) using more modular equations and modular forms. Though rather tricky to implement, his approach is very fast in practice. Recently, Couveignes and Morain showed how to use powers of small Elkies primes [12] All these ideas are also described in [48] and were implemented [2, 27, 44, 43] The results are striking, the record being that of the computation of the cardinality of a curve modulo a prime p of 500 digits (see [30] The only remaining problem was that these ideas could not work when p = 2 (and more generally p ) As a matter of ....

Schoof, R. Counting points on elliptic curves over finite fields. To appear in Proc. Journ'ees Arithm'etiques 93, Jan. 1995.

....algorithmic problems to be solved is the efficient computation of the cardinality of elliptic curves over finite fields. It was not until recently that Schoof s polynomial time algorithm for solving this problem could be efficiently used, due to the work of Atkin [1, 2] and Elkies [19] see also [48, 42] and the results of the implementations given in [42, 34, 30, 45] It gave satisfactory results in the large characteristic case, and only very recently was it possible to make it work as well in the small characteristic case, using Couveignes s thesis [12] The aim of this paper is to explain ....

....his own solution to the problem of computing g (X) using more modular equations and modular forms. Though rather tricky to implement, his approach is very fast in practice. Recently, Couveignes and Morain showed how to use powers of small Elkies primes [15] All these ideas are also described in [48] and were implemented [2, 30, 42, 41, 45] The results are striking, the record being that of the computation of the cardinality of a curve modulo a prime p of 500 digits (see [34] The only remaining problem was that these ideas could not work when p = 2 (and more generally p ) As a matter ....

Schoof, R. Counting points on elliptic curves over finite fields. To appear in Proc. Journ'ees Arithm'etiques 93, Jan. 1995.

....that we are looking for one isogeny of degree . Let F consist of the point at infinity and the points of abscissa (r 1 = 1 r ( Gamma 1) 2. The curve E is in fact C = 1 = 2 ) It can be shown that the coefficients of E can be deduced in a rational way from that of E (see [21] for instance) The functions and are related via (z) I( z) COMPUTATION OF ISOGENIES 5 At this point, we can find the isogeny I using Stark s method [23] i.e. developping the Weierstrass function of E as a continued fraction in that of E. Since we know the degree of the ....

....of I, which is really what we are looking for. We need about coefficients of and to be able to perform the computations. Therefore, the method works as well when char(K ) 0, provided char(K ) 3 and char(K ) AE . Computing I takes O( 2 ) operations and uses O( memory. We refer to [2, 10, 5, 3, 21, 19, 20] for more details. The method is very efficient in practice and the record is the computation of the cardinality of a curve modulo a prime of 500 decimal digits [19] 6. The small characteristic case 6.1. The problems to be solved. We assume from now on that K is a finite field of characteristic ....

Schoof, R. Counting points on elliptic curves over finite fields. J. Th'eor. Nombres Bordeaux 7 (1995), 219--254.

....and the law, denoted #, has OE = 0 : 1 : 0] as neutral element. We denote by fn the n th division polynomial in x. The degree of fn is (n 2 1) 2 if n is odd. The group of n torsion points, E[n] P # E( F q ) nP = OE can be represented by F q [x, y] f n (x) F(x, y) see [18]) The morphism # : E( F q ) # E( F q ) x, y) ## (x q , y q ) of E satisfies # 2 t# q = 0 over E( F q ) with t # Z, satisfying t # 2 # q. Recall : #E(F q ) q 1 t. When # is an odd prime number (see [6] for # = 2) we consider the restriction # # of # to E[#] ....

R. Schoof, Counting points on elliptic curves over finite fields, J. Theor. Nombres Bordeaux 7 (1995), 219-254. MR 97i:11070

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R. Schoof. Counting points on elliptic curves over finite fields. Journal de Theorie des nombres de Bordeaux, 7:219--254, 1995. Available at http://www.cirm.univ-mrs.fr/EMIS/journals/JTNB/.

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R. Schoof. Counting Points on Elliptic Curves over Finite Fields. In J. Theor. Nombres Bordeaux, 7:219--254, 1995.

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R. Schoof, Counting points on elliptic curves over finite fields, J. Theorie des Nombres de Bordeaux (1995), no. 7, 219--254.

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R. Schoof. Counting points on elliptic curves over finite fields. J. Theor. Nombres Bordeaux, 7(1):219--254, 1995.

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Schoof, R. Counting points on elliptic curves over finite fields. J. Th'eor. Nombres Bordeaux 7 (1995), 219--254.

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