A.O.L. Atkin. The number of points on an elliptic curve modulo a prime, 1992. Email on the NMBRTHRY mailing list.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... F.W.O. research 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 ....

A.O.L. Atkin. The number of points on an elliptic curve modulo a prime. Series of e-mails to the NMBRTHRY mailing list, 1992.

....) 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 ....

....7 778v 7 881) Table 2. Factors of f . 3. The foundations of the SEA algorithm What we have just described over Q has an impact on what happens modulo p, when we reduce a CM curve E modulo a prime p of good reduction that splits completely in t . We take the following result from [1] (see also [25] Let and p be two distinct primes, and E=F p an elliptic curve. Put #E = p 1 U , D = 4p U . We denote the splitting type of a squarefree polynomial P (X) by the degrees of its factors. For instance, a polynomial of degree 4 having two linear factors and one quadratic factor ....

A. O. L. Atkin. The number of points on an elliptic curve modulo a prime (II). Draft. Available on http://listserv.nodak.edu/archives/nmbrthry.html, 1992.

.... less practical 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 ....

A.O.L. Atkin. The number of points on an elliptic curve modulo a prime. Series of e-mails to the NMBRTHRY mailing list, 1992.

....over F q if 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 ....

A. O. L. Atkin, The number of points on an elliptic curve modulo a prime 2, Preprint (1992)

....isogenous over F q if 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 ....

A. O. L. Atkin, The number of points on an elliptic curve modulo a prime, Preprint (1988)

....y 1 if P = Q Table 1: Addition Formulae for the Curve y 2 = x 3 ax b Example 1: Let m 2 Z. Then the map [m] E(F p ) E(F p ) Q 7 mQ is an endomorphism of E(F p ) This is called the multiplication by m endomorphism and applies to any group. Observe that [0] is the zero map while [1] is the identity. Example 2: Since we are working with fields of characteristice p, we see that (x 1 x 2 ) p = x p 1 x p 2 and (x 1 x 2 ) p = x p 1 x p 2 for any field elements x 1 ; x 2 2 F p . This allows us to define the p th power Frobenius endomorphism OE p as OE p : E(F p ....

....m 2 Zwith p 6 jm. Then [m] is separable and deg[m] deg s [m] m 2 2) Let e 1 be an integer. Then [p e ] is inseparable with deg[p e ] p 2e and deg s [p e ] deg i [p e ] p e 3) The Frobenius endomorphism OE is purely inseparable with deg OE = deg i OE = p 4) The endomorphism [1] Gamma OE (which will come into play later) is a separable endomorphism. 5) From remarks 1) and 2) and the fact that [m] 6= Gammam] if m 6= 0, we see that [m] n] implies that m = n. The reason for introducing the degree function is shown by the following theorem. 3 Theorem 1 Let 2 ....

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A.O.L. Atkin. The Number of Points on an Elliptic Curve Modulo a Prime (ii). preprint, 1992.

....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 Menezes [JM99] ....

....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 factor can be found by ....

A. O. L. Atkin. The number of points on an elliptic curve modulo a prime. Series of e-mails to the NMBRTHRY mailing list, 1992.

....modulo the division polynomials. The time required for point counting over F q with Schoof s algorithm is O(log 5 q) with asymptotically fast algorithms for arithmetic (or O(log 8 q) with naive arithmetic) To reduce 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 ....

A. O. L. Atkin. The number of points on an elliptic curve modulo a prime. Series of e-mails to the NMBRTHRY mailing list, 1992.

....= e n r (V 0 ; W 0 ) and its order n s = ord( 5) If r s = d, then return (r; s) Otherwise, go to 1) We provide explanation of each step in the above method. In Step 1, we compute N 1 in polynomial time, using the Schoof Elkies Atkin algorithm and its variants [26] 5] 16] 6] 1][2][8] 22] 17] 7] 12] As described earlier, N Miller is regarded as Miller s algorithm that nds the group structure of the n primary part E(F q k ) n of E(F q k ) In 2) note that the multiplication by N k =n d map [N k =n d ] E(F q k ) E(F q k ) is an abelian group homomorphism and ....

A. O. L. Atkin, \The number of points on an elliptic curve modulo a prime (ii)," Draft, 1992.

....Compute = e n r (V 0 ; W 0 ) and its order n s = ord( 5) If r s = d, then return (r; s) Otherwise, go to 1) We provide explanation of each step in the above method. In Step 1, we compute N 1 in polynomial time, using the Schoof Elkies Atkin algorithm and its variants [26] 5] 16] 6][1][2] 8] 22] 17] 7] 12] As described earlier, N Miller is regarded as Miller s algorithm that nds the group structure of the n primary part E(F q k ) n of E(F q k ) In 2) note that the multiplication by N k =n d map [N k =n d ] E(F q k ) E(F q k ) is an abelian group homomorphism ....

A. O. L. Atkin, \The number of points on an elliptic curve modulo a prime", Draft, 1988.

....choose a prime q between 10 l and 10 l 1 , then randomly chose a; b 2 (F q ) and check whether 4a 3 27b 2 6= 0 mod q. If this is the case, we use our implementation for the group structure computation [17] or, for primes q 10 9 , the implementation [8] of an algorithm of Atkin [1], to compute the order n of E a;b (F q ) Finally we factor n to find p and k. Having built up this file, for k = 3; 4; 13 we go through the following algorithm: 1. Read 6 tuple (q; a; b; n; p; k) from file. 2. Use the algorithm for group structure computation to find a group element g ....

O. Atkin. The number of points on an elliptic curve modulo a prime. Manuscript.

....(t 1 ; t 2 ) It is now easy to compute the opposite of P , simply noting that this opposite is the third point of intersection of the line joining P and O E with E . Precisely, if GammaP = t 0 ; s 0 ) one has t 0 = t Gamma1 a 1 t a 3 s : 9) When t 1 = t 2 , we get F d (t 1 ) [2]t 1 = 2 t 1 Gamma a 1 t 2 1 Gamma 2 a 2 t 3 1 (a 1 a 2 Gamma 7 a 3 ) t 4 1 O(t 5 1 ) 10) 4.3. The Hasse invariant. Using Proposition , we see that [p] t) Gamma F p (t=S(t) G p (t=S(t) Gamma1=S(t) p ; S( p] t) Gamma 1 G p (t=S(t) Gamma1=S(t) p : 11) ....

....1 to p r 1 =2 and N prime to p do (a) compute (M(t) S (M(t) N ] ffi (U(t) S(U(t) b) use IsogenyTest to test whether M comes from an isogeny; if yes, stop. Note that we compute (M(t) S (M(t) using a formal addition between the preceding computed value and (U(t) S(U(t) or [2](U(t) S(U(t) The cost of the second approach is the cost of finding one morphism, O(L 1=2 ) multiplications, plus O(L) times the cost of an addition in the formal group O(L ) multiplications plus O(L) times the cost of the isogeny test of cost O(L 2 ) So, the complexity of ....

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Atkin, A. O. L. The number of points on an elliptic curve modulo a prime (ii). Draft. Available on http://listserv.nodak.edu/archives/nmbrthry.html, 1992.

....idea works for nearly half the primes #, called Elkies primes. For such an #, the algorithm has just to compute an eigenvalue of # acting on E[#] Atkin [1] had given in 1988 the sort and match method used now for bad primes #. Then he made the algorithm practical for very large finite fields [2] and the method became the SEA (for Schoof Elkies Atkin) algorithm. For the last improvements in this scope, see [5] 6] and [12] and for the case p small, see [7] and the implementation in [13] In this article we compute, for an Elkies prime #, the complexity of the best asymptotic method used ....

.... isogenies are defined over F q and if we let E[#] # be an eigenspace with P # a generator, we have h # (x) # d i=1 (x x(iP # ) # F q [x] and E[#] # = #P # # = F q [x, y] h # (x) F(x, y) Let # # (x, y) 0 be the canonical equation of the modular curve X 0 (#) see [2], 15] for a simpler equation) We know that # is an Elkies prime if and only if # # (j(E) x) 0 has a root in F q . For p #= 2, 3 and # an Elkies prime, the formulas of Atkin [2] 15] give, from a root of # # (j(E) x) 0 in F q , the value of p 1 = # d i=1 x(iP # ) and the coe#cients of ....

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A. O. L. Atkin, The number of points on an elliptic curve modulo a prime (II). Draft, 1992.

....[10] showed how to perform computations in the kernel of an isogeny of degree #, by computing a factor of degree d = # 1) 2 of f # . This idea works for nearly half the primes #, called Elkies primes. For such an #, the algorithm has just to compute an eigenvalue of # acting on E[#] Atkin [1] had given in 1988 the sort and match method used now for bad primes #. Then he made the algorithm practical for very large finite fields [2] and the method became the SEA (for Schoof Elkies Atkin) algorithm. For the last improvements in this scope, see [5] 6] and [12] and for the case p ....

A. O. L. Atkin, The number of points on an elliptic curve modulo a prime (I). Draft, 1988.

.... y Reynald LERCIER z Antoine.Joux ens.fr lercier celar.fr April 6, 1999 Abstract A classical way to compute the number of points of elliptic curves defined over finite fields from partial data obtained in SEA (Schoof Elkies Atkin) algorithm is a so called Match and Sort method due to Atkin [Atk88, Atk91]. This method is a baby giant steps way to find the number of points among C candidates with O( p C) elliptic curve additions. Observing that the partial information modulo Atkin s primes is redundant, we propose to take advantage of this redundancy to eliminate the usual elliptic curve ....

....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 arithmetic) In finite fields of small characteristic, we are indebted to Couveignes [Cou94] for filling remaining gaps and ....

A. O. L. Atkin. The number of points on an elliptic curve modulo a prime, 1991. Email on the Number Theory Mailing List.

.... y Reynald LERCIER z Antoine.Joux ens.fr lercier celar.fr April 6, 1999 Abstract A classical way to compute the number of points of elliptic curves defined over finite fields from partial data obtained in SEA (Schoof Elkies Atkin) algorithm is a so called Match and Sort method due to Atkin [Atk88, Atk91]. This method is a baby giant steps way to find the number of points among C candidates with O( p C) elliptic curve additions. Observing that the partial information modulo Atkin s primes is redundant, we propose to take advantage of this redundancy to eliminate the usual elliptic curve ....

....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 arithmetic) In finite fields of small characteristic, we are indebted to Couveignes [Cou94] for filling remaining gaps and ....

[Article contains additional citation context not shown here]

A. O. L. Atkin. The number of points on an elliptic curve modulo a prime, 1988. Email on the Number Theory Mailing List.

No context found.

A.O.L. Atkin. The number of points on an elliptic curve modulo a prime, 1992. Email on the NMBRTHRY mailing list.

No context found.

O. Atkin. The number of points on an elliptic curve modulo a prime. Manuscript.

No context found.

A.O.L. Atkin. The number of points on an elliptic curve modulo a prime. Series of e-mails to the NMBRTHRY mailing list, 1992.

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A. O. L. Atkin, The number of points on an elliptic curve modulo a prime, Series of e-mails to the NUMBERTHRY mailing list, 1992.

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A.O. Atkin, The number of points on an elliptic curve modulo a prime, Draft, 1998.

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A.O.L. Atkin. The number of points on an elliptic curve modulo a prime. Series of e-mails to the NMBRTHRY mailing list, 1992.

No context found.

A.O.L. Atkin. The number of points on an elliptic curve modulo a prime. Series of e-mails to the NMBRTHRY mailing list, 1992.

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Atkin (A. O. L.). -- The number of points on an elliptic curve modulo a prime. -- Preprint, 1988.

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Atkin, A. O. L. The number of points on an elliptic curve modulo a prime. Draft, 1988.

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