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12
Algorithms for computing isogenies between elliptic curves
 Math. Comp
, 2000
"... Abstract. The heart of the improvements by Elkies to Schoof’s algorithm for computing the cardinality of elliptic curves over a finite field is the ability to compute isogenies between curves. Elkies ’ approach is well suited for the case where the characteristic of the field is large. Couveignes sh ..."
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Cited by 31 (6 self)
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Abstract. The heart of the improvements by Elkies to Schoof’s algorithm for computing the cardinality of elliptic curves over a finite field is the ability to compute isogenies between curves. Elkies ’ approach is well suited for the case where the characteristic of the field is large. Couveignes showed how to compute isogenies in small characteristic. The aim of this paper is to describe the first successful implementation of Couveignes’s algorithm. In particular, we describe the use of fast algorithms for performing incremental operations on series. We also insist on the particular case of the characteristic 2. 1.
An extension of Satoh's algorithm and its implementation
 J. RAMANUJAN MATH. SOC
, 2000
"... We describe a fast algorithm for counting points on elliptic curves defined over finite fields of small characteristic, following Satoh. Our main contribution is an extension to characteristics two and three. We give a detailed description with the optimisations necessary for an efficient implementa ..."
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Cited by 15 (3 self)
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We describe a fast algorithm for counting points on elliptic curves defined over finite fields of small characteristic, following Satoh. Our main contribution is an extension to characteristics two and three. We give a detailed description with the optimisations necessary for an efficient implementation. Finally we give the number of points we have computed on a "random" curve defined over the field F q with q = 2 8009 .
Computational Aspects of Curves of Genus at Least 2
 Algorithmic number theory. 5th international symposium. ANTSII
, 1996
"... . This survey discusses algorithms and explicit calculations for curves of genus at least 2 and their Jacobians, mainly over number fields and finite fields. Miscellaneous examples and a list of possible future projects are given at the end. 1. Introduction An enormous number of people have per ..."
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Cited by 14 (3 self)
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. This survey discusses algorithms and explicit calculations for curves of genus at least 2 and their Jacobians, mainly over number fields and finite fields. Miscellaneous examples and a list of possible future projects are given at the end. 1. Introduction An enormous number of people have performed an enormous number of computations on elliptic curves, as one can see from even a perfunctory glance at [29]. A few years ago, the same could not be said for curves of higher genus, even though the theory of such curves had been developed in detail. Now, however, polynomialtime algorithms and sometimes actual programs are available for solving a wide variety of problems associated with such curves. The genus 2 case especially is becoming accessible: in light of recent work, it seems reasonable to expect that within a few years, packages will be available for doing genus 2 computations analogous to the elliptic curve computations that are currently possible in PARI, MAGMA, SIMATH, apec...
Remarks on the SchoofElkiesAtkin algorithm
 Math. Comp
, 1998
"... Abstract. Schoof’s algorithm computes the number m of points on an elliptic curve E defined over a finite field Fq. Schoof determines m modulo small primes ℓ using the characteristic equation of the Frobenius of E and polynomials of degree O(ℓ 2). With the works of Elkies and Atkin, we have just to ..."
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Cited by 12 (0 self)
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Abstract. Schoof’s algorithm computes the number m of points on an elliptic curve E defined over a finite field Fq. Schoof determines m modulo small primes ℓ using the characteristic equation of the Frobenius of E and polynomials of degree O(ℓ 2). With the works of Elkies and Atkin, we have just to compute, when ℓ is a “good ” prime, an eigenvalue of the Frobenius using polynomials of degree O(ℓ). In this article, we compute the complexity of Müller’s algorithm, which is the best known method for determining one eigenvalue and we improve the final step in some cases. Finally, when ℓ is “bad”, we describe how to have polynomials of small degree and how to perform computations, in Schoof’s algorithm, on xvalues only. 1.
Isogeny volcanoes and the SEA algorithm
 In ANTSV, volume 2369 of LNCS
, 2000
"... . In 1985, Schoof gave a deterministic polynomial time algorithm to compute the cardinality of an elliptic curve over a nite eld. His algorithm computes the cardinality modulo small primes and builds the answer using the Chinese Remaindering Theorem. The improvements of Atkin and Elkies made the com ..."
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Cited by 11 (1 self)
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. In 1985, Schoof gave a deterministic polynomial time algorithm to compute the cardinality of an elliptic curve over a nite eld. His algorithm computes the cardinality modulo small primes and builds the answer using the Chinese Remaindering Theorem. The improvements of Atkin and Elkies made the computations easier and practical. Couveignes and Morain showed how to extend the ideas of Elkies to the case of prime powers, in eect walking through the Tate module associated to the curve, via rational isogenies. This latter algorithm works well if the Frobenius of the curve has two distinct eigenvalues and there remained to tackle the case of repeated eigenvalues. The purpose of this work is to solve this problem, explaining how this relates to the computation of the endomorphism ring of the curve, as worked out by Kohel. 1. Introduction Let E be an elliptic curve dened over a nite eld F q , where q = p r with p prime. By Hasse's theorem, the Frobenius of the curve is an endomorp...
Finding Secure Curves with the SatohFGH Algorithm and an EarlyAbort Strategy
 in B. P (ed), Advances in Cryptology  EUROCRYPT 2001, Lecture Notes in Computer Science 2045
, 2001
"... The use of elliptic curves in cryptography relies on the ability to count the number of points on a given curve. Before 1999, the SEA algorithm was the only ecient method known for random curves. Then Satoh proposed a new algorithm based on the canonical padic lift of the curve for p 5. In an ..."
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Cited by 5 (2 self)
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The use of elliptic curves in cryptography relies on the ability to count the number of points on a given curve. Before 1999, the SEA algorithm was the only ecient method known for random curves. Then Satoh proposed a new algorithm based on the canonical padic lift of the curve for p 5. In an earlier paper, the authors extended Satoh's method to the case of characteristics two and three. This paper presents an implementation of the SatohFGH algorithm and its application to the problem of nding curves suitable for cryptography. By combining SatohFGH and an earlyabort strategy based on SEA, we are able to nd secure random curves in characteristic two in much less time than previously reported. In particular we can generate curves widely considered to be as secure as RSA1024 in less than one minute each on a fast workstation.
Counting points on elliptic curves over F p n using Couveignes's algorithm
, 1995
"... The heart of the improvements of Elkies to Schoof's algorithm for computing the cardinality of elliptic curves over a finite field is the ability to compute isogenies between curves. Elkies' approach was well suited for the case where the characteristic of the field is large. Couveignes showed how t ..."
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Cited by 1 (0 self)
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The heart of the improvements of Elkies to Schoof's algorithm for computing the cardinality of elliptic curves over a finite field is the ability to compute isogenies between curves. Elkies' approach was well suited for the case where the characteristic of the field is large. Couveignes showed how to compute isogenies in small characteristic. The aim of this paper is to describe the first successful implementation of Couveignes's algorithm and to give numerous computational examples. In particular, we describe the use of fast algorithms for performing incremental operations on series. We will also insist on the particular case of the characteristic 2. 1 Introduction Elliptic curves have been used successfully to factor integers [25, 34], and prove the primality of large integers [4, 18, 3]. Moreover they turned out to be an interesting alternative to the use of Z=NZin cryptographical schemes. The first schemes were presented in [33, 23] and followed by many more (see for instance [31...
Computing isogenies in F 2 n
, 1996
"... . Contrary to what happens over prime fields of large characteristic, the main cost when counting the number of points of an elliptic curve E over F2 n is the computation of isogenies of prime degree `. The best method so far is due to Couveignes and needs asymptotically O(` 3 ) field operations. ..."
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Cited by 1 (0 self)
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. Contrary to what happens over prime fields of large characteristic, the main cost when counting the number of points of an elliptic curve E over F2 n is the computation of isogenies of prime degree `. The best method so far is due to Couveignes and needs asymptotically O(` 3 ) field operations. We outline in this article some nice properties satisfied by these isogenies and show how we can get from them a new algorithm that seems to perform better in practice than Couveignes's though of the same complexity. On a representative problem, we gain a speedup of 5 for the whole computation. 1 Introduction Many number theoretic algorithms are based on elliptic curves, among which integer factorization [5] or primality testing [1]. More directly, counting the number of points on these curves is essential to design secure cryptographical public schemes [8]. Algorithms to compute the cardinality of elliptic curves defined over finite fields of large characteristic give now satisfying resul...
A Survey of Elliptic Curve Cryptosystems, Part I: Introductory
, 2003
"... The theory of elliptic curves is a classical topic in many branches of algebra and number theory, but recently it is receiving more attention in cryptography. An elliptic curve is a twodimensional (planar) curve defined by an equation involving a cubic power of coordinate x and a square power of co ..."
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The theory of elliptic curves is a classical topic in many branches of algebra and number theory, but recently it is receiving more attention in cryptography. An elliptic curve is a twodimensional (planar) curve defined by an equation involving a cubic power of coordinate x and a square power of coordinate y. One class of these curves is