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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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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.
Elliptic Curves and their use in Cryptography
 DIMACS Workshop on Unusual Applications of Number Theory
, 1997
"... The security of many cryptographic protocols depends on the difficulty of solving the socalled "discrete logarithm" problem, in the multiplicative group of a finite field. Although, in the general case, there are no polynomial time algorithms for this problem, constant improvements are ..."
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The security of many cryptographic protocols depends on the difficulty of solving the socalled "discrete logarithm" problem, in the multiplicative group of a finite field. Although, in the general case, there are no polynomial time algorithms for this problem, constant improvements are being made  with the result that the use of these protocols require much larger key sizes, for a given level of security, than may be convenient. An abstraction of these protocols shows that they have analogues in any group. The challenge presents itself: find some other groups for which there are no good attacks on the discrete logarithm, and for which the group operations are sufficiently economical. In 1985, the author suggested that the groups arising from a particular mathematical object known as an "elliptic curve" might fill the bill. In this paper I review the general cryptographic protocols which are involved, briefly describe elliptic curves and review the possible attacks again...
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 sh ..."
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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...