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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 41 (7 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.
Implementation Of The AtkinGoldwasserKilian Primality Testing Algorithm
 RAPPORT DE RECHERCHE 911, INRIA, OCTOBRE
, 1988
"... We describe a primality testing algorithm, due essentially to Atkin, that uses elliptic curves over finite fields and the theory of complex multiplication. In particular, we explain how the use of class fields and genus fields can speed up certain phases of the algorithm. We sketch the actual implem ..."
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Cited by 9 (7 self)
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We describe a primality testing algorithm, due essentially to Atkin, that uses elliptic curves over finite fields and the theory of complex multiplication. In particular, we explain how the use of class fields and genus fields can speed up certain phases of the algorithm. We sketch the actual implementation of this test and its use on testing large primes, the records being two numbers of more than 550 decimal digits. Finally, we give a precise answer to the question of the reliability of our computations, providing a certificate of primality for a prime number.
On the importance of securing your bins: The garbagemaninthemiddle attack
, 1997
"... In this paper, we address the following problem: " Is it possible to weaken/attack a scheme when a (provably) secure cryptosystem is used? ". The answer is yes. We exploit weak errorhandling methods. Our attack relies on the cryptanalyst being able to modify some ciphertext and then getti ..."
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Cited by 8 (2 self)
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In this paper, we address the following problem: " Is it possible to weaken/attack a scheme when a (provably) secure cryptosystem is used? ". The answer is yes. We exploit weak errorhandling methods. Our attack relies on the cryptanalyst being able to modify some ciphertext and then getting access to the decryption of this modified ciphertext. Moreover, it applies on many cryptosystems, including RSA, Rabin, LUC, KMOV, Demytko, ElGamal and its analogues, 3pass system, knapsack scheme, etc. . .
Elliptic Curves, Primality Proving And Some Titanic Primes
, 1989
"... We describe how to generate large primes using the primality proving algorithm of Atkin. Figure 1: The Titanic . 1. Introduction. During the last ten years, primality testing evolved at great speed. Motivated by the RSA cryptosystem [3], the first deterministic primality proving algorithm was de ..."
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Cited by 5 (3 self)
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We describe how to generate large primes using the primality proving algorithm of Atkin. Figure 1: The Titanic . 1. Introduction. During the last ten years, primality testing evolved at great speed. Motivated by the RSA cryptosystem [3], the first deterministic primality proving algorithm was designed by Adleman, Pomerance and Rumely [2] and made practical by Cohen, H. W. Lenstra and A. K. Lenstra (see [9, 10] and more recently [5]). It was then proved that the time needed to test an arbitrary integer N for primality is O((log N) c log log log N ) for some positive constant c ? 0. When implemented on a huge computer, the algorithm was able to test 200 digit numbers in about 10 minutes of CPU time. A few years ago, Goldwasser and Kilian [11], used the theory of elliptic curves over finite fields to give the first primality proving algorithm whose running time is polynomial in log N (assuming a plausible conjecture in number theory). Atkin [4] used the theory of complex multiplicat...
DISTRIBUTED PRIMALITY PROVING AND THE PRIMALITY OF (2^3539+ 1)/3
, 1991
"... We explain how the Elliptic Curve Primality Proving algorithm can be implemented in a distributed way. Applications are given to the certification of large primes (more than 500 digits). As a result, we describe the successful attempt at proving the primality of the lO65digit (2^3539+ l)/3, the fir ..."
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Cited by 2 (1 self)
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We explain how the Elliptic Curve Primality Proving algorithm can be implemented in a distributed way. Applications are given to the certification of large primes (more than 500 digits). As a result, we describe the successful attempt at proving the primality of the lO65digit (2^3539+ l)/3, the first ordinary Titanic prime.
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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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...
Pseudoprimes: A Survey Of Recent Results
, 1992
"... this paper, we aim at presenting the most recent results achieved in the theory of pseudoprime numbers. First of all, we make a list of all pseudoprime varieties existing so far. This includes Lucaspseudoprimes and the generalization to sequences generated by integer polynomials modulo N , elliptic ..."
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this paper, we aim at presenting the most recent results achieved in the theory of pseudoprime numbers. First of all, we make a list of all pseudoprime varieties existing so far. This includes Lucaspseudoprimes and the generalization to sequences generated by integer polynomials modulo N , elliptic pseudoprimes. We discuss the making of tables and the consequences on the design of very fast primality algorithms for small numbers. Then, we describe the recent work of Alford, Granville and Pomerance, in which they prove that there