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Elliptic Curves And Primality Proving
 Math. Comp
, 1993
"... The aim of this paper is to describe the theory and implementation of the Elliptic Curve Primality Proving algorithm. ..."
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Cited by 162 (22 self)
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The aim of this paper is to describe the theory and implementation of the Elliptic Curve Primality Proving algorithm.
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.
Solving Equations Of Small Degree Modulo Large Primes
"... Introduction Atkin's algorithm [11] requires finding roots of polynomials modulo large primes (several hundreds decimal digits). However, the theory tells us that these polynomials split completely in the field Z=pZ we are interested in. The most straightforward approach is to use the probabi ..."
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Introduction Atkin's algorithm [11] requires finding roots of polynomials modulo large primes (several hundreds decimal digits). However, the theory tells us that these polynomials split completely in the field Z=pZ we are interested in. The most straightforward approach is to use the probabilistic algorithm of Berlekamp (the "folk method" as described in [8]). In the general case, this method carries out a distinct degree factorization of a given polynom. The reader is referred to the recent work by Shoup [15] for the most recent progress in polynomial factorization. On the other hand, we have often to find roots of special polynomials. For instance, extracting qth roots (q prime) in Z=pZ has been studied in [14] for q = 2 and generalized in [18] for q odd. Extracting roots in more general fields is described in [1]. We can also wonder whether we