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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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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...
New experimental results concerning the Goldbach conjecture
 ALGORITHMIC NUMBER THEORY (THIRD INTERNATIONAL SYMPOSIUM, ANTSIII
, 1998
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Analysis of DPA Countermeasures Based on Randomizing the Binary Algorithm
, 2003
"... One of the major threats to the security of cryptosystems nowadays is the information leaked through side channels. For instance, power analysis attacks have been successfully mounted on cryptosystems embedded into small devices such as smart cards. ..."
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One of the major threats to the security of cryptosystems nowadays is the information leaked through side channels. For instance, power analysis attacks have been successfully mounted on cryptosystems embedded into small devices such as smart cards.
The Similarities (and Differences) between Polynomials and Integers
, 1994
"... The purpose of this paper is to examine the two domains of the integers and the polynomials, in an attempt to understand the nature of complexity in these very basic situations. Can we formalize the integer algorithms which shed light on the polynomial domain, and vice versa? When will the casti ..."
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The purpose of this paper is to examine the two domains of the integers and the polynomials, in an attempt to understand the nature of complexity in these very basic situations. Can we formalize the integer algorithms which shed light on the polynomial domain, and vice versa? When will the casting of one in the other speed up an existing algorithm? Why do some problems not lend themselves to this kind of speedup? We give several simple and natural theorems that show how problems in one domain can be embedded in the other, and we examine the complexitytheoretic consequences of these embeddings. We also prove several results on the impossibility of solving integer problems by mimicking their polynomial counterparts. 1 Introduction It is a fact frequently remarked upon that polynomials and integers share a number of characteristics. Usually the Fast Fourier Transform is then Supported by NSF grants DMS8807202 and CCR9204630. y Supported by NSF grant CCR9207797. 1 giv...
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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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.
Probabilistic primality testing
 Analysis of Algorithms Seminar I. INRIA Research Report XXX
, 1992
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Atkin's test: news from the front
 IN ADVANCES IN CRYPTOLOGY
, 1990
"... We make an attempt to compare the speed of some primality testing algorithms for certifying 100digit prime numbers. ..."
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We make an attempt to compare the speed of some primality testing algorithms for certifying 100digit prime numbers.
Computational Methods in Public Key Cryptology
, 2002
"... These notes informally review the most common methods from computational number theory that have applications in public key cryptology. ..."
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These notes informally review the most common methods from computational number theory that have applications in public key cryptology.
Easy numbers for the Elliptic Curve Primality Proving Algorithm
, 1992
"... We present some new classes of numbers that are easier to test for primality with the Elliptic Curve Primality Proving algorithm than average numbers. It is shown that this is the case for about half the numbers of the Cunningham project. Computational examples are given. ..."
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We present some new classes of numbers that are easier to test for primality with the Elliptic Curve Primality Proving algorithm than average numbers. It is shown that this is the case for about half the numbers of the Cunningham project. Computational examples are given.