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A computational approach to pocklington certificates in type theory
 In Proc. of the 8th Int. Symp. on Functional and Logic Programming, volume 3945 of LNCS
, 2006
"... Abstract. Pocklington certificates are known to provide short proofs of primality. We show how to perform this in the framework of formal, mechanically checked, proofs. We present an encoding of certificates for the proof system Coq which yields radically improved performances by relying heavily on ..."
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Abstract. Pocklington certificates are known to provide short proofs of primality. We show how to perform this in the framework of formal, mechanically checked, proofs. We present an encoding of certificates for the proof system Coq which yields radically improved performances by relying heavily on computations inside and outside of the system (twolevel approach). 1 Formal Computational Proofs 1.1 Machines and the Quest for Correctness It is generally considered that modern mathematical logic was born towards the end of 19 th century, with the work of logicians like Frege, Peano, Russell or Zermelo, which lead to the precise definition of the notion of logical deduction and to formalisms like arithmetic, set theory or early type theory. From then on, a mathematical proof could be understood as a mathematical object itself, whose correction obeys some welldefined syntactical rules. In most formalisms, a formal proof is viewed as some treestructure; in natural deduction for instance, given to formal proofs σA and σB respectively of propositions A and B, these can be combined in order to build a proof of A ∧ B: σA σB ⊢ A ⊢ B ⊢ A ∧ B To sum things up, the logical point of view is that a mathematical statement holds in a given formalism if there exists a formal proof of this statement which follows the syntactical rules of the formalism. A traditional mathematical text can then be understood as an informal description of the formal proof. Things changed in the 1960ties, when N.G. de Bruijn’s team started to use computers to actually build formal proofs and verify their correctness. Using the fact that datastructures like formal proofs are very naturally represented in a computer’s memory, they delegated the proofverification work to the machine; their software Automath is considered as the first proofsystem and is the common
Efficient Generation of Prime Numbers
, 2000
"... The generation of prime numbers underlies the use of most publickey schemes, essentially as a major primitive needed for the creation of key pairs or as a computation stage appearing during various cryptographic setups. Surprisingly, despite decades of intense mathematical studies on primality test ..."
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The generation of prime numbers underlies the use of most publickey schemes, essentially as a major primitive needed for the creation of key pairs or as a computation stage appearing during various cryptographic setups. Surprisingly, despite decades of intense mathematical studies on primality testing and an observed progressive intensification of cryptographic usages, prime number generation algorithms remain scarcely investigated and most reallife implementations are of rather poor performance. Common generators typically output a nbit prime in heuristic average complexity O(n^4) or O(n^4/log n) and these figures, according to experience, seem impossible to improve significantly: this paper rather shows a simple way to substantially reduce the value of hidden constants to provide much more efficient prime generation algorithms. We apply our...
Fast Generation of Prime Numbers of Portable Devices: An Update
 Proceedings of CHES 2006, LNCS 4249
, 2006
"... Abstract. The generation of prime numbers underlies the use of most publickey cryptosystems, essentially as a primitive needed for the creation of RSA key pairs. Surprisingly enough, despite decades of intense mathematical studies on primality testing and an observed progressive intensification of ..."
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Abstract. The generation of prime numbers underlies the use of most publickey cryptosystems, essentially as a primitive needed for the creation of RSA key pairs. Surprisingly enough, despite decades of intense mathematical studies on primality testing and an observed progressive intensification of cryptography, prime number generation algorithms remain scarcely investigated and most reallife implementations are of dramatically poor performance. We show simple techniques that substantially improve all algorithms previously suggested or extend their capabilities. We derive fast implementations on appropriately equipped portable devices like smartcards embedding a cryptographic coprocessor. This allows onboard generation of RSA keys featuring a very attractive (average) processing time. Our motivation here is to help transferring this task from terminals where this operation usually took place so far, to portable devices themselves in near future for more confidence, security, and compliance with networkscaled distributed protocols such as electronic cash or mobile commerce.
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.
Some Numbertheoretic Conjectures and Their Relation to the Generation of Cryptographic Primes
, 1992
"... . The purpose of this paper is to justify the claim that a method for generating primes presented at EUROCRYPT'89 generates primes with virtually uniform distribution. Using convincing heuristic arguments, the conditional probability distributions of the size of the largest prime factor p 1 (n) ..."
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Cited by 1 (0 self)
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. The purpose of this paper is to justify the claim that a method for generating primes presented at EUROCRYPT'89 generates primes with virtually uniform distribution. Using convincing heuristic arguments, the conditional probability distributions of the size of the largest prime factor p 1 (n) of a number n on the order of N is derived, given that n satisfies one of the conditions 2n+1 is prime, 2an+1 is prime for a given a, or the d integers u 1 ; : : : ; u d , where u 1 = 2a 1 n + 1 and u t = 2a t u t\Gamma1 + 1 for 2 t d, are all primes for a given list of integers a 1 ; : : : ; a d . In particular, the conditional probabilities that n is itself a prime, or is of the form "k times a prime" for k = 2; 3; : : : ; is treated for the above conditions. It is shown that although for all k these probabilities strongly depend on the condition placed on n, the probability distribution of the relative size oe 1 (n) = log N p 1 (n) of the largest prime factor of n is virtually independent...
A GENERALIZATION OF MILLER’S PRIMALITY THEOREM PEDRO BERRIZBEITIA AND AURORA OLIVIERI
"... Abstract. For any integer r we show that the notion of ωprime to base a introduced by Berrizbeitia and Berry, 2000, leads to a primality test for numbers n congruent to 1 modulo r, which runs in polynomial time assuming the Extended Riemann Hypothesis (ERH). For r = 2 we obtain Miller’s classical r ..."
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Abstract. For any integer r we show that the notion of ωprime to base a introduced by Berrizbeitia and Berry, 2000, leads to a primality test for numbers n congruent to 1 modulo r, which runs in polynomial time assuming the Extended Riemann Hypothesis (ERH). For r = 2 we obtain Miller’s classical result. 1.
A Fermatlike sequence and primes of the form 2h.3^n +1
, 1995
"... Fermat numbers are a classical topic in elementary number theory. Fermat introduced them and claimed that all these numbers are prime. This claim was disproofed by Euler who gave a property on the eventual divisors of the Fermat numbers. In this article we exhibit another serie whose definition is ..."
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Fermat numbers are a classical topic in elementary number theory. Fermat introduced them and claimed that all these numbers are prime. This claim was disproofed by Euler who gave a property on the eventual divisors of the Fermat numbers. In this article we exhibit another serie whose definition is close to the one of Fermat numbers and which exhibit similar properties. This problem will lead us to the sets of covering congruences for numbers 2h:3 n + 1 as similarly Fermat numbers lead to Sierpinski's problem.
A Fermatlike Sequence . . .
, 1995
"... Fermat numbers are a classical topic in elementary number theory. Fermat introduced them and claimed that all these numbers are prime. This claim was disproofed by Euler who gave a property on the eventual divisors of the Fermat numbers. In this article we exhibit another serie whose definition is ..."
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Fermat numbers are a classical topic in elementary number theory. Fermat introduced them and claimed that all these numbers are prime. This claim was disproofed by Euler who gave a property on the eventual divisors of the Fermat numbers. In this article we exhibit another serie whose definition is close to the one of Fermat numbers and which exhibit similar properties. This problem will lead us to the sets of covering congruences for numbers 2h:3 n + 1 as similarly Fermat numbers lead to Sierpinski's problem.
Article 01.2.3 Prime Pythagorean triangles
"... A prime Pythagorean triangle has three integer sides of which the hypotenuse and one leg are primes. In this article we investigate their properties and distribution. We are also interested in finding chains of such triangles, where the hypotenuse of one triangle is the leg of the next in the sequen ..."
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A prime Pythagorean triangle has three integer sides of which the hypotenuse and one leg are primes. In this article we investigate their properties and distribution. We are also interested in finding chains of such triangles, where the hypotenuse of one triangle is the leg of the next in the sequence. We exhibit a chain of seven prime Pythagorean triangles and we include a brief discussion of primality proofs for the larger elements (up to 2310 digits) of the associated set of eight primes.