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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.
Primality testing using elliptic curves
 Journal of the ACM
, 1999
"... Abstract. We present a primality proving algorithm—a probabilistic primality test that produces short certificates of primality on prime inputs. We prove that the test runs in expected polynomial time for all but a vanishingly small fraction of the primes. As a corollary, we obtain an algorithm for ..."
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Cited by 22 (0 self)
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Abstract. We present a primality proving algorithm—a probabilistic primality test that produces short certificates of primality on prime inputs. We prove that the test runs in expected polynomial time for all but a vanishingly small fraction of the primes. As a corollary, we obtain an algorithm for generating large certified primes with distribution statistically close to uniform. Under the conjecture that the gap between consecutive primes is bounded by some polynomial in their size, the test is shown to run in expected polynomial time for all primes, yielding a Las Vegas primality test. Our test is based on a new methodology for applying group theory to the problem of prime certification, and the application of this methodology using groups generated by elliptic curves over finite fields. We note that our methodology and methods have been subsequently used and improved upon, most notably in the primality proving algorithm of Adleman and Huang using hyperelliptic curves and
Fast Generation of Prime Numbers and Secure PublicKey Cryptographic Parameters
, 1995
"... A very efficient recursive algorithm for generating nearly random provable primes is presented. The expected time for generating a prime is only slightly greater than the expected time required for generating a pseudoprime of the same size that passes the MillerRabin test for only one base. The ..."
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Cited by 21 (0 self)
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A very efficient recursive algorithm for generating nearly random provable primes is presented. The expected time for generating a prime is only slightly greater than the expected time required for generating a pseudoprime of the same size that passes the MillerRabin test for only one base. Therefore our algorithm is even faster than presentlyused algorithms for generating only pseudoprimes because several MillerRabin tests with independent bases must be applied for achieving a sufficient confidence level. Heuristic arguments suggest that the generated primes are close to uniformly distributed over the set of primes in the specified interval. Security constraints on the prime parameters of certain cryptographic systems are discussed, and in particular a detailed analysis of the iterated encryption attack on the RSA publickey cryptosystem is presented. The prime generation algorithm can easily be modified to generate nearly random primes or RSAmoduli that satisfy t...
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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Cited by 14 (4 self)
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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
New Fibonacci and Lucas primes
 Math. Comp
, 1999
"... Abstract. Extending previous searches for prime Fibonacci and Lucas numbers, all probable prime Fibonacci numbers Fn have been determined for 6000
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Cited by 6 (0 self)
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Abstract. Extending previous searches for prime Fibonacci and Lucas numbers, all probable prime Fibonacci numbers Fn have been determined for 6000 <n≤50000 and all probable prime Lucas numbers Ln have been determined for 1000 <n≤50000. A rigorous proof of primality is given for F9311
A Problem Concerning a Character Sum (Extended Abstract)
"... ? ) E. Teske 1 and H.C. Williams ??2 1 Technische Universitat Darmstadt Institut fur Theoretische Informatik Alexanderstrae 10, 64283 Darmstadt Germany 2 University of Manitoba Dept. of Computer Science Winnipeg, MB Canada R3T 2N2 Abstract. Let p be a prime congruent to 1 modulo 4, n p ..."
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? ) E. Teske 1 and H.C. Williams ??2 1 Technische Universitat Darmstadt Institut fur Theoretische Informatik Alexanderstrae 10, 64283 Darmstadt Germany 2 University of Manitoba Dept. of Computer Science Winnipeg, MB Canada R3T 2N2 Abstract. Let p be a prime congruent to 1 modulo 4, n p the Legendre symbol and S(k) = P p 1 n=1 n k n p . The problem of nding a prime p such that S(3) > 0 was one of the motivating forces behind the development of several of Shanks' ideas for computing in algebraic number elds, although neither he nor D. H. and Emma Lehmer were ever successful in nding such a p. In this extended abstract we summarize some techniques which were successful in producing, for each k such that 3 k 2000, a value for p such that S(k) > 0. 1 Introduction Let d denote a fundamental discriminant of an imaginary quadratic eld IK = Q( p d ) and let h(d) denote the class number of IK. Let p be a prime ( 3(mod 4)), n p the Legendre symbol and S...