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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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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 complete Vinogradov 3primes theorem under the Riemann hypothesis
 ERA Am. Math. Soc
, 1997
"... Abstract. We outline a proof that if the Generalized Riemann Hypothesis holds, then every odd number above 5 is a sum of three prime numbers. The proof involves an asymptotic theorem covering all but a finite number of cases, an intermediate lemma, and an extensive computation. 1. ..."
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Abstract. We outline a proof that if the Generalized Riemann Hypothesis holds, then every odd number above 5 is a sum of three prime numbers. The proof involves an asymptotic theorem covering all but a finite number of cases, an intermediate lemma, and an extensive computation. 1.
Primality Testing Revisited
, 1992
"... . Rabin's algorithm is commonly used in computer algebra systems and elsewhere for primality testing. This paper presents an experience with this in the Axiom* computer algebra system. As a result of this experience, we suggest certain strengthenings of the algorithm. Introduction It is customary ..."
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. Rabin's algorithm is commonly used in computer algebra systems and elsewhere for primality testing. This paper presents an experience with this in the Axiom* computer algebra system. As a result of this experience, we suggest certain strengthenings of the algorithm. Introduction It is customary in computer algebra to use the algorithm presented by Rabin [1980] to determine if numbers are prime (and primes are needed throughout algebraic algorithms). As is well known, a single iteration of Rabin's algorithm, applied to the number N , has probability at most 0.25 of reporting "N is probably prime", when in fact N is composite. For most N , the probability is much less than 0.25. Here, "probability" refers to the fact that Rabin's algorithm begins with the choice of a "random" seed x, not congruent to 0 modulo N . In practice, however, true randomness is hard to achieve, and computer algebra systems often use a fixed set of x  for example Axiom release 1 uses the set f3; 5; 7; 11;...
The RabinMonier theorem for Lucas pseudoprimes
 Math. Comp
, 1997
"... Abstract. We give bounds on the number of pairs (P, Q)with0≤P, Q < n such that a composite number n is a strong Lucas pseudoprime with respect to the parameters (P, Q). 1. ..."
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Abstract. We give bounds on the number of pairs (P, Q)with0≤P, Q < n such that a composite number n is a strong Lucas pseudoprime with respect to the parameters (P, Q). 1.
New experimental results concerning the Goldbach conjecture
 Algorithmic Number Theory (Third International Symposium, ANTSIII
, 1998
"... and their applications. SMC is sponsored by the Netherlands Organization for Scientific Research (NWO). CWI is a member of ..."
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and their applications. SMC is sponsored by the Netherlands Organization for Scientific Research (NWO). CWI is a member of
Finding strong pseudoprimes to several bases. II,Math
 Department of Mathematics, Anhui Normal University
"... Abstract. Define ψm to be the smallest strong pseudoprime to all the first m prime bases. If we know the exact value of ψm, we will have, for integers n<ψm, a deterministic efficient primality testing algorithm which is easy to implement. Thanks to Pomerance et al. and Jaeschke, the ψm are known for ..."
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Abstract. Define ψm to be the smallest strong pseudoprime to all the first m prime bases. If we know the exact value of ψm, we will have, for integers n<ψm, a deterministic efficient primality testing algorithm which is easy to implement. Thanks to Pomerance et al. and Jaeschke, the ψm are known for 1 ≤ m ≤ 8. Upper bounds for ψ9,ψ10 and ψ11 were first given by Jaeschke, and those for ψ10 and ψ11 were then sharpened by the first author in his previous paper (Math. Comp. 70 (2001), 863–872). In this paper, we first follow the first author’s previous work to use biquadratic residue characters and cubic residue characters as main tools to tabulate all strong pseudoprimes (spsp’s) n < 1024 to the first five or six prime bases, which have the form n = pq with p, q odd primes and q − 1= k(p−1),k =4/3, 5/2, 3/2, 6; then we tabulate all Carmichael numbers < 1020, to the first six prime bases up to 13, which have the form n = q1q2q3 with each prime factor qi ≡ 3 mod 4. There are in total 36 such Carmichael numbers, 12 numbers of which are also spsp’s to base 17; 5 numbers are spsp’s to bases 17 and 19; one number is an spsp to the first 11 prime bases up to 31. As a result the upper bounds for ψ9,ψ10 and ψ11 are lowered from 20 and 22decimaldigit numbers to a 19decimaldigit number: ψ9 ≤ ψ10 ≤ ψ11 ≤ Q11 = 3825 12305 65464 13051 (19 digits) = 149491 · 747451 · 34233211. We conjecture that ψ9 = ψ10 = ψ11 = 3825 12305 65464 13051, and give reasons to support this conjecture. The main idea for finding these Carmichael numbers is that we loop on the largest prime factor q3 and propose necessary conditions on n to be a strong pseudoprime to the first 5 prime bases. Comparisons of effectiveness with Arnault’s, Bleichenbacher’s, Jaeschke’s, and Pinch’s methods for finding (Carmichael) numbers with three prime factors, which are strong pseudoprimes to the first several prime bases, are given. 1.
Improved Bounds for Goldback Conjecture
"... : Goldach's conjecture states that every even integer greater or equal to 6 is the sum of two prime numbers. This result is still unproved. This conjecture has been numerically checked up to 4:10 11 on an IBM 3083 mainframe. We describe here an implementation on a less powerful machine which raise ..."
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: Goldach's conjecture states that every even integer greater or equal to 6 is the sum of two prime numbers. This result is still unproved. This conjecture has been numerically checked up to 4:10 11 on an IBM 3083 mainframe. We describe here an implementation on a less powerful machine which raises the bound to 10 12 . Keywords: Prime numbers; Goldbach's problem (R'esum'e : tsvp) CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE Centre National de la Recherche Scientifique Institut National de Recherche en Informatique (URA 227) Universit e de Rennes 1  Insa de Rennes et en Automatique  unit e de recherche de Rennes Am'eliorations de bornes au sujet de la conjecture de Goldbach (premi`ere version) R'esum'e : La conjecture de Goldbach stipule que tout nombre pair sup'erieur ou 'egal `a 6 est somme de deux nombres premiers. Ce r'esultat est `a ce jour non d'emontr'e. Il a 'et'e v'erifi'e num'eriquement jusqu'`a 4:10 11 sur un IBM 3083. Nous d'ecrivons ici une impl'ementation...
TWO KINDS OF STRONG PSEUDOPRIMES UP TO 10 36
"... Abstract. Let n>1 be an odd composite integer. Write n − 1=2sd with d odd. If either bd ≡ 1modnor b2rd ≡−1modnfor some r =0, 1,...,s − 1, then we say that n isastrongpseudoprimetobaseb, or spsp(b) forshort. Define ψt to be the smallest strong pseudoprime to all the first t prime bases. If we know th ..."
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Abstract. Let n>1 be an odd composite integer. Write n − 1=2sd with d odd. If either bd ≡ 1modnor b2rd ≡−1modnfor some r =0, 1,...,s − 1, then we say that n isastrongpseudoprimetobaseb, or spsp(b) forshort. Define ψt to be the smallest strong pseudoprime to all the first t prime bases. If we know the exact value of ψt, we will have, for integers n<ψt, a deterministic efficient primality testing algorithm which is easy to implement. Thanks to Pomerance et al. and Jaeschke, the ψt are known for 1 ≤ t ≤ 8. Conjectured values of ψ9,...,ψ12 were given by us in our previous papers (Math. Comp. 72 (2003), 2085–2097; 74 (2005), 1009–1024). The main purpose of this paper is to give exact values of ψ ′ t for 13 ≤ t ≤ 19; to give a lower bound of ψ ′ 20: ψ ′ 20> 1036; and to give reasons and numerical evidence of K2 and C3spsp’s < 1036 to support the following conjecture: ψt = ψ ′ t <ψ′ ′ t for any t ≥ 12, where ψ ′ t (resp. ψ′ ′ t) is the smallest K2 (resp. C3) strong pseudoprime to all the first t prime bases. For this purpose we describe procedures for computing and enumerating the two kinds of spsp’s < 1036 to the first 9 prime bases. The entire calculation took about 4000 hours on a PC Pentium IV/1.8GHz. (Recall that a K2spsp is an spsp of the form: n = pq with p, q primes and q − 1=2(p−1); and that a C3spsp is an spsp and a Carmichael number of the form: n = q1q2q3 with each prime factor qi ≡ 3mod4.) 1.
NOTES ON SOME NEW KINDS OF PSEUDOPRIMES
"... Abstract. J. Browkin defined in his recent paper (Math. Comp. 73 (2004), pp. 1031–1037) some new kinds of pseudoprimes, called Sylow ppseudoprimes and elementary Abelian ppseudoprimes. He gave examples of strong pseudoprimes to many bases which are not Sylow ppseudoprime to two bases only, where ..."
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Abstract. J. Browkin defined in his recent paper (Math. Comp. 73 (2004), pp. 1031–1037) some new kinds of pseudoprimes, called Sylow ppseudoprimes and elementary Abelian ppseudoprimes. He gave examples of strong pseudoprimes to many bases which are not Sylow ppseudoprime to two bases only, where p = 2 or 3. In this paper, in contrast to Browkin’s examples, we give facts and examples which are unfavorable for Browkin’s observation to detect compositeness of odd composite numbers. In Section 2, we tabulate and compare counts of numbers in several sets of pseudoprimes and find that most strong pseudoprimes are also Sylow 2pseudoprimes to the same bases. In Section 3, we give examples of Sylow ppseudoprimes to the first several prime bases for the first several primes p. We especially give an example of a strong pseudoprime to the first six prime bases, which is a Sylow ppseudoprime to the same bases for all p ∈{2, 3, 5, 7, 11, 13}. In Section 4, we define n to be a kfold Carmichael Sylow pseudoprime, ifitisaSylowppseudoprime to all bases prime to n for all the first k smallest odd prime factors p of n − 1. We find and tabulate all three 3fold Carmichael Sylow pseudoprimes < 1016. In Section 5, we define a positive odd composite n to be a Sylow uniform pseudoprime to bases b1,...,bk, or a Sylupsp(b1,...,bk) for short, if it is a Sylppsp(b1,...,bk) for all the first ω(n − 1) − 1 small prime factors p of n − 1, where ω(n − 1) is the number of distinct prime factors of n − 1. We find and tabulate all the 17 Sylupsp(2, 3, 5)’s < 1016 and some Sylupsp(2, 3, 5, 7, 11)’s < 1024. Comparisons of effectiveness of Browkin’s observation with Miller tests to detect compositeness of odd composite numbers are given in Section 6. 1.
VERIFYING THE GOLDBACH CONJECTURE UP TO 4 · 10 14
"... Abstract. Using a carefully optimized segmented sieve and an efficient checking algorithm, the Goldbach conjecture has been verified and is now known to be true up to 4 · 10 14. The program was distributed to various workstations. It kept track of maximal values of the smaller prime p in the minimal ..."
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Abstract. Using a carefully optimized segmented sieve and an efficient checking algorithm, the Goldbach conjecture has been verified and is now known to be true up to 4 · 10 14. The program was distributed to various workstations. It kept track of maximal values of the smaller prime p in the minimal partition of the even numbers, where a minimal partition is a representation 2n = p + q with 2n − p ′ being composite for all p ′ <p. The maximal prime p needed in the considered interval was found to be 5569 and is needed for the partition 389965026819938 = 5569 + 389965026814369.