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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 custo ..."
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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
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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 ..."
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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.
MO419 – Probabilistic Algorithms – Flávio K. Miyazawa – IC/UNICAMP 2010 A survey on Probabilistic Algorithms to Primality Test
"... One of the longstanding problems in using encryption to encode messages is that the recipient of the message needs to know the key in order to decrypt the message. Clearly we somehow have to get the key to the participants so they can use it. We can’t send the key to them without encrypting *it*, or ..."
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One of the longstanding problems in using encryption to encode messages is that the recipient of the message needs to know the key in order to decrypt the message. Clearly we somehow have to get the key to the participants so they can use it. We can’t send the key to them without encrypting *it*, or someone might “eavesdrop ” and get it. But this puts us in an infinite loop: the
Article electronically published on November 2, 2004 FINDING C3STRONG PSEUDOPRIMES
"... Abstract. Let q1 <q2 <q3 be odd primes and N = q1q2q3. Put d =gcd(q1−1,q2 − 1,q3 − 1) and hi = qi−1,i=1, 2, 3. d Then we call d the kernel, the triple (h1,h2,h3)thesignature, andH = h1h2h3 the height of N, respectively. We call N a C3number if it is a Carmichael numberwitheachprimefactorqi≡3 ..."
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Abstract. Let q1 <q2 <q3 be odd primes and N = q1q2q3. Put d =gcd(q1−1,q2 − 1,q3 − 1) and hi = qi−1,i=1, 2, 3. d Then we call d the kernel, the triple (h1,h2,h3)thesignature, andH = h1h2h3 the height of N, respectively. We call N a C3number if it is a Carmichael numberwitheachprimefactorqi≡3 mod 4. If N is a C3number and a strong pseudoprime to the t bases bi for 1 ≤ i ≤ t, wecallNaC3spsp(b1,b2,...,bt). Since C3numbers have probability of error 1/4 (the upper bound of that for the RabinMiller test), they often serve as the exact values or upper bounds of ψm (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. In this paper, we first describe an algorithm for finding C3spsp(2)’s, to a given limit, with heights bounded. There are in total 21978 C3spsp(2)’s < 1024 with heights < 109. We then give an overview of the 21978 C3spsp(2)’s and tabulate 54 of them, which are C3spsp’s to the first 8 prime bases up to 19; three numbers are spsp’s to the first 11 prime bases up to 31. No C3spsp’s < 1024 to the first 12 prime bases with heights < 109 were found. We conjecture that there exist no C3spsp’s < 1024 to the first 12 prime bases with heights ≥ 109 and so that
Article electronically published on February 17, 2000 FINDING STRONG PSEUDOPRIMES TO SEVERAL BASES
"... Dedicated to the memory of P. Erdős (1913–1996) 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 primality testing algorithm which is not only easier to implement but also f ..."
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Dedicated to the memory of P. Erdős (1913–1996) 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 primality testing algorithm which is not only easier to implement but also faster than either the Jacobi sum test or the elliptic curve test. Thanks to Pomerance et al. and Jaeschke, ψm are known for 1 ≤ m ≤ 8. Upper bounds for ψ9,ψ10 and ψ11 were given by Jaeschke. In this paper we tabulate all strong pseudoprimes (spsp’s) n<1024 to the first ten prime bases 2, 3, ·· · , 29, which have the form n = pq with p, q odd primes and q −1 =k(p −1),k=2, 3, 4. There are in total 44 such numbers, six of which are also spsp(31), and three numbers are spsp’s to both bases 31 and 37. As a result the upper bounds for ψ10 and ψ11 are lowered from 28 and 29decimaldigit numbers to 22decimaldigit numbers, and a 24decimaldigit upper bound for ψ12 is obtained. The main tools used in our methods are the biquadratic residue characters and cubic residue characters. We propose necessary conditions for n to be a strong pseudoprime to one or to several prime bases. Comparisons of effectiveness with both Jaeschke’s and Arnault’s methods are given. 1.