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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...
Reductions of an elliptic curve with almost prime orders
"... 1 Let E be an elliptic curve over Q. For a prime p of good reduction, let Ep be the reduction of E modulo p. We investigate Koblitz’s Conjecture about the number of primes p for which Ep(Fp) has prime order. More precisely, our main result is that if E is with Complex Multiplication, then there exis ..."
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1 Let E be an elliptic curve over Q. For a prime p of good reduction, let Ep be the reduction of E modulo p. We investigate Koblitz’s Conjecture about the number of primes p for which Ep(Fp) has prime order. More precisely, our main result is that if E is with Complex Multiplication, then there exist infinitely many primes p for which #Ep(Fp) has at most 5 prime factors. We also obtain upper bounds for the number of primes p ≤ x for which #Ep(Fp) is a prime. 1
ON CONJUGACY CLASSES AND DERIVED LENGTH
, 905
"... Abstract. Let G be a finite group and A, B and D be conjugacy classes of G with D ⊆ AB = {xy  x ∈ A, y ∈ B}. Denote by η(AB) the number of distinct conjugacy classes such that AB is the union of those. Set CG(A) = {g ∈ G  x g = x for all x ∈ A}. If AB = D then CG(D)/(CG(A) ∩ CG(B)) is an abelian ..."
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Abstract. Let G be a finite group and A, B and D be conjugacy classes of G with D ⊆ AB = {xy  x ∈ A, y ∈ B}. Denote by η(AB) the number of distinct conjugacy classes such that AB is the union of those. Set CG(A) = {g ∈ G  x g = x for all x ∈ A}. If AB = D then CG(D)/(CG(A) ∩ CG(B)) is an abelian group. If, in addition, G is supersolvable, then the derived length of CG(D)/(CG(A) ∩ CG(B)) is bounded above by 2η(AB). 1.
Article 13.8.8 On the Ratio of the Sum of Divisors and Euler’s Totient Function I
"... We prove that the only solutions to the equation σ(n) = 2 · φ(n) with at most three distinct prime factors are 3, 35 and 1045. Moreover, there exist at most a finite number of solutions to σ(n) = 2·φ(n) with Ω(n) ≤ k, and there are at most 22k +k −k squarefree solutions to φ(n) ∣ ∣σ(n) if ω(n) ..."
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We prove that the only solutions to the equation σ(n) = 2 · φ(n) with at most three distinct prime factors are 3, 35 and 1045. Moreover, there exist at most a finite number of solutions to σ(n) = 2·φ(n) with Ω(n) ≤ k, and there are at most 22k +k −k squarefree solutions to φ(n) ∣ ∣σ(n) if ω(n) = k. Lastly the number of solutions to φ(n) ∣ ∣σ(n) as x → ∞ is O ( xexp ( −1 √)) 2 logx. 1
Ramanujan and I
, 1998
"... Perhaps the title ‘Ramanujan and the birth of Probabilistic Number Theory ’ would have been more appropriate and personal, but since Ramanujan’s work influenced me greatly in other subjects too, I decided on this somewhat immodest title. Perhaps I should start at the beginning and relate how I first ..."
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Perhaps the title ‘Ramanujan and the birth of Probabilistic Number Theory ’ would have been more appropriate and personal, but since Ramanujan’s work influenced me greatly in other subjects too, I decided on this somewhat immodest title. Perhaps I should start at the beginning and relate how I first found out about Ramanujan’s existence. In March, 1931 I found a simple proof of the following old and wellknown theorem of Tchebychev: “Given any integer n, there is always a prime p such that n ≤ p < 2n. ” My paper was not very well written. Kalmar rewrote my paper and said in the introduction that Ramanujan found a somewhat similar proof. In fact the two proofs were very similar; my proof had perhaps the advantage of being more arithmetical. He asked me to look it up in the Collected Works of Ramanujan which I immediately read with great interest. I very much enjoyed the beautiful obituary of Hardy in this volume [23]. I am not competent to write about much of Ramanujan’s work on identities and on the τ–function since I never was good at finding identities. So I will ignore this aspect of Ramanujan’s work here and many of my colleagues who are much more competent to write about it than I will do so. I will therefore write about Ramanujan’s work on partitions and on prime numbers and here too I will restrict myself to the asymptotic theory. My paper [7] on Tchebychev’s theorem, which was actually my very first, appeared in 1932. One of the key lemmas was the proof that – p≤
THE RANGE OF CARMICHAEL’S λFUNCTION
"... ABSTRACT. In this paper, we show that the counting function of the set of values of the Carmichael λfunction is x/(log x)η+o(1), where η = 1 − (1 + log log 2)/(log 2) = 0.08607.... 1 ..."
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ABSTRACT. In this paper, we show that the counting function of the set of values of the Carmichael λfunction is x/(log x)η+o(1), where η = 1 − (1 + log log 2)/(log 2) = 0.08607.... 1