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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 the Order of Finitely Generated Subgroups of Q*(mod ρ) and Divisors of ρ1
, 1996
"... Introduction Let r be a positive integer. We say that r nonzero integers a 1 , ..., a r are multiplicatively independent if whenever there exist m 1 , ..., m r # Z such that r =1, it follows that m 1 =}}}=m r =0. We assume that none of a 1 , ..., a r is a perfect square or \1; let 1 denote t ..."
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Introduction Let r be a positive integer. We say that r nonzero integers a 1 , ..., a r are multiplicatively independent if whenever there exist m 1 , ..., m r # Z such that r =1, it follows that m 1 =}}}=m r =0. We assume that none of a 1 , ..., a r is a perfect square or \1; let 1 denote the subgroup of Q* generated by a 1 , ..., a r and let 1 p  denote the order of such a group 1 (mod p). In the case r=1, 1=(a), let ord p (a) denote the order of a (mod p). The famous Artin Conjecture for primitive roots (see [1]) states that ord p (a)=p&1 for infinitely many primes p. Artin's Conjecture has been proved under the assumption of the Generalized Riemann Hypothesis by C. Hooley (See [13]). In his paper it is implicitly shown (unconditionally) that ord p (a)>p#log p (1.1) for all but O(x#log x) primes p#x. article no. 0044 207 0022314X#96 #18.00 Copyright # 1996 by Academic Press, Inc. All rights of reproduction in any form reserved. * Supported in part by C.I.C.M.A.
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