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Two contradictory conjectures concerning Carmichael numbers
"... Erdös [8] conjectured that there are x 1;o(1) Carmichael numbers up to x, whereas Shanks [24] was skeptical as to whether one might even nd an x up to which there are more than p x Carmichael numbers. Alford, Granville and Pomerance [2] showed that there are more than x 2=7 Carmichael numbers up to ..."
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Erdös [8] conjectured that there are x 1;o(1) Carmichael numbers up to x, whereas Shanks [24] was skeptical as to whether one might even nd an x up to which there are more than p x Carmichael numbers. Alford, Granville and Pomerance [2] showed that there are more than x 2=7 Carmichael numbers up to x, and gave arguments which even convinced Shanks (in persontoperson discussions) that Erdös must be correct. Nonetheless, Shanks's skepticism stemmed from an appropriate analysis of the data available to him (and his reasoning is still borne out by Pinch's extended new data [14,15]), and so we herein derive conjectures that are consistent with Shanks's observations, while tting in with the viewpoint of Erdös [8] and the results of [2,3].
Security analysis of the GennaroHaleviRabin signature scheme
 IN PROCEEDINGS OF EUROCRYPT 2000
, 2000
"... We exhibit an attack against a signature scheme recently proposed by Gennaro, Halevi and Rabin [9]. The scheme’s security is based on two assumptions namely the strong RSA assumption and the existence of a divisionintractable hashfunction. For the latter, the authors conjectured a security level ..."
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We exhibit an attack against a signature scheme recently proposed by Gennaro, Halevi and Rabin [9]. The scheme’s security is based on two assumptions namely the strong RSA assumption and the existence of a divisionintractable hashfunction. For the latter, the authors conjectured a security level exponential in the hashfunction’s digest size whereas our attack is subexponential with respect to the digest size. Moreover, since the new attack is optimal, the length of the hash function can now be rigorously fixed. In particular, to get a security level equivalent to 1024bit RSA, one should use a digest size of approximately 1024 bits instead of the 512 bits suggested in [9].
Reciprocals of certain large additive functions
 225—231; MR0619450 (82k:10053). involving Arithmetric Functions 12
, 1981
"... 1. Introduction and statement of results ..."
On Positive Integers ≤x with Prime Factors ≤t log x
"... . It is not difficult to estimate the function /(x; y), which counts integers x, free of prime factors ? y, by "smooth" functions whenever y log 1=2 x or y is a fixed power of x. This can be extended to y ! log 3=4 x, and y ? log 2+" x under the assumption of the Riemann Hypothesis. The re ..."
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. It is not difficult to estimate the function /(x; y), which counts integers x, free of prime factors ? y, by "smooth" functions whenever y log 1=2 x or y is a fixed power of x. This can be extended to y ! log 3=4 x, and y ? log 2+" x under the assumption of the Riemann Hypothesis. The real difficulty lies when y is a fixed multiple of log x and, in this paper, we investigate the set of integers x, free of prime factors ? t log x, by estimating various functions related to /(x; t log x). 1. INTRODUCTION. Define S(x; y) to be the set of positive integers x, composed only of prime factors y. The cardinality of this set, /(x; y), is called the DickmanDe Bruijn function and has been extensively investigated by many authors (see [14] for a review). In this section we will give some wellknown results about /(x; y) and sketch proofs of smooth asymptotic estimates when y ! log 1=2 x and when y is a fixed power of x. We also indicate how, in the literature, these have been ...
DISCRETE LOGARITHMS, DIFFIEHELLMAN, AND REDUCTIONS
"... Abstract. We consider the OnePrimeNotp and AllPrimesButp variants of the Discrete Logarithm (DL) problem in a group of prime order p. We give reductions to the DiffieHellman (DH) problem that do not depend on any unproved conjectures about smooth or prime numbers in short intervals. We show t ..."
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Abstract. We consider the OnePrimeNotp and AllPrimesButp variants of the Discrete Logarithm (DL) problem in a group of prime order p. We give reductions to the DiffieHellman (DH) problem that do not depend on any unproved conjectures about smooth or prime numbers in short intervals. We show that the OnePrimeNotpDL problem reduces to DH in time roughly Lp(1/2); the AllPrimesButpDL problem reduces to DH in time roughly Lp(2/5); and the AllPrimesButpDL problem reduces to the DH plus Integer Factorization problems in polynomial time. We also prove that under the Riemann Hypothesis, with ε log p queries to a yesorno oracle one can reduce DL to DH in time roughly Lp(1/2); and under a conjecture about smooth numbers, with εlog p queries to a yesorno oracle one can reduce DL to DH in polynomial time. 1.