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Fully distributed threshold RSA under standard assumptions
 ADVANCES IN CRYPTOLOGY — ASIACRYPT 2001, VOLUME ??? OF LNCS
, 2001
"... The aim of this article is to propose a fully distributed environment for the RSA scheme. What we have in mind is highly sensitive applications and even if we are ready to pay a price in terms of efficiency, we do not want any compromise of the security assumptions that we make. Recently Shoup propo ..."
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Cited by 23 (3 self)
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The aim of this article is to propose a fully distributed environment for the RSA scheme. What we have in mind is highly sensitive applications and even if we are ready to pay a price in terms of efficiency, we do not want any compromise of the security assumptions that we make. Recently Shoup proposed a practical RSA threshold signature scheme that allows to share the ability to sign between a set of players. This scheme can be used for decryption as well. However, Shoup’s protocol assumes a trusted dealer to generate and distribute the keys. This comes from the fact that the scheme needs a special assumption on the RSA modulus and this kind of RSA moduli cannot be easily generated in an efficient way with many players. Of course, it is still possible to call theoretical results on multiparty computation, but we cannot hope to design efficient protocols. The only practical result to generate RSA moduli in a distributive manner is Boneh and Franklin’s protocol but it seems difficult to modify it in order to generate the kind of RSA moduli that Shoup’s protocol requires. The present work takes a different path by proposing a method to enhance the key generation with some additional properties and revisits Shoup’s protocol to work with the resulting RSA moduli. Both of these enhancements decrease the performance of the basic protocols. However, we think that in the applications we target, these enhancements provide practical solutions. Indeed, the key generation protocol is usually run only once and the number of players used to sign or decrypt is not very large. Moreover, these players have time to perform their task so that the communication or time complexity are not overly important.
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].
The distribution of Lucas and elliptic pseudoprimes
, 2001
"... Let L(x) denote the counting function for Lucas pseudoprimes, and E(x) denote the elliptic pseudoprime counting function. We prove that, for large x, L(x) ≤ x L(x) −1/2 and E(x) ≤ x L(x) −1/3, where L(x) = exp(log xlog log log x / log log x). ..."
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Cited by 9 (1 self)
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Let L(x) denote the counting function for Lucas pseudoprimes, and E(x) denote the elliptic pseudoprime counting function. We prove that, for large x, L(x) ≤ x L(x) −1/2 and E(x) ≤ x L(x) −1/3, where L(x) = exp(log xlog log log x / log log x).
Finding Four Million Large Random Primes
 In Crypto '90, LNCS 537
, 1991
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Values of the Euler Function in Various Sequences
 MONATSH. MATH. 146, 1–19
, 2005
"... Let ’ðnÞ and ðnÞ denote the Euler and Carmichael functions, respectively. In this paper, we investigate the equation ’ðnÞ r ðnÞ s,wherer5s51are fixed positive integers. We also study those positive integers n, not equal to a prime or twice a prime, such that ’ðnÞ p 1 holds with some prime p, as wel ..."
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Cited by 4 (3 self)
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Let ’ðnÞ and ðnÞ denote the Euler and Carmichael functions, respectively. In this paper, we investigate the equation ’ðnÞ r ðnÞ s,wherer5s51are fixed positive integers. We also study those positive integers n, not equal to a prime or twice a prime, such that ’ðnÞ p 1 holds with some prime p, as well as those positive integers n such that the equation ’ðnÞ f ðmÞ holds with some integer m, where f is a fixed polynomial with integer coefficients and degree deg f> 1.
Abstract
, 2008
"... We present an algorithm to invert the Euler function ϕ(m). The algorithm, for a given integer n ≥ 1, in polynomial time “on average”, finds the set Ψ(n) of all solutions m to the equation ϕ(m) = n. In fact, in the worst case the set Ψ(n) is exponentially large and cannot be constructed by a polynom ..."
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We present an algorithm to invert the Euler function ϕ(m). The algorithm, for a given integer n ≥ 1, in polynomial time “on average”, finds the set Ψ(n) of all solutions m to the equation ϕ(m) = n. In fact, in the worst case the set Ψ(n) is exponentially large and cannot be constructed by a polynomial time algorithm. In the opposite direction, we show, under some widely accepted number theoretic conjecture, that the Partition Problem, an NPcomplete problem, can be reduced, in polynomial time, to the problem of deciding whether ϕ(m) = n has a solution, for polynomially (in the input size of the Partition problem) many values of n. In fact, the following problem is NPcomplete: Given a set of positive integers S, decide whether there is an n ∈ S satisfying ϕ(m) = n, for some integer m. Finally, we establish close links between the problem of inverting the Euler function and the integer factorisation problem. 1
Article electronically published on January 23, 2006 COMPLEXITY OF INVERTING THE EULER FUNCTION
"... Abstract. Given an integer n, how hard is it to find the set of all integers m such that ϕ(m) =n, whereϕ is the Euler totient function? We present a certain basic algorithm which, given the prime number factorization of n, in polynomial time “on average ” (that is, (log n) O(1)), finds the set of al ..."
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Abstract. Given an integer n, how hard is it to find the set of all integers m such that ϕ(m) =n, whereϕ is the Euler totient function? We present a certain basic algorithm which, given the prime number factorization of n, in polynomial time “on average ” (that is, (log n) O(1)), finds the set of all such solutions m. In fact, in the worst case this set of solutions is exponential in log n, and so cannot be constructed by a polynomial time algorithm. In the opposite direction, we show, under a widely accepted number theoretic conjecture, that the Partition Problem, anNPcomplete problem, can be reduced in polynomial (in the input size) time to the problem of deciding whether ϕ(m) =n has a solution, for polynomially (in the input size of the Partition Problem)manyvaluesofn (where the prime factorizations of these n are given). What this means is that the problem of deciding whether there even exists a solution m to ϕ(m)=n, let alone finding any or all such solutions, is very likely to be intractable. Finally, we establish close links between the problem of inverting the Euler function and the integer factorization problem. 1.
POSITIVE INTEGERS n SUCH THAT na σ(n) − 1
"... Abstract. For a positive integer n let σ(n) be the sum of divisors function of n. In this note, we fix a positive integer a and we investigate the positive integers n such that na σ(n) − 1. We also show that under a plausible hypothesis related to the distribution of prime numbers there exist infi ..."
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Abstract. For a positive integer n let σ(n) be the sum of divisors function of n. In this note, we fix a positive integer a and we investigate the positive integers n such that na σ(n) − 1. We also show that under a plausible hypothesis related to the distribution of prime numbers there exist infinitely many positive integers n such that na σ(n) − 1 holds for all integers a coprime to n.