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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 22 (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.
The Carmichael Numbers up to 10^15
, 1992
"... There are 105212 Carmichael numbers up to 10 : we describe the calculations. ..."
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Cited by 18 (7 self)
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There are 105212 Carmichael numbers up to 10 : we describe the calculations.
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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Cited by 12 (0 self)
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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].
Implementation Of The AtkinGoldwasserKilian Primality Testing Algorithm
 Rapport de Recherche 911, INRIA, Octobre
, 1988
"... . We describe a primality testing algorithm, due essentially to Atkin, that uses elliptic curves over finite fields and the theory of complex multiplication. In particular, we explain how the use of class fields and genus fields can speed up certain phases of the algorithm. We sketch the actual impl ..."
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Cited by 9 (7 self)
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. We describe a primality testing algorithm, due essentially to Atkin, that uses elliptic curves over finite fields and the theory of complex multiplication. In particular, we explain how the use of class fields and genus fields can speed up certain phases of the algorithm. We sketch the actual implementation of this test and its use on testing large primes, the records being two numbers of more than 550 decimal digits. Finally, we give a precise answer to the question of the reliability of our computations, providing a certificate of primality for a prime number. IMPLEMENTATION DU TEST DE PRIMALITE D' ATKIN, GOLDWASSER, ET KILIAN R'esum'e. Nous d'ecrivons un algorithme de primalit'e, principalement du `a Atkin, qui utilise les propri'et'es des courbes elliptiques sur les corps finis et la th'eorie de la multiplication complexe. En particulier, nous expliquons comment l'utilisation du corps de classe et du corps de genre permet d'acc'el'erer les calculs. Nous esquissons l'impl'ementati...
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
"... e theory also suggests that pseudoprimes are rare. On the basis of extensive experience and analysis, Pomerance [5, 8] conjectures that the number of pseudoprimes less than n is at most n=L(n) 1+o(1) (2) where L(n) = exp log n log log log n log log n ! : Supported by NSF grant CCR8914428 ..."
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Cited by 5 (0 self)
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e theory also suggests that pseudoprimes are rare. On the basis of extensive experience and analysis, Pomerance [5, 8] conjectures that the number of pseudoprimes less than n is at most n=L(n) 1+o(1) (2) where L(n) = exp log n log log log n log log n ! : Supported by NSF grant CCR8914428, and RSA Data Security. email address: rivest@theory.lcs.mit.edu If this conjecture is correct, and we make the (unjustied) additional assumption that the o(1) in conjecture (2) can be ignored, then the number of pseudoprimes less than 2 256 is conjectured to be at most 4 10 52 whereas the number of 256bit primes is approximately 6:5 10 74 : Thus, if Pomerance's conjecture
Some Primality Testing Algorithms
 Notices of the AMS
, 1993
"... We describe the primality testing algorithms in use in some popular computer algebra systems, and give some examples where they break down in practice. 1 Introduction In recent years, fast primality testing algorithms have been a popular subject of research and some of the modern methods are now i ..."
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We describe the primality testing algorithms in use in some popular computer algebra systems, and give some examples where they break down in practice. 1 Introduction In recent years, fast primality testing algorithms have been a popular subject of research and some of the modern methods are now incorporated in computer algebra systems (CAS) as standard. In this review I give some details of the implementations of these algorithms and a number of examples where the algorithms prove inadequate. The algebra systems reviewed are Mathematica, Maple V, Axiom and Pari/GP. The versions we were able to use were Mathematica 2.1 for Sparc, copyright dates 19881992; Maple V Release 2, copyright dates 19811993; Axiom Release 1.2 (version of February 18, 1993); Pari/GP 1.37.3 (Sparc version, dated November 23, 1992). The tests were performed on Sparc workstations. Primality testing is a large and growing area of research. For further reading and comprehensive bibliographies, the interested re...
The Pseudoprimes up to 10^13
, 1995
"... . There are 38975 Fermat pseudoprimes (base 2) up to 10 11 , 101629 up to 10 12 and 264239 up to 10 13 : we describe the calculations and give some statistics. The numbers were generated by a variety of strategies, the most important being a backtracking search for possible prime factorisatio ..."
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. There are 38975 Fermat pseudoprimes (base 2) up to 10 11 , 101629 up to 10 12 and 264239 up to 10 13 : we describe the calculations and give some statistics. The numbers were generated by a variety of strategies, the most important being a backtracking search for possible prime factorisations, and the computations checked by a sieving technique. 1 Introduction A (Fermat) pseudoprime (base 2) is a composite number N with the property that 2 N \Gamma1 j 1 mod N . For background on pseudoprimes and primality tests in general we refer to Bressoud [1], Brillhart et al [2], Koblitz [4], Ribenboim [12] and [13] or Riesel [14]. Previous tables of pseudoprimes were computed by Pomerance, Selfridge and Wagstaff [11]. We have shown that there are 38975 pseudoprimes up to 10 11 , 101629 up to 10 12 and 264239 up to 10 13 ; all have at most 9 prime factors. Let P (X) denote the number of pseudoprimes less than X and let P (d; X) denote the number with exactly d prime factors. In ...
The Carmichael numbers up to 10 20
"... We extend our previous computations to show that there are 8220777 Carmichael numbers up to 10 20. As before, the numbers were generated by a backtracking search for possible prime factorisations together with a “large prime variation”. We present further statistics on the distribution of Carmichae ..."
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We extend our previous computations to show that there are 8220777 Carmichael numbers up to 10 20. As before, the numbers were generated by a backtracking search for possible prime factorisations together with a “large prime variation”. We present further statistics on the distribution of Carmichael
The Carmichael Numbers up to 10^16
 Math. Comp
, 1993
"... We extend our previous computations to show that there are 246683 Carmichael numbers up to 10 16 . As before, the numbers were generated by a backtracking search for possible prime factorisations together with a "large prime variation". We present further statistics on the distribution of Carmich ..."
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We extend our previous computations to show that there are 246683 Carmichael numbers up to 10 16 . As before, the numbers were generated by a backtracking search for possible prime factorisations together with a "large prime variation". We present further statistics on the distribution of Carmichael numbers. 1.