Results 21  30
of
80
Improving Privacy in Cryptographic Elections
, 1986
"... This report describes two simple extensions to the paper A Robust and Verifiable Cryptographically Secure Election Scheme presented in the 1985 Symposium on the Foundations of Computer Science. The first extension allows the "government" to be divided into an arbitrary number of "tellers". With ..."
Abstract

Cited by 12 (0 self)
 Add to MetaCart
This report describes two simple extensions to the paper A Robust and Verifiable Cryptographically Secure Election Scheme presented in the 1985 Symposium on the Foundations of Computer Science. The first extension allows the "government" to be divided into an arbitrary number of "tellers". With this extension, trust in any one teller is sufficient to assure privacy, even if the remaining tellers conspire in an attempt to breach privacy. The second extension allows a government to reveal (and convince voters of) the winner in an election without releasing the actual tally. Combining these two extensions in a uniform manner remains an open problem. 1 Introduction and Background In [CoFi85], a protocol was presented which gives a method of holding a mutually verifiable secretballot election. The participants are the voters, a government, a trusted "beacon" which generates publically readable random bits, and a trusted global clock. The protocol has four basic phases. In phas...
It Is Easy to Determine Whether a Given Integer Is Prime
, 2004
"... The problem of distinguishing prime numbers from composite numbers, and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be super ..."
Abstract

Cited by 12 (1 self)
 Add to MetaCart
The problem of distinguishing prime numbers from composite numbers, and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be superfluous to discuss the problem at length. Nevertheless we must confess that all methods that have been proposed thus far are either restricted to very special cases or are so laborious and difficult that even for numbers that do not exceed the limits of tables constructed by estimable men, they try the patience of even the practiced calculator. And these methods do not apply at all to larger numbers... It frequently happens that the trained calculator will be sufficiently rewarded by reducing large numbers to their factors so that it will compensate for the time spent. Further, the dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated... It is in the nature of the problem
A Lower Bound for Primality
, 1999
"... Recent work by Bernasconi, Damm and Shparlinski proved lower bounds on the circuit complexity of the squarefree numbers, and raised as an open question if similar (or stronger) lower bounds could be proved for the set of prime numbers. In this short note, we answer this question affirmatively, by s ..."
Abstract

Cited by 11 (5 self)
 Add to MetaCart
Recent work by Bernasconi, Damm and Shparlinski proved lower bounds on the circuit complexity of the squarefree numbers, and raised as an open question if similar (or stronger) lower bounds could be proved for the set of prime numbers. In this short note, we answer this question affirmatively, by showing that the set of prime numbers (represented in the usual binary notation) is not contained in AC 0 [p] for any prime p. Similar lower bounds are presented for the set of squarefree numbers, and for the problem of computing the greatest common divisor of two numbers. 1 Introduction What is the computational complexity of the set of prime numbers? There is a large body of work presenting important upper bounds on the complexity of the set of primes (including [AH87, APR83, Mil76, R80, SS77]), but  Supported in part by NSF grant CCR9734918. y Supported in part by NSF grant CCR9700239. z Supported in part by ARC grant A69700294. as was pointed out recently in [BDS98a, BDS9...
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 ..."
Abstract

Cited by 9 (7 self)
 Add to MetaCart
. 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...
Sharpening PRIMES is in P for a large family of numbers
 Math. Comp
, 2005
"... We present algorithms that are deterministic primality tests for a large family of integers, namely, integers n ≡ 1 (mod 4) for which an integer a is given such that the Jacobi symbol ( a) = −1, and n integers n ≡ −1 (mod 4) for which an integer a is given such that ( a 1−a) = ( ) = −1. The algo ..."
Abstract

Cited by 9 (0 self)
 Add to MetaCart
We present algorithms that are deterministic primality tests for a large family of integers, namely, integers n ≡ 1 (mod 4) for which an integer a is given such that the Jacobi symbol ( a) = −1, and n integers n ≡ −1 (mod 4) for which an integer a is given such that ( a 1−a) = ( ) = −1. The algorithms n n we present run in 2 − min(k,[2 log log n]) Õ(log n) 6 time, where k = ν2(n − 1) is the exact power of 2 dividing n − 1 when n ≡ 1 (mod 4) and k = ν2(n + 1) if n ≡ −1 (mod 4). The complexity of our algorithms improves up to Õ(log n)4 when k ≥ [2 log log n]. We also give tests for more general family of numbers and study their complexity.
Reducing lattice bases to find smallheight values of univariate polynomials
 in [13] (2007). URL: http://cr.yp.to/papers.html#smallheight. Citations in this document: §A
, 2004
"... Abstract. This paper generalizes several previous results on finding divisors in residue classes (Lenstra, Konyagin, Pomerance, Coppersmith, HowgraveGraham, Nagaraj), finding divisors in intervals (Rivest, Shamir, Coppersmith, HowgraveGraham), finding modular roots (Hastad, Vallée, Girault, Toffin ..."
Abstract

Cited by 8 (4 self)
 Add to MetaCart
Abstract. This paper generalizes several previous results on finding divisors in residue classes (Lenstra, Konyagin, Pomerance, Coppersmith, HowgraveGraham, Nagaraj), finding divisors in intervals (Rivest, Shamir, Coppersmith, HowgraveGraham), finding modular roots (Hastad, Vallée, Girault, Toffin, Coppersmith, HowgraveGraham), finding highpower divisors (Boneh, Durfee, HowgraveGraham), and finding codeword errors beyond half distance (Sudan, Guruswami, Goldreich, Ron, Boneh) into a unified algorithm that, given f and g, finds all rational numbers r such that f(r) and g(r) both have small height. 1.
The second largest prime divisor of an odd perfect number exceeds ten thousand
 Math. Comp
, 1999
"... Abstract. Let σ(n) denote the sum of positive divisors of the natural number n. Such a number is said to be perfect if σ(n) =2n. It is well known that a number is even and perfect if and only if it has the form 2 p−1 (2 p − 1) where 2 p − 1isprime. No odd perfect numbers are known, nor has any proof ..."
Abstract

Cited by 8 (1 self)
 Add to MetaCart
Abstract. Let σ(n) denote the sum of positive divisors of the natural number n. Such a number is said to be perfect if σ(n) =2n. It is well known that a number is even and perfect if and only if it has the form 2 p−1 (2 p − 1) where 2 p − 1isprime. No odd perfect numbers are known, nor has any proof of their nonexistence ever been given. In the meantime, much work has been done in establishing conditions necessary for their existence. One class of necessary conditions would be lower bounds for the distinct prime divisors of an odd perfect number. For example, Cohen and Hagis have shown that the largest prime divisor of an odd perfect number must exceed 10 6, and Hagis showed that the second largest must exceed 10 3. In this paper, we improve the latter bound. In particular, we prove the statement in the title of this paper. 1.
Square free values of the order function
 MR2028173 (2004i:11116), Zbl 1066.11044
, 2003
"... Given a 0}, we consider the problem of enumerating the integers m coprime to a such that the order of a modulo m is square free. This question is raised in analogy to a result recently proved jointly with F. Saidak and I. Shparlinski where square free values of the Carmichael function are studie ..."
Abstract

Cited by 7 (0 self)
 Add to MetaCart
Given a 0}, we consider the problem of enumerating the integers m coprime to a such that the order of a modulo m is square free. This question is raised in analogy to a result recently proved jointly with F. Saidak and I. Shparlinski where square free values of the Carmichael function are studied. The technique is the one of Hooley that uses the Chebotarev Density Theorem to enumerate primes for which the index i p (a) of a modulo p is divisible by a given integer. 1.