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49
Some results on pseudosquares
 Math. Comp
, 1996
"... Abstract. If p is an odd prime, the pseudosquare Lp is defined to be the least positive nonsquare integer such that Lp ≡ 1 (mod 8) and the Legendre symbol (Lp/q) = 1 for all odd primes q ≤ p. In this paper we first discuss the connection between pseudosquares and primality testing. We then describe ..."
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Abstract. If p is an odd prime, the pseudosquare Lp is defined to be the least positive nonsquare integer such that Lp ≡ 1 (mod 8) and the Legendre symbol (Lp/q) = 1 for all odd primes q ≤ p. In this paper we first discuss the connection between pseudosquares and primality testing. We then describe a new numerical sieving device which was used to extend the table of known pseudosquares up to L271. We also present several numerical results concerning the growth rate of the pseudosquares, results which so far confirm that Lp √ e p/2, an inequality that must hold under the extended Riemann Hypothesis. 1.
Computational Aspects of Curves of Genus at Least 2
 Algorithmic number theory. 5th international symposium. ANTSII
, 1996
"... . This survey discusses algorithms and explicit calculations for curves of genus at least 2 and their Jacobians, mainly over number fields and finite fields. Miscellaneous examples and a list of possible future projects are given at the end. 1. Introduction An enormous number of people have per ..."
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. This survey discusses algorithms and explicit calculations for curves of genus at least 2 and their Jacobians, mainly over number fields and finite fields. Miscellaneous examples and a list of possible future projects are given at the end. 1. Introduction An enormous number of people have performed an enormous number of computations on elliptic curves, as one can see from even a perfunctory glance at [29]. A few years ago, the same could not be said for curves of higher genus, even though the theory of such curves had been developed in detail. Now, however, polynomialtime algorithms and sometimes actual programs are available for solving a wide variety of problems associated with such curves. The genus 2 case especially is becoming accessible: in light of recent work, it seems reasonable to expect that within a few years, packages will be available for doing genus 2 computations analogous to the elliptic curve computations that are currently possible in PARI, MAGMA, SIMATH, apec...
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 ..."
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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
Kronecker’s and Newton’s approaches to solving: A First Comparison
, 1999
"... In these pages we make a first attempt to compute efficiency of symbolic and numerical analysis procedures that solve systems of multivariate polynomial equations. In particular, we compare Kronecker’s solution (from the symbolic approach) with approximate zero theory (introduced by M. Shub & S ..."
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In these pages we make a first attempt to compute efficiency of symbolic and numerical analysis procedures that solve systems of multivariate polynomial equations. In particular, we compare Kronecker’s solution (from the symbolic approach) with approximate zero theory (introduced by M. Shub & S. Smale as a foundation of numerical analysis). To this purpose we show upper and lower bounds of the bit length of approximate zeros. We also introduce efficient procedures that transform local Kronecker’s solution into approximate zeros and conversely. As an application of our study we exhibit an efficient procedure to compute splitting fields and Lagrange resolvent of univariate polynomial equations. We remark that this procedure is obtained by a convenient combination of both approaches (numeric and symbolic) to multivariate polynomial solving.
ZeroKnowledge Arguments and PublicKey Cryptography
, 1995
"... In this work we consider the DiffieHellman Publickey model in which an additional short random string is shared by all users. This, which we call PublicKey PublicRandomness (PKPR) model, is very powerful as we show that it supports simple noninteractive implementations of important cryptographi ..."
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Cited by 11 (2 self)
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In this work we consider the DiffieHellman Publickey model in which an additional short random string is shared by all users. This, which we call PublicKey PublicRandomness (PKPR) model, is very powerful as we show that it supports simple noninteractive implementations of important cryptographic primitives. We give a noninteractive implementation of Oblivious Transfer in the PKPR model. Our implementation is secure against receivers with unlimited computational power. Building on this result, we show that all languages in NP have Perfect ZeroKnowledge Arguments in the PKPR model. 1 Introduction In PrivateKey Cryptography interaction is an essential resource. If two parties want to communicate secretly, they have to meet and agree on a common secret key. The need to establish a common key, before any communication could take place, severely limits the usefulness of this paradigm. Nonetheless it had been considered necessary for any form of secret communication. The PublicKey...
Primality proving via one round in ECPP and one iteration in AKS
 Advances in Cryptology – CRYPTO 2003
, 2003
"... On August 2002, Agrawal, Kayal and Saxena announced the first deterministic and polynomial time primality testing algorithm. For an input n, the AKS algorithm runs in heuristic time Õ(log6 n). Verification takes roughly the same amount of time. On the other hand, the Elliptic Curve Primality Proving ..."
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On August 2002, Agrawal, Kayal and Saxena announced the first deterministic and polynomial time primality testing algorithm. For an input n, the AKS algorithm runs in heuristic time Õ(log6 n). Verification takes roughly the same amount of time. On the other hand, the Elliptic Curve Primality Proving algorithm (ECPP), runs in random heuristic time Õ(log6 n) ( Õ(log5 n) if the fast multiplication is used), and generates certificates which can be easily verified. More recently, Berrizbeitia gave a variant of the AKS algorithm, in which some primes cost much less time to prove than a general prime does. Building on these celebrated results, this paper explores the possibility of designing a more efficient algorithm. A random primality proving algorithm with heuristic time complexity Õ(log4 n) is presented. It generates a certificate of primality which is O(log n) bits long and can be verified in deterministic time Õ(log 4 n). The reduction in time complexity is achieved by first generalizing Berrizbeitia’s algorithm to one which has higher density of easilyproved primes. For a general prime, one round of ECPP is deployed to reduce its primality proof to the proof of a random easilyproved prime. 1
The role of smooth numbers in number theoretic algorithms
 In International Congress of Mathematicians
, 1994
"... A smooth number is a number with only small prime factors. In particular, a positive integer is ysmooth if it has no prime factor exceeding y. Smooth numbers are a useful tool in number theory because they not only have a simple multiplicative structure, but are also fairly numerous. These twin pr ..."
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A smooth number is a number with only small prime factors. In particular, a positive integer is ysmooth if it has no prime factor exceeding y. Smooth numbers are a useful tool in number theory because they not only have a simple multiplicative structure, but are also fairly numerous. These twin properties of smooth numbers
On the Complexity of Breaking the DiffieHellman Protocol
 Computer Science Department
, 1996
"... It is shown that for a class of finite groups, breaking the DiffieHellman protocol is polynomialtime equivalent to computing discrete logarithms. Let G be a cyclic group with generator g and order jGj whose prime factorization is known. When for each large prime factor p of jGj an auxiliary group ..."
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It is shown that for a class of finite groups, breaking the DiffieHellman protocol is polynomialtime equivalent to computing discrete logarithms. Let G be a cyclic group with generator g and order jGj whose prime factorization is known. When for each large prime factor p of jGj an auxiliary group H p defined over GF (p) with smooth order is given, then breaking the DiffieHellman protocol for G and computing discrete logarithms in G are polynomialtime equivalent. Possible auxiliary groups H p are elliptic curves over GF (p) or over an extension field of GF (p), certain subgroups of the multiplicative group of such an extension field, and the Jacobian of a hyperelliptic curve. For a list of expressions in p, including p \Gamma 1, p + 1, and the cyclotomic polynomials of low degree in p, it is shown that an appropriate group H p can efficiently be constructed if one of the expressions in the list is smooth. Furthermore, efficient constructions of DiffieHellman groups with provable e...