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Close to Uniform Prime Number Generation With Fewer Random Bits
"... Abstract. In this paper we analyze a simple method for generating prime numbers with fewer random bits. Assuming the Extended Riemann Hypothesis, we can prove that our method generates primes according to a distribution that can be made arbitrarily close to uniform. This is unlike the PRIMEINC algor ..."
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Abstract. In this paper we analyze a simple method for generating prime numbers with fewer random bits. Assuming the Extended Riemann Hypothesis, we can prove that our method generates primes according to a distribution that can be made arbitrarily close to uniform. This is unlike the PRIMEINC algorithm studied by Brandt and Damg˚aard and its many variants implemented in numerous software packages, which reduce the number of random bits used at the price of a distribution easily distinguished from uniform. Our new method is also no more computationally expensive than the ones in current use, and opens up interesting options for prime number generation in constrained environments. Keywords: Publickey cryptography, prime number generation, RSA, efficient implementations, random bits. 1
ACE Encrypt: The Advanced Cryptographic Engine’s Public Key Encryption Scheme
, 2000
"... This document describes the part of the Advanced Cryptographic Engine (ACE) pertaining to public key encryption. It specifies a public key encryption scheme with enough detail to ensure interoperability between different implementations. This scheme is almost as efficient as commercially used scheme ..."
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This document describes the part of the Advanced Cryptographic Engine (ACE) pertaining to public key encryption. It specifies a public key encryption scheme with enough detail to ensure interoperability between different implementations. This scheme is almost as efficient as commercially used schemes, yet unlike such schemes, can be proven secure under reasonable and welldefined intractability assumptions. A concrete security analysis of the scheme is presented.
A Sublinear Time Parallel GCD Algorithm for the EREW PRAM
, 2009
"... We present a parallel algorithm that computes the greatest common divisor of two integers of n bits in length that takes O(n log log n / logn) expected time using n 6+ǫ processors on the EREW PRAM parallel model of computation. We believe this to be the first sublinear time algorithm on the EREW PRA ..."
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We present a parallel algorithm that computes the greatest common divisor of two integers of n bits in length that takes O(n log log n / logn) expected time using n 6+ǫ processors on the EREW PRAM parallel model of computation. We believe this to be the first sublinear time algorithm on the EREW PRAM for this problem.
Factoring Polynomials Modulo Composites
, 1997
"... This paper characterizes all the factorizations of a polynomial with coefficients in the ring Z n where n is a composite number. We give algorithms to compute such factorizations along with algebraic classifications. Contents 1 Introduction 3 1.1 Circuit complexity theory . . . . . . . . . . . . ..."
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This paper characterizes all the factorizations of a polynomial with coefficients in the ring Z n where n is a composite number. We give algorithms to compute such factorizations along with algebraic classifications. Contents 1 Introduction 3 1.1 Circuit complexity theory . . . . . . . . . . . . . . . . . . . . . . 3 2 Some Important Tools in Z n [x] 4 2.1 The Z n [x] phenomena . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 The Chinese Remainder Theorem . . . . . . . . . . . . . . . . . . 5 2.3 Irreducibility criteria in Z p k [x] . . . . . . . . . . . . . . . . . . . 7 2.4 Hensel's Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.5 A naive approach to factoring . . . . . . . . . . . . . . . . . . . . 11 3 The Case of Small Discriminants 12 3.1 The padic numbers . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 Resultants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.3 The correspondence to factoring over the padics . . . . ....
Signature Schemes Based on the Strong RSA Assumption \Lambda
, 1999
"... Abstract We describe and analyze a new digital signature scheme. The new scheme is quite efficient, does not require the the signer to maintain any state, and can be proven secure against adaptive chosen message attack under a reasonable intractability assumption, the socalled strong RSA assumption ..."
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Abstract We describe and analyze a new digital signature scheme. The new scheme is quite efficient, does not require the the signer to maintain any state, and can be proven secure against adaptive chosen message attack under a reasonable intractability assumption, the socalled strong RSA assumption. Moreover, a hash function can be incorporated into the scheme in such a way that it is also secure in the random oracle model under the standard RSA assumption. 1 Introduction We describe new, efficient digital signature schemes whose security is based on the strong RSA assumption. By security, we mean security against an adaptive chosen message attack, as defined in [11]. To prove that our new schemes are secure, we need to make the strong RSA assumption, recently introduced by [2]. We also need a collisionresistant hash functionactually, as we shall see, a universal oneway hash function [16] is sufficient.
Multitrapdoor Commitments and their Applications to NonMalleable Protocols*
, 2004
"... Abstract We introduce the notion of multitrapdoor commitments which is a stronger form of trapdoorcommitment schemes. We then construct two very efficient instantiations of multitrapdoor commitment schemes, based on the Strong RSA Assumption and the recently introduced StrongDiffieHellman Assumpt ..."
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Abstract We introduce the notion of multitrapdoor commitments which is a stronger form of trapdoorcommitment schemes. We then construct two very efficient instantiations of multitrapdoor commitment schemes, based on the Strong RSA Assumption and the recently introduced StrongDiffieHellman Assumption. The main applications of our result are nonmalleable trapdoor commtiments and a compilerthat takes any proof of knowledge and transforms it into one which is secure against a concurrent maninthemiddle attack. Such a proof of knowledge immediately yields concurrently secureidentification protocols. When using our numbertheoretic istantiations, the nonmalleable commitment and the compiler are very efficient (require no more than four exponentiations). The latter also maintains the round complexity of the original proof of knowledge; it works in the common reference stringmodel, which in any case is necessary to prove security of proofs of knowledge under this kind of attacks. Compared to previously known efficient solutions, ours is a factor of two faster. 1 Introduction A commitment scheme is the cryptographic equivalent of an envelope. Consider the classic example of sealed bid auctions. Parties who want to bid on an item, place their bids in an envelope, in order to maintain secrecy until the end of the bidding period. At that time all bids are revealed by opening the envelopes, which in particular means that parties cannot alter bids at this point. A commitment scheme plays the role of the envelope: it's a cryptographic protocol composed of two phases: the committing phase and the opening phase. At the end of the first phase, a sender has committed to a message which however remains secret; in the opening phase the sender can only reveal that fixed message.
Genetic Algorithms for the Extended GCD Problem
, 1998
"... The extended greatest common divisor (GCD) problem is, given a vector a = (a 1 ; : : : ; a n ) of positive integers, compute g, the greatest common divisor of these integers, and find a vector x = (x 1 ; : : : ; x n ) of integer coefficients such that g = n X i=1 a i x i : It is desirable to fin ..."
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The extended greatest common divisor (GCD) problem is, given a vector a = (a 1 ; : : : ; a n ) of positive integers, compute g, the greatest common divisor of these integers, and find a vector x = (x 1 ; : : : ; x n ) of integer coefficients such that g = n X i=1 a i x i : It is desirable to find a solution vector x where kxk is small. We present several genetic algorithms for this problem. Our algorithms search among small multisubsets of fa 1 ; : : : ; a n g; a solution for a particular multisubset is extended to a complete solution by padding x with zeros. We also present the results of our implementations of these methods. 1 Introduction We present several genetic algorithms for solving the extended greatest common divisor problem. After defining the problem and discussing previous work, we will state our results. The extended greatest common divisor (GCD) problem is, given a vector a = (a 1 ; : : : ; a n ) of positive integers, compute g, the greatest common divisor of thes...