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The secure remote password protocol
 In Proceedings of the 1998 Internet Society Network and Distributed System Security Symposium
, 1998
"... This paper presents a new password authentication and keyexchange protocol suitable for authenticating users and exchanging keys over an untrusted network. The new protocol resists dictionary attacks mounted by either passive or active network intruders, allowing, in principle, even weak passphrase ..."
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Cited by 185 (5 self)
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This paper presents a new password authentication and keyexchange protocol suitable for authenticating users and exchanging keys over an untrusted network. The new protocol resists dictionary attacks mounted by either passive or active network intruders, allowing, in principle, even weak passphrases to be used safely. It also o ers perfect forward secrecy, which protects past sessions and passwords against future compromises. Finally, user passwords are stored in a form that is not plaintextequivalent to the password itself, so an attacker who captures the password database cannot use it directly to compromise security and gain immediate access to the host. This new protocol combines techniques of zeroknowledge proofs with asymmetric key exchange protocols and o ers signi cantly improved performance over comparably strong extended methods that resist stolenveri er attacks such as Augmented EKE or BSPEKE. 1
Solving Large Sparse Linear Systems Over Finite Fields
, 1991
"... Many of the fast methods for factoring integers and computing discrete logarithms require the solution of large sparse linear systems of equations over finite fields. This paper presents the results of implementations of several linear algebra algorithms. It shows that very large sparse systems can ..."
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Cited by 79 (2 self)
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Many of the fast methods for factoring integers and computing discrete logarithms require the solution of large sparse linear systems of equations over finite fields. This paper presents the results of implementations of several linear algebra algorithms. It shows that very large sparse systems can be solved efficiently by using combinations of structured Gaussian elimination and the conjugate gradient, Lanczos, and Wiedemann methods. 1. Introduction Factoring integers and computing discrete logarithms often requires solving large systems of linear equations over finite fields. General surveys of these areas are presented in [14, 17, 19]. So far there have been few implementations of discrete logarithm algorithms, but many of integer factoring methods. Some of the published results have involved solving systems of over 6 \Theta 10 4 equations in more than 6 \Theta 10 4 variables [12]. In factoring, equations have had to be solved over the field GF (2). In that situation, ordinary...
Discrete logarithms in gf(p) using the number field sieve
 SIAM J. Discrete Math
, 1993
"... Recently, several algorithms using number field sieves have been given to factor a number n in heuristic expected time Ln[1/3; c], where Ln[v; c] = exp{(c + o(1))(log n) v (log log n) 1−v}, for n → ∞. In this paper we present an algorithm to solve the discrete logarithm problem for GF (p) with heur ..."
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Cited by 68 (1 self)
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Recently, several algorithms using number field sieves have been given to factor a number n in heuristic expected time Ln[1/3; c], where Ln[v; c] = exp{(c + o(1))(log n) v (log log n) 1−v}, for n → ∞. In this paper we present an algorithm to solve the discrete logarithm problem for GF (p) with heuristic expected running time Lp[1/3; 3 2/3]. For numbers of a special form, there is an asymptotically slower but more practical version of the algorithm.
On DiffieHellman Key Agreement with Short Exponents
 Proc. Eurocrypt '96, LNCS 1070
, 1996
"... The difficulty of computing discrete logarithms known to be "short" is examined, motivated by recent practical interest in using DiftieHellman key agreement with short exponents (e.g. over Zp with 160bit exponents and 1024bit primes p). A new divideandconquer algorithm for discret ..."
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Cited by 60 (0 self)
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The difficulty of computing discrete logarithms known to be "short" is examined, motivated by recent practical interest in using DiftieHellman key agreement with short exponents (e.g. over Zp with 160bit exponents and 1024bit primes p). A new divideandconquer algorithm for discrete logarithms is presented, combining Pollard's lambda method with a partial PohhgHellman decomposition. For random Diftie Hellman primes p, examination reveals this partial decomposition itself allows recovery of short exponents in many cases, while the new technique dramatically extends the range. Use of subgroups of large prime order precludes the attack at essentially no cost, and is the recommended solution.
Discrete Logarithms: the Effectiveness of the Index Calculus Method
, 1996
"... . In this article we survey recent developments concerning the discrete logarithm problem. Both theoretical and practical results are discussed. We emphasize the case of finite fields, and in particular, recent modifications of the index calculus method, including the number field sieve and the func ..."
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Cited by 26 (1 self)
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. In this article we survey recent developments concerning the discrete logarithm problem. Both theoretical and practical results are discussed. We emphasize the case of finite fields, and in particular, recent modifications of the index calculus method, including the number field sieve and the function field sieve. We also provide a sketch of the some of the cryptographic schemes whose security depends on the intractibility of the discrete logarithm problem. 1 Introduction Let G be a cyclic group generated by an element t. The discrete logarithm problem in G is to compute for any b 2 G the least nonnegative integer e such that t e = b. In this case, we write log t b = e. Our purpose, in this paper, is to survey recent work on the discrete logarithm problem. Our approach is twofold. On the one hand, we consider the problem from a purely theoretical perspective. Indeed, the algorithms that have been developed to solve it not only explore the fundamental nature of one of the basic s...
Practical Approaches to Attaining Security Against Adaptively Chosen Ciphertext Attacks
 In Advances in Cryptology–Crypto ’92
, 1992
"... Abstract. This paper presents three methods for strengthening public key cryptosystems in such a way that they become secure against adaptively chosen ciphertext attacks. In an adaptively chosen ciphertext attack, an attacker can query the deciphering algorithm with any ciphertexts, except for the e ..."
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Cited by 25 (2 self)
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Abstract. This paper presents three methods for strengthening public key cryptosystems in such a way that they become secure against adaptively chosen ciphertext attacks. In an adaptively chosen ciphertext attack, an attacker can query the deciphering algorithm with any ciphertexts, except for the exact object ciphertext to be cryptanalyzed. The rst strengthening method is based on the use of oneway hash functions, the second on the use of universal hash functions and the third on the use of digital signature schemes. Each method is illustrated by an example ofapublickey cryptosystem based on the intractability ofcomputing discrete logarithms in nite elds. Two other issues, namely applications of the methods to public key cryptosystems based on other intractable problems and enhancement of information authentication capability to the cryptosystems, are also discussed. 1
Elliptic Curve Discrete Logarithms and the Index Calculus
"... . The discrete logarithm problem forms the basis of numerous cryptographic systems. The most effective attack on the discrete logarithm problem in the multiplicative group of a finite field is via the index calculus, but no such method is known for elliptic curve discrete logarithms. Indeed, Miller ..."
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Cited by 25 (4 self)
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. The discrete logarithm problem forms the basis of numerous cryptographic systems. The most effective attack on the discrete logarithm problem in the multiplicative group of a finite field is via the index calculus, but no such method is known for elliptic curve discrete logarithms. Indeed, Miller [23] has given a brief heuristic argument as to why no such method can exist. IN this note we give a detailed analysis of the index calculus for elliptic curve discrete logarithms, amplifying and extending miller's remarks. Our conclusions fully support his contention that the natural generalization of the index calculus to the elliptic curve discrete logarithm problem yields an algorithm with is less efficient than a bruteforce search algorithm. 0. Introduction The discrete logarithm problem for the multiplicative group F q of a finite field can be solved in subexponential time using the Index Calculus method, which appears to have been first discovered by Kraitchik [14, 15] in the 192...
Crypto for Tiny Objects
, 2004
"... This work presents the first known implementation of elliptic curve cryptography for sensor networks, motivated by those networks' need for an e#cient, secure mechanism for shared cryptographic keys' distribution and redistribution among nodes. Through instrumentation of UC Berkeley's ..."
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Cited by 21 (2 self)
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This work presents the first known implementation of elliptic curve cryptography for sensor networks, motivated by those networks' need for an e#cient, secure mechanism for shared cryptographic keys' distribution and redistribution among nodes. Through instrumentation of UC Berkeley's TinyOS, this work demonstrates that secretkey cryptography is already viable on the MICA2 mote. Through analyses of another's implementation of modular exponentiation and of its own implementation of elliptic curves, this work concludes that publickey infrastructure may also be tractable in 4 kilobytes of primary memory on this 8bit, 7.3828MHz device.
Designing and detecting trapdoors for discrete log cryptosystems
 ADVANCES IN CRYPTOLOGY CRYPTO '92
, 1993
"... Using a number field sieve, discrete logarithms modulo primes of special forms can be found faster than standard primes. This has raised concerns about trapdoors in discrete log cryptosystems, such as the Digital Signature Standard. This paper discusses the practical impact of these trapdoors, and ..."
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Cited by 16 (0 self)
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Using a number field sieve, discrete logarithms modulo primes of special forms can be found faster than standard primes. This has raised concerns about trapdoors in discrete log cryptosystems, such as the Digital Signature Standard. This paper discusses the practical impact of these trapdoors, and how to avoid them.
Improvements to the general number field sieve for discrete logarithms in prime fields
 Mathematics of Computation
, 2003
"... Abstract. In this paper, we describe many improvements to the number field sieve. Our main contribution consists of a new way to compute individual logarithms with the number field sieve without solving a very large linear system for each logarithm. We show that, with these improvements, the number ..."
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Cited by 15 (1 self)
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Abstract. In this paper, we describe many improvements to the number field sieve. Our main contribution consists of a new way to compute individual logarithms with the number field sieve without solving a very large linear system for each logarithm. We show that, with these improvements, the number field sieve outperforms the gaussian integer method in the hundred digit range. We also illustrate our results by successfully computing discrete logarithms with GNFS in a large prime field. 1.