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PolynomialTime Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
 SIAM J. on Computing
, 1997
"... A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time by at most a polynomial factor. This may not be true when quantum mechanics is taken into consideration. ..."
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Cited by 941 (2 self)
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A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time by at most a polynomial factor. This may not be true when quantum mechanics is taken into consideration. This paper considers factoring integers and finding discrete logarithms, two problems which are generally thought to be hard on a classical computer and which have been used as the basis of several proposed cryptosystems. Efficient randomized algorithms are given for these two problems on a hypothetical quantum computer. These algorithms take a number of steps polynomial in the input size, e.g., the number of digits of the integer to be factored.
Guide to Elliptic Curve Cryptography
, 2004
"... Elliptic curves have been intensively studied in number theory and algebraic geometry for over 100 years and there is an enormous amount of literature on the subject. To quote the mathematician Serge Lang: It is possible to write endlessly on elliptic curves. (This is not a threat.) Elliptic curves ..."
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Cited by 439 (18 self)
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Elliptic curves have been intensively studied in number theory and algebraic geometry for over 100 years and there is an enormous amount of literature on the subject. To quote the mathematician Serge Lang: It is possible to write endlessly on elliptic curves. (This is not a threat.) Elliptic curves also figured prominently in the recent proof of Fermat's Last Theorem by Andrew Wiles. Originally pursued for purely aesthetic reasons, elliptic curves have recently been utilized in devising algorithms for factoring integers, primality proving, and in publickey cryptography. In this article, we aim to give the reader an introduction to elliptic curve cryptosystems, and to demonstrate why these systems provide relatively small block sizes, highspeed software and hardware implementations, and offer the highest strengthperkeybit of any known publickey scheme.
Simulating Physics with Computers
 SIAM Journal on Computing
, 1982
"... A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time of at most a polynomial factor. This may not be true when quantum mechanics is taken into consideration. ..."
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Cited by 428 (1 self)
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A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time of at most a polynomial factor. This may not be true when quantum mechanics is taken into consideration. This paper considers factoring integers and finding discrete logarithms, two problems which are generally thought to be hard on a classical computer and have been used as the basis of several proposed cryptosystems. Efficient randomized algorithms are given for these two problems on a hypothetical quantum computer. These algorithms take a number of steps polynomial in the input size, e.g., the number of digits of the integer to be factored. AMS subject classifications: 82P10, 11Y05, 68Q10. 1 Introduction One of the first results in the mathematics of computation, which underlies the subsequent development of much of theoretical computer science, was the distinction between computable and ...
A PublicKey Infrastructure for Key Distribution in TinyOS Based on Elliptic Curve Cryptography
, 2004
"... We present the first known implementation of elliptic curve cryptography over F2 p for sensor networks based on the 8bit, 7.3828MHz MICA2 mote. Through instrumentation of UC Berkeley's TinySec module, we argue that, although secretkey cryptography has been tractable in this domain for some ..."
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Cited by 210 (3 self)
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We present the first known implementation of elliptic curve cryptography over F2 p for sensor networks based on the 8bit, 7.3828MHz MICA2 mote. Through instrumentation of UC Berkeley's TinySec module, we argue that, although secretkey cryptography has been tractable in this domain for some time, there has remained a need for an efficient, secure mechanism for distribution of secret keys among nodes. Although publickey infrastructure has been thought impractical, we argue, through analysis of our own implementation for TinyOS of multiplication of points on elliptic curves, that publickey infrastructure is, in fact, viable for TinySec keys' distribution, even on the MICA2. We demonstrate that public keys can be generated within 34 seconds, and that shared secrets can be distributed among nodes in a sensor network within the same, using just over 1 kilobyte of SRAM and 34 kilobytes of ROM.
The gapproblems: a new class of problems for the security of cryptographic schemes
 Proceedings of PKC 2001, volume 1992 of LNCS
, 1992
"... Abstract. This paper introduces a novel class of computational problems, the gap problems, which can be considered as a dual to the class of the decision problems. We show the relationship among inverting problems, decision problems and gap problems. These problems find a nice and rich practical ins ..."
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Cited by 132 (11 self)
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Abstract. This paper introduces a novel class of computational problems, the gap problems, which can be considered as a dual to the class of the decision problems. We show the relationship among inverting problems, decision problems and gap problems. These problems find a nice and rich practical instantiation with the DiffieHellman problems. Then, we see how the gap problems find natural applications in cryptography, namely for proving the security of very efficient schemes, but also for solving a more than 10year old open security problem: the Chaum’s undeniable signature.
The XTR public key system
, 2000
"... This paper introduces the XTR public key system. XTR is based on a new method to represent elements of a subgroup of a multiplicative group of a finite field. Application of XTR in cryptographic protocols leads to substantial savings both in communication and computational overhead without compromis ..."
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Cited by 85 (13 self)
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This paper introduces the XTR public key system. XTR is based on a new method to represent elements of a subgroup of a multiplicative group of a finite field. Application of XTR in cryptographic protocols leads to substantial savings both in communication and computational overhead without compromising security.
Computing discrete logarithms in real quadratic congruence function fields of large genus
 Math. Comp
, 1999
"... Abstract. The discrete logarithm problem in various finite abelian groups is the basis for some well known public key cryptosystems. Recently, real quadratic congruence function fields were used to construct a public key distribution system. The security of this public key system is based on the dif ..."
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Cited by 42 (8 self)
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Abstract. The discrete logarithm problem in various finite abelian groups is the basis for some well known public key cryptosystems. Recently, real quadratic congruence function fields were used to construct a public key distribution system. The security of this public key system is based on the difficulty of a discrete logarithm problem in these fields. In this paper, we present a probabilistic algorithm with subexponential running time that computes such discrete logarithms in real quadratic congruence function fields of sufficiently large genus. This algorithm is a generalization of similar algorithms for real quadratic number fields. 1.
Generating ElGamal signatures without knowing the secret key
, 1996
"... . We present a new method to forge ElGamal signatures if the public parameters of the system are not chosen properly. Since the secret key is hereby not found this attack shows that forging ElGamal signatures is sometimes easier than the underlying discrete logarithm problem. 1 Introduction ElGamal ..."
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Cited by 40 (0 self)
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. We present a new method to forge ElGamal signatures if the public parameters of the system are not chosen properly. Since the secret key is hereby not found this attack shows that forging ElGamal signatures is sometimes easier than the underlying discrete logarithm problem. 1 Introduction ElGamal's digital signature scheme [4] relies on the difficulty of computing discrete logarithms in the multiplicative group IF p and can therefore be broken if the computation of discrete logarithms is feasible. However, the converse has never been proved. In this paper we show that it is sometimes possible to forge signatures without breaking the underlying discrete logarithm problem. This shows that the ElGamal signature scheme and some variants of the scheme must be used very carefully. The paper is organized as follows. Section 2 describes the ElGamal signature scheme. In Section 3 we present a method to forge signatures if some additional information on the generator is known. We show that...
The function field sieve in the medium prime case
 Advances in Cryptology – EUROCRYPT 2006, LNCS 4004 (2006
"... Abstract. In this paper, we study the application of the function field sieve algorithm for computing discrete logarithms over finite fields of the form Fqn when q is a mediumsized prime power. This approach is an alternative to a recent paper of Granger and Vercauteren for computing discrete logar ..."
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Cited by 33 (9 self)
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Abstract. In this paper, we study the application of the function field sieve algorithm for computing discrete logarithms over finite fields of the form Fqn when q is a mediumsized prime power. This approach is an alternative to a recent paper of Granger and Vercauteren for computing discrete logarithms in tori, using efficient torus representations. We show that when q is not too large, a very efficient L(1/3) variation of the function field sieve can be used. Surprisingly, using this algorithm, discrete logarithms computations over some of these fields are even easier than computations in the prime field and characteristic two field cases. We also show that this new algorithm has security implications on some existing cryptosystems, such as torus based cryptography in T30, short signature schemes in characteristic 3 and cryptosystems based on supersingular abelian varieties. On the other hand, cryptosystems involving larger basefields and smaller extension degrees, typically of degree at most 6, such as LUC, XTR or T6 torus cryptography, are not affected. 1
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...