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39
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 610 (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.
Constructive and destructive facets of Weil descent on elliptic curves
 JOURNAL OF CRYPTOLOGY
, 2002
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An algorithm for solving the discrete log problem on hyperelliptic curves
, 2000
"... Abstract. We present an indexcalculus algorithm for the computation of discrete logarithms in the Jacobian of hyperelliptic curves defined over finite fields. The complexity predicts that it is faster than the Rho method for genus greater than 4. To demonstrate the efficiency of our approach, we de ..."
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Cited by 92 (7 self)
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Abstract. We present an indexcalculus algorithm for the computation of discrete logarithms in the Jacobian of hyperelliptic curves defined over finite fields. The complexity predicts that it is faster than the Rho method for genus greater than 4. To demonstrate the efficiency of our approach, we describe our breaking of a cryptosystem based on a curve of genus 6 recently proposed by Koblitz. 1
A double large prime variation for small genus hyperelliptic index calculus
 Mathematics of Computation
, 2004
"... Abstract. In this article, we examine how the index calculus approach for computing discrete logarithms in small genus hyperelliptic curves can be improved by introducing a double large prime variation. Two algorithms are presented. The first algorithm is a rather natural adaptation of the double la ..."
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Cited by 66 (14 self)
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Abstract. In this article, we examine how the index calculus approach for computing discrete logarithms in small genus hyperelliptic curves can be improved by introducing a double large prime variation. Two algorithms are presented. The first algorithm is a rather natural adaptation of the double large prime variation to the intended context. On heuristic and experimental grounds, it seems to perform quite well but lacks a complete and precise analysis. Our second algorithm is a considerably simplified variant, which can be analyzed easily. The resulting complexity improves on the fastest known algorithms. Computer experiments show that for hyperelliptic curves of genus three, our first algorithm surpasses Pollard’s Rho method even for rather small field sizes. 1.
A General Framework for Subexponential Discrete Logarithm Algorithms in Groups of Unknown Order
, 2000
"... We develop a generic framework for the computation of logarithms in nite class groups. The model allows to formulate a probabilistic algorithm based on collecting relations in an abstract way independently of the specific type of group to which it is applied, and to prove a subexponential running ti ..."
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Cited by 62 (10 self)
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We develop a generic framework for the computation of logarithms in nite class groups. The model allows to formulate a probabilistic algorithm based on collecting relations in an abstract way independently of the specific type of group to which it is applied, and to prove a subexponential running time if a certain smoothness assumption is verified. The algorithm proceeds in two steps: First, it determines the abstract group structure as a product of cyclic groups; second, it computes an explicit isomorphism, which can be used to extract discrete logarithms.
Applications of Arithmetical Geometry to Cryptographic Constructions
 Proceedings of the Fifth International Conference on Finite Fields and Applications
"... Public key cryptosystems are very important tools for data transmission. Their performance and security depend on the underlying crypto primitives. In this paper we describe one such primitive: The Discrete Logarithm (DL) in cyclic groups of prime order (Section 1). To construct DLsystems we use me ..."
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Cited by 48 (1 self)
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Public key cryptosystems are very important tools for data transmission. Their performance and security depend on the underlying crypto primitives. In this paper we describe one such primitive: The Discrete Logarithm (DL) in cyclic groups of prime order (Section 1). To construct DLsystems we use methods from algebraic and arithmetic geometry and especially the theory of abelian varieties over finite fields. It is explained why Jacobian varieties of hyperelliptic curves of genus 4 are candidates for cryptographically "good" abelian varieties (Section 2). In the third section we describe the (constructive and destructive) role played by Galois theory: Local and global Galois representation theory is used to count points on abelian varieties over finite fields and we give some applications of Weil descent and Tate duality.
A cryptographic application of Weil descent
 CODES AND CRYPTOGRAPHY, LNCS 1746
, 1999
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ARITHMETIC ON SUPERELLIPTIC CURVES
, 2000
"... This paper is concerned with algorithms for computing in the divisor class group of a nonsingular plane curve of the form yn = c(x) which has only one point at infinity. Divisors are represented as ideals, and an ideal reduction algorithm based on lattice reduction is given. We obtain a unique rep ..."
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Cited by 45 (4 self)
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This paper is concerned with algorithms for computing in the divisor class group of a nonsingular plane curve of the form yn = c(x) which has only one point at infinity. Divisors are represented as ideals, and an ideal reduction algorithm based on lattice reduction is given. We obtain a unique representative for each divisor class and the algorithms for addition and reduction of divisors run in polynomial time. An algorithm is also given for solving the discrete logarithm problem when the curve is defined over a finite field.
Computing discrete logarithms in highgenus hyperelliptic Jacobians in provably subexponential time
 Mathematics of Computation
, 1999
"... Abstract. We provide a subexponential algorithm for solving the discrete logarithm problem in Jacobians of highgenus hyperelliptic curves over finite fields. Its expected running time for instances with genus g and underlying finite field Fq satisfying g ≥ ϑ log q for a positive constant ϑ is given ..."
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Cited by 43 (8 self)
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Abstract. We provide a subexponential algorithm for solving the discrete logarithm problem in Jacobians of highgenus hyperelliptic curves over finite fields. Its expected running time for instances with genus g and underlying finite field Fq satisfying g ≥ ϑ log q for a positive constant ϑ is given by