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26
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 369 (17 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.
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 78 (6 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 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 54 (9 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.
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 51 (10 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.
Hyperelliptic Curve Cryptosystems: Closing the Performance Gap to Elliptic Curves
 Workshop on Cryptographic Hardware and Embedded Systems — CHES 2003
, 2003
"... For most of the time since they were proposed, it was widely believed that hyperelliptic curve cryptosystems (HECC) carry a substantial performance penalty compared to elliptic curve cryptosystems (ECC) and are, thus, not too attractive for practical applications. Only quite recently improvements ha ..."
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Cited by 42 (13 self)
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For most of the time since they were proposed, it was widely believed that hyperelliptic curve cryptosystems (HECC) carry a substantial performance penalty compared to elliptic curve cryptosystems (ECC) and are, thus, not too attractive for practical applications. Only quite recently improvements have been made, mainly restricted to curves of genus 2. The work at hand advances the stateoftheart considerably in several aspects. First, we generalize and improve the closed formulae for the group operation of genus 3 for HEC defined over fields of characteristic two. For certain curves we achieve over 50% complexity improvement compared to the best previously published results. Second, we introduce a new complexity metric for ECC and HECC defined over characteristic two fields which allow performance comparisons of practical relevance. It can be shown that the HECC performance is in the range of the performance of an ECC; for specific parameters HECC can even possess a lower complexity than an ECC at the same security level. Third, we describe the first implementation of a HEC cryptosystem on an embedded (ARM7) processor. Since HEC are particularly attractive for constrained environments, such a case study should be of relevance.
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 41 (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.
Low Cost Security: Explicit Formulae for Genus 4 Hyperelliptic Curves
, 2003
"... It is widely believed that genus four hyperelliptic curve cryptosystems (HECC) are not attractive for practical applications because of their complexity compared to systems based on lower genera, especially elliptic curves. Our contribution shows that for low cost security applications genus4 hyper ..."
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Cited by 25 (12 self)
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It is widely believed that genus four hyperelliptic curve cryptosystems (HECC) are not attractive for practical applications because of their complexity compared to systems based on lower genera, especially elliptic curves. Our contribution shows that for low cost security applications genus4 hyperelliptic curves (HEC) can outperform genus2 HEC and that we can achieve a performance similar to genus3 HEC. Furthermore our implementation results show that a genus4 HECC is an alternative cryptosystem to systems based on elliptic curves. In the work at hand...
The arithmetic of Jacobian groups of superelliptic cubics
 MR2085899 (2005f:11126) GENERIC APPROACH TO SEARCHING FOR JACOBIANS 505
"... Abstract. We present two algorithms for the arithmetic of cubic curves with a totally ramified prime at infinity. The first algorithm, inspired by Cantor’s reduction for hyperelliptic curves, is easily implemented with a few lines of code, making use of a polynomial arithmetic package. We prove expl ..."
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Cited by 22 (2 self)
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Abstract. We present two algorithms for the arithmetic of cubic curves with a totally ramified prime at infinity. The first algorithm, inspired by Cantor’s reduction for hyperelliptic curves, is easily implemented with a few lines of code, making use of a polynomial arithmetic package. We prove explicit reducedness criteria for superelliptic curves of genus 3 and 4, which show the correctness of the algorithm. The second approach, quite general in nature and applicable to further classes of curves, uses the FGLM algorithm for switching between Gröbner bases for different orderings. Carrying out the computations symbolically, we obtain explicit reduction formulae in terms of the input data. 1.
Solving Elliptic Curve Discrete Logarithm Problems Using Weil Descent
 JOURNAL OF THE RAMANUJAN MATHEMATICAL SOCIETY
, 2001
"... We provide the first cryptographically interesting instance of the elliptic curve discrete logarithm problem which resists all previously known attacks, but which can be solved with modest computer resources using the Weil descent attack methodology of Frey. We report on our implementation of index ..."
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Cited by 18 (3 self)
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We provide the first cryptographically interesting instance of the elliptic curve discrete logarithm problem which resists all previously known attacks, but which can be solved with modest computer resources using the Weil descent attack methodology of Frey. We report on our implementation of indexcalculus methods for hyperelliptic curves over characteristic two finite fields, and discuss the cryptographic implications of our results.