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128
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 382 (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.
Constructive And Destructive Facets Of Weil Descent On Elliptic Curves
 JOURNAL OF CRYPTOLOGY
, 2000
"... In this paper we look in detail at the curves which arise in the method of Galbraith and Smart for producing curves in the Weil restriction of an elliptic curve over a finite field of characteristic two of composite degree. We explain how this method can be used to construct hyperelliptic cryptosys ..."
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Cited by 140 (12 self)
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In this paper we look in detail at the curves which arise in the method of Galbraith and Smart for producing curves in the Weil restriction of an elliptic curve over a finite field of characteristic two of composite degree. We explain how this method can be used to construct hyperelliptic cryptosystems which could be as secure as a cryptosystems based on the original elliptic curve. On the other hand, we show that this may provide a way of attacking the original elliptic curve cryptosystem using recent advances in the study of the discrete logarithm problem on hyperelliptic curves. We examine the resulting higher genus curves in some detail and propose an additional check on elliptic curve systems defined over fields of characteristic two so as to make them immune from the methods in this paper. 1. Introduction In this paper we address two problems: How to construct hyperelliptic cryptosystems and how to attack elliptic curve cryptosystems defined over fields of even characteristic ...
Efficient Pairing Computation on Supersingular Abelian Varieties
 Designs, Codes and Cryptography
, 2004
"... We present a general technique for the efficient computation of pairings on supersingular Abelian varieties. As particular cases, we describe efficient pairing algorithms for elliptic and hyperelliptic curves in characteristic 2. The latter is faster than all previously known pairing algorithms, and ..."
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Cited by 130 (23 self)
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We present a general technique for the efficient computation of pairings on supersingular Abelian varieties. As particular cases, we describe efficient pairing algorithms for elliptic and hyperelliptic curves in characteristic 2. The latter is faster than all previously known pairing algorithms, and as a bonus also gives rise to faster conventional Jacobian arithmetic.
Supersingular curves in cryptography
, 2001
"... Frey and Rück gave a method to map the discrete logarithm problem in the divisor class group of a curve over ¢¡ into a finite field discrete logarithm problem in some extension. The discrete logarithm problem in the divisor class group can therefore be solved as long ¥ as is small. In the elliptic ..."
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Cited by 87 (8 self)
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Frey and Rück gave a method to map the discrete logarithm problem in the divisor class group of a curve over ¢¡ into a finite field discrete logarithm problem in some extension. The discrete logarithm problem in the divisor class group can therefore be solved as long ¥ as is small. In the elliptic curve case it is known that for supersingular curves one ¥§¦© ¨ has. In this paper curves of higher genus are studied. Bounds on the possible values ¥ for in the case of supersingular curves are given. Ways to ensure that a curve is not supersingular are also given. 1.
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 80 (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
Counting Points on Hyperelliptic Curves over Finite Fields
"... . We describe some algorithms for computing the cardinality of hyperelliptic curves and their Jacobians over finite fields. They include several methods for obtaining the result modulo small primes and prime powers, in particular an algorithm `a la Schoof for genus 2 using Cantor 's division pol ..."
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Cited by 58 (7 self)
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. We describe some algorithms for computing the cardinality of hyperelliptic curves and their Jacobians over finite fields. They include several methods for obtaining the result modulo small primes and prime powers, in particular an algorithm `a la Schoof for genus 2 using Cantor 's division polynomials. These are combined with a birthday paradox algorithm to calculate the cardinality. Our methods are practical and we give actual results computed using our current implementation. The Jacobian groups we handle are larger than those previously reported in the literature. Introduction In recent years there has been a surge of interest in algorithmic aspects of curves. When presented with any curve, a natural task is to compute the number of points on it with coordinates in some finite field. When the finite field is large this is generally difficult to do. Ren'e Schoof gave a polynomial time algorithm for counting points on elliptic curves i.e., those of genus 1, in his ground...
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 55 (11 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.
Formulae for Arithmetic on Genus 2 Hyperelliptic Curves
 Applicable Algebra in Engineering, Communication and Computing
, 2003
"... The ideal class group of hyperelliptic curves can be used in cryptosystems based on the discrete logarithm problem. In this article we present explicit formulae to perform the group operations for genus 2 curves. The formulae are completely general but to achieve the lowest number of operations we t ..."
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Cited by 49 (3 self)
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The ideal class group of hyperelliptic curves can be used in cryptosystems based on the discrete logarithm problem. In this article we present explicit formulae to perform the group operations for genus 2 curves. The formulae are completely general but to achieve the lowest number of operations we treat odd and even characteristic separately. We present 3 different coordinate systems which are suitable for different environments, e. g. on a smart card we should avoid inversions while in software a limited number is acceptable. The presented formulae render genus two hyperelliptic curves very useful in practice. The first system are affine coordinates where each group operation needs one inversion. Then we consider projective coordinates avoiding inversions on the cost of more multiplications and a further coordinate. Finally, we introduce a new system of coordinates and state algorithms showing that doublings are comparably cheap and no inversions are needed. A comparison between the systems concludes the paper.
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 41 (12 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.