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62
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 396 (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.
Optimal Extension Fields for Fast Arithmetic in PublicKey Algorithms
, 1998
"... Abstract. This contribution introduces a class of Galois field used to achieve fast finite field arithmetic which we call an Optimal Extension Field (OEF). This approach is well suited for implementation of publickey cryptosystems based on elliptic and hyperelliptic curves. Whereas previous reported ..."
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Cited by 66 (14 self)
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Abstract. This contribution introduces a class of Galois field used to achieve fast finite field arithmetic which we call an Optimal Extension Field (OEF). This approach is well suited for implementation of publickey cryptosystems based on elliptic and hyperelliptic curves. Whereas previous reported optimizations focus on finite fields of the form GF (p) and GF (2 m), an OEF is the class of fields GF (p m), for p a prime of special form and m a positive integer. Modern RISC workstation processors are optimized to perform integer arithmetic on integers of size up to the word size of the processor. Our construction employs wellknown techniques for fast finite field arithmetic which fully exploit the fast integer arithmetic found on these processors. In this paper, we describe our methods to perform the arithmetic in an OEF and the methods to construct OEFs. We provide a list of OEFs tailored for processors with 8, 16, 32, and 64 bit word sizes. We report on our application of this approach to construction of elliptic curve cryptosystems and demonstrate a substantial performance improvement over all previous reported software implementations of Galois field arithmetic for elliptic curves.
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 54 (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.
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 37 (7 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
Construction of secure random curves of genus 2 over prime fields
 Advances in Cryptology – EUROCRYPT 2004, volume 3027 of Lecture Notes in Comput. Sci
, 2004
"... Abstract. For counting points of Jacobians of genus 2 curves defined over large prime fields, the best known method is a variant of Schoof’s algorithm. We present several improvements on the algorithms described by Gaudry and Harley in 2000. In particular we rebuild the symmetry that had been broken ..."
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Cited by 37 (12 self)
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Abstract. For counting points of Jacobians of genus 2 curves defined over large prime fields, the best known method is a variant of Schoof’s algorithm. We present several improvements on the algorithms described by Gaudry and Harley in 2000. In particular we rebuild the symmetry that had been broken by the use of Cantor’s division polynomials and design a faster division by 2 and a division by 3. Combined with the algorithm by Matsuo, Chao and Tsujii, our implementation can count the points on a Jacobian of size 164 bits within about one week on a PC. 1
Aspects of Hyperelliptic Curves over Large Prime Fields in Software Implementations
, 2004
"... Abstract. We present an implementation of elliptic curves and of hyperelliptic curves of genus 2 and 3 over prime fields. To achieve a fair comparison between the different types of groups, we developed an adhoc arithmetic library, designed to remove most of the overheads that penalize implementati ..."
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Cited by 37 (5 self)
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Abstract. We present an implementation of elliptic curves and of hyperelliptic curves of genus 2 and 3 over prime fields. To achieve a fair comparison between the different types of groups, we developed an adhoc arithmetic library, designed to remove most of the overheads that penalize implementations of curvebased cryptography over prime fields. These overheads get worse for smaller fields, and thus for larger genera for a fixed group size. We also use techniques for delaying modular reductions to reduce the amount of modular reductions in the formulae for the group operations. The result is that the performance of hyperelliptic curves of genus 2 over prime fields is much closer to the performance of elliptic curves than previously thought. For groups of 192 and 256 bits the difference is about 14 % and 15 % respectively.
Efficient Arithmetic on Genus 2 Hyperelliptic Curves over Finite Fields via Explicit Formulae
 In Cryptology ePrint archive, Report 2002/121
, 2002
"... We extend the explicit formulae for arithmetic on genus two curves of [13, 21] to fields of even characteristic and to arbitrary equation of the curve. These formulae can be evaluated faster than the more general Cantor algorithm and allow to obtain faster arithmetic on a hyperelliptic genus 2 curve ..."
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Cited by 30 (4 self)
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We extend the explicit formulae for arithmetic on genus two curves of [13, 21] to fields of even characteristic and to arbitrary equation of the curve. These formulae can be evaluated faster than the more general Cantor algorithm and allow to obtain faster arithmetic on a hyperelliptic genus 2 curve than on elliptic curves. We give timings for implementations using various libraries for the field arithmetic.
Genus Two Hyperelliptic Curve Coprocessor
 In Workshop on Cryptographic Hardware and Embedded Systems  CHES 2002
, 2002
"... Abstract. Hyperelliptic curve cryptography with genus larger than one has not been seriously considered for cryptographic purposes because many existing implementations are significantly slower than elliptic curve versions with the same level of security. In this paper, the first ever complete hardw ..."
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Cited by 20 (0 self)
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Abstract. Hyperelliptic curve cryptography with genus larger than one has not been seriously considered for cryptographic purposes because many existing implementations are significantly slower than elliptic curve versions with the same level of security. In this paper, the first ever complete hardware implementation of a hyperelliptic curve coprocessor is described. This coprocessor is designed for genus two curves over F 2 113. Additionally, a modification to the Extended Euclidean Algorithm is presented for the GCD calculation required by Cantor’s algorithm. On average, this new method computes the GCD in onefourth the time required by the Extended Euclidean Algorithm. 1