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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 594 (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.
On the performance of hyperelliptic cryptosystems
, 1999
"... In this paper we discuss various aspects of cryptosystems based on hyperelliptic curves. In particular we cover the implementation of the group law on such curves and how to generate suitable curves for use in cryptography. This paper presents a practical comparison between the performance of ellip ..."
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Cited by 30 (4 self)
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In this paper we discuss various aspects of cryptosystems based on hyperelliptic curves. In particular we cover the implementation of the group law on such curves and how to generate suitable curves for use in cryptography. This paper presents a practical comparison between the performance of elliptic curve based digital signature schemes and schemes based on hyperelliptic curves. We conclude that, at present, hyperelliptic curves offer no performance advantage over elliptic curves.
Deformation theory and the computation of zeta functions
 Proc. London Math. Soc
, 2004
"... An attractive and challenging problem in computational number theory is to count in an e cient manner the number of solutions to a multivariate polynomial equation over a nite eld. One desires an algorithm whose time complexity is a small polynomial function of some appropriate measure of the size o ..."
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Cited by 29 (1 self)
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An attractive and challenging problem in computational number theory is to count in an e cient manner the number of solutions to a multivariate polynomial equation over a nite eld. One desires an algorithm whose time complexity is a small polynomial function of some appropriate measure of the size of the
Counting points on varieties over finite fields of small characteristic
 ALGORITHMIC NUMBER THEORY
, 2008
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An Extension of Kedlaya's Algorithm to Hyperelliptic Curves in Characteristic 2
, 2002
"... We present an algorithm for computing the zeta function of an arbitrary hyperelliptic curve over a finite field Fq of characteristic 2, thereby extending the algorithm of Kedlaya for odd characteristic. For a genus g hyperelliptic curve defined over F2 n , the averagecase time complexity is O(g ) a ..."
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Cited by 18 (5 self)
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We present an algorithm for computing the zeta function of an arbitrary hyperelliptic curve over a finite field Fq of characteristic 2, thereby extending the algorithm of Kedlaya for odd characteristic. For a genus g hyperelliptic curve defined over F2 n , the averagecase time complexity is O(g ) and the averagecase space complexity is O(g ), whereas the worstcase time and space complexities are O(g ) and ) respectively.
Computing zeta functions over finite fields
 Contemporary Math
, 1999
"... Abstract. In this report, we discuss the problem of computing the zeta function of an algebraic variety defined over a finite field, with an emphasis on computing the reduction modulo p m of the zeta function of a hypersurface, where p is the characteristic of the finite field. 1991 Mathematics Subj ..."
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Cited by 17 (3 self)
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Abstract. In this report, we discuss the problem of computing the zeta function of an algebraic variety defined over a finite field, with an emphasis on computing the reduction modulo p m of the zeta function of a hypersurface, where p is the characteristic of the finite field. 1991 Mathematics Subject Classification: 11Y16, 11T99, 14Q15. 1.
Fast genus two hyperelliptic curve cryptosystems
, 2001
"... Abstract: Most current hyperelliptic curve cryptosystems (HECC) use the Cantor algorithm in hyperelliptic additions for encryption and decryption. It has been reported that hyperelliptic curve cryptosystems are more than several times slower than elliptic curve cryptosystems (ECC). It is then an int ..."
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Cited by 16 (1 self)
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Abstract: Most current hyperelliptic curve cryptosystems (HECC) use the Cantor algorithm in hyperelliptic additions for encryption and decryption. It has been reported that hyperelliptic curve cryptosystems are more than several times slower than elliptic curve cryptosystems (ECC). It is then an interesting and challenging question that if the HECC could be faster than the ECC. Recently, Harley proposed a faster algorithm of addition on genus two hyperelliptic curves. In the first part of this paper, we show an improvement of the Harley algorithm which needs less computation then faster than the elliptic addition. The second part of the paper is a tentative implementation of the genus two HECC using the improved Harley algorithm, comparing with the ECC, which shown that the performance of the HECC is actually equal to that of the ECC.
Computational Aspects of Curves of Genus at Least 2
 Algorithmic number theory. 5th international symposium. ANTSII
, 1996
"... . This survey discusses algorithms and explicit calculations for curves of genus at least 2 and their Jacobians, mainly over number fields and finite fields. Miscellaneous examples and a list of possible future projects are given at the end. 1. Introduction An enormous number of people have per ..."
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Cited by 15 (2 self)
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. This survey discusses algorithms and explicit calculations for curves of genus at least 2 and their Jacobians, mainly over number fields and finite fields. Miscellaneous examples and a list of possible future projects are given at the end. 1. Introduction An enormous number of people have performed an enormous number of computations on elliptic curves, as one can see from even a perfunctory glance at [29]. A few years ago, the same could not be said for curves of higher genus, even though the theory of such curves had been developed in detail. Now, however, polynomialtime algorithms and sometimes actual programs are available for solving a wide variety of problems associated with such curves. The genus 2 case especially is becoming accessible: in light of recent work, it seems reasonable to expect that within a few years, packages will be available for doing genus 2 computations analogous to the elliptic curve computations that are currently possible in PARI, MAGMA, SIMATH, apec...
Quantum computation of zeta functions of curves
 Computational Complexity
"... We exhibit a quantum algorithm for determining the zeta function of a genus g curve over a finite field Fq, which is polynomial in g and log(q). This amounts to giving an algorithm to produce provably random elements of the class group of a curve, plus a recipe for recovering a Weil polynomial from ..."
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Cited by 14 (0 self)
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We exhibit a quantum algorithm for determining the zeta function of a genus g curve over a finite field Fq, which is polynomial in g and log(q). This amounts to giving an algorithm to produce provably random elements of the class group of a curve, plus a recipe for recovering a Weil polynomial from enough of its cyclic resultants. The latter effectivizes a result of Fried in a restricted setting. 1
Algorithmic theory of zeta functions over finite fields
 ALGORITHMIC NUMBER THEORY
, 2008
"... We give an introductory account of the general algorithmic theory of the zeta function of an algebraic set defined over a finite field. ..."
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Cited by 11 (3 self)
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We give an introductory account of the general algorithmic theory of the zeta function of an algebraic set defined over a finite field.