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23
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.
List Decoding of AlgebraicGeometric Codes
 IEEE Trans. on Information Theory
, 1999
"... We generalize Sudan's results for ReedSolomon codes to the class of algebraicgeometric codes, designing algorithms for list decoding of algebraic geometric codes which can decode beyond the conventional errorcorrection bound (d\Gamma1)=2, d being the minimumdistance of the code. Our main algorith ..."
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Cited by 41 (3 self)
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We generalize Sudan's results for ReedSolomon codes to the class of algebraicgeometric codes, designing algorithms for list decoding of algebraic geometric codes which can decode beyond the conventional errorcorrection bound (d\Gamma1)=2, d being the minimumdistance of the code. Our main algorithm is based on an interpolation scheme and factorization of polynomials over algebraic function fields. For the latter problem we design a polynomialtime algorithm and show that the resulting overall listdecoding algorithm runs in polynomial time under some mild conditions. Several examples are included.
Arithmetic On Superelliptic Curves
 Math. Comp
, 2000
"... This paper is concerned with algorithms for computing in the divisor class group of a nonsingular plane curve of the form y n = 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 repre ..."
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Cited by 37 (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 y n = 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.
Linear algebra algorithms for divisors on an algebraic curve
 Mathematics of Computation
"... Abstract. We use an embedding of the symmetric dth power of any algebraic curve C of genus g into a Grassmannian space to give algorithms for working with divisors on C, using only linear algebra in vector spaces of dimension O(g), and matrices of size O(g 2) × O(g). When the base field k is finite ..."
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Cited by 21 (1 self)
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Abstract. We use an embedding of the symmetric dth power of any algebraic curve C of genus g into a Grassmannian space to give algorithms for working with divisors on C, using only linear algebra in vector spaces of dimension O(g), and matrices of size O(g 2) × O(g). When the base field k is finite, or if C has a rational point over k, these give algorithms for working on the Jacobian of C that require O(g 4) field operations, arising from the Gaussian elimination. Our point of view is strongly geometric, and our representation of points on the Jacobian is fairly simple to deal with; in particular, none of our algorithms involves arithmetic with polynomials. We note that our algorithms have the same asymptotic complexity for general curves as the more algebraic algorithms in Florian Hess ’ 1999 Ph.D. thesis, which workswithfunctionfieldsasextensionsofk[x]. However, for special classes of curves, Hess ’ algorithms are asymptotically more efficient than ours, generalizing other known efficient algorithms for special classes of curves, such as hyperelliptic curves (Cantor 1987), superelliptic curves (Galbraith, Paulus, and Smart 2002), and Cab curves (Harasawa and Suzuki 2000); in all those cases, one can attain a complexity of O(g 2). 1.
Decoding AlgebraicGeometric Codes Beyond the ErrorCorrection Bound
, 1998
"... Generalizing the highnoise decoding methods of [1, 19] to the class of algebraicgeometric codes, we design the first polynomialtime algorithms to decode algebraicgeometric codes significantly beyond the conventional errorcorrection bound. Applying our results to codes obtained from curves with m ..."
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Cited by 13 (4 self)
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Generalizing the highnoise decoding methods of [1, 19] to the class of algebraicgeometric codes, we design the first polynomialtime algorithms to decode algebraicgeometric codes significantly beyond the conventional errorcorrection bound. Applying our results to codes obtained from curves with many rational points, we construct arbitrarily long, constantrate linear codes over a fixed field F q such that a codeword is efficiently, nonuniquely reconstructible after a majority of its letters have been arbitrarily corrupted. We also construct codes such that a codeword is uniquely and efficiently reconstructible after a majority of its letters have been corrupted by noise which is random in a specified sense. We summarize our results in terms of bounds on asymptotic parameters, giving a new characterization of decoding beyond the errorcorrection bound. 1 Introduction Errorcorrecting codes, originally designed to accommodate reliable transmission of information through unreliable ...
Fast arithmetic on Jacobians of Picard curves
 PUBLIC KEY CRYPTOGRAPHY  PKC 2004, VOLUME 2947 OF LNCS
, 2004
"... In this paper we present a fast addition algorithm in the Jacobian of a Picard curve over a finite field Fq of characteristic different from 3. This algorithm has a nice geometric interpretation, comparable to the classic "chord and tangent" law for the elliptic curves. Computational cost for addit ..."
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Cited by 12 (4 self)
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In this paper we present a fast addition algorithm in the Jacobian of a Picard curve over a finite field Fq of characteristic different from 3. This algorithm has a nice geometric interpretation, comparable to the classic "chord and tangent" law for the elliptic curves. Computational cost for addition is 144M + 12SQ + 2I and 158M + 16SQ + 2I for doubling.
Asymptotically fast group operations on Jacobians of general curves
 Mathematics of Computation
, 2007
"... Abstract. Let C be a curve of genus g over a field k. We describe probabilistic algorithms for addition and inversion of the classes of rational divisors in the Jacobian of C. After a precomputation, which is done only once for the curve C, the algorithms use only linear algebra in vector spaces of ..."
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Cited by 11 (1 self)
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Abstract. Let C be a curve of genus g over a field k. We describe probabilistic algorithms for addition and inversion of the classes of rational divisors in the Jacobian of C. After a precomputation, which is done only once for the curve C, the algorithms use only linear algebra in vector spaces of dimension at most O(g log g), and so take O(g 3+ɛ) field operations in k, using Gaussian elimination. Using fast algorithms for the linear algebra, one can improve this time to O(g 2.376). This represents a significant improvement over the previous record of O(g 4) field operations (also after a precomputation) for general curves of genus g. 1.
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 10 (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