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97
IdentityBased Encryption from the Weil Pairing
, 2001
"... We propose a fully functional identitybased encryption scheme (IBE). The scheme has chosen ciphertext security in the random oracle model assuming an elliptic curve variant of the computational DiffieHellman problem. Our system is based on bilinear maps between groups. The Weil pairing on elliptic ..."
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Cited by 1123 (24 self)
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We propose a fully functional identitybased encryption scheme (IBE). The scheme has chosen ciphertext security in the random oracle model assuming an elliptic curve variant of the computational DiffieHellman problem. Our system is based on bilinear maps between groups. The Weil pairing on elliptic curves is an example of such a map. We give precise definitions for secure identity based encryption schemes and give several applications for such systems.
Short signatures from the Weil pairing
, 2001
"... Abstract. We introduce a short signature scheme based on the Computational DiffieHellman assumption on certain elliptic and hyperelliptic curves. The signature length is half the size of a DSA signature for a similar level of security. Our short signature scheme is designed for systems where signa ..."
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Cited by 560 (31 self)
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Abstract. We introduce a short signature scheme based on the Computational DiffieHellman assumption on certain elliptic and hyperelliptic curves. The signature length is half the size of a DSA signature for a similar level of security. Our short signature scheme is designed for systems where signatures are typed in by a human or signatures are sent over a lowbandwidth channel. 1
Elliptic Curves And Primality Proving
 Math. Comp
, 1993
"... The aim of this paper is to describe the theory and implementation of the Elliptic Curve Primality Proving algorithm. ..."
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Cited by 162 (22 self)
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The aim of this paper is to describe the theory and implementation of the Elliptic Curve Primality Proving algorithm.
Counting Points on Elliptic Curves Over Finite Fields
, 1995
"... . We describe three algorithms to count the number of points on an elliptic curve over a finite field. The first one is very practical when the finite field is not too large; it is based on Shanks's babystepgiantstep strategy. The second algorithm is very efficient when the endomorphism ring of ..."
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Cited by 82 (0 self)
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. We describe three algorithms to count the number of points on an elliptic curve over a finite field. The first one is very practical when the finite field is not too large; it is based on Shanks's babystepgiantstep strategy. The second algorithm is very efficient when the endomorphism ring of the curve is known. It exploits the natural lattice structure of this ring. The third algorithm is based on calculations with the torsion points of the elliptic curve [18]. This deterministic polynomial time algorithm was impractical in its original form. We discuss several practical improvements by Atkin and Elkies. 1. Introduction. Let p be a large prime and let E be an elliptic curve over F p given by a Weierstraß equation Y 2 = X 3 +AX +B for some A, B 2 F p . Since the curve is not singular we have that 4A 3 + 27B 2 6j 0 (mod p). We describe several methods to count the rational points on E, i.e., methods to determine the number of points (x; y) on E with x; y 2 F p . Most o...
Algorithms for computing isogenies between elliptic curves
 Math. Comp
, 2000
"... Abstract. The heart of the improvements by Elkies to Schoof’s algorithm for computing the cardinality of elliptic curves over a finite field is the ability to compute isogenies between curves. Elkies ’ approach is well suited for the case where the characteristic of the field is large. Couveignes sh ..."
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Cited by 31 (6 self)
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Abstract. The heart of the improvements by Elkies to Schoof’s algorithm for computing the cardinality of elliptic curves over a finite field is the ability to compute isogenies between curves. Elkies ’ approach is well suited for the case where the characteristic of the field is large. Couveignes showed how to compute isogenies in small characteristic. The aim of this paper is to describe the first successful implementation of Couveignes’s algorithm. In particular, we describe the use of fast algorithms for performing incremental operations on series. We also insist on the particular case of the characteristic 2. 1.
Short Programs for functions on Curves
 IBM THOMAS J. WATSON RESEARCH CENTER
, 1986
"... The problem of deducing a function on an algebraic curve having a given divisor is important in the field of indefinite integration. Indeed, it is the main computational step in determining whether an algebraic function posseses an indefinite integral. It has also become important recently in th ..."
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Cited by 29 (1 self)
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The problem of deducing a function on an algebraic curve having a given divisor is important in the field of indefinite integration. Indeed, it is the main computational step in determining whether an algebraic function posseses an indefinite integral. It has also become important recently in the study of discrete elliptic logarithms in cryptography, and in the construction of the new class of errorcorrecting codes which exceed the VarshamovGilbert bound. It can also be used to give a partial answer to a question raised by Schoof in his paper on computing the exact number of points on an elliptic curve over a finite field. Heretofore,
KMS states and complex multiplication
 the proceedings of the Abel Symposium
, 2005
"... The following problem in operator algebra has been open for several years. Problem 1.1. For some number field K (other than Q) exhibit an explicit quantum statistical mechanical system (A, σt) with the following properties: (1) The partition function Z(β) is the Dedekind zeta function of K. ..."
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Cited by 23 (6 self)
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The following problem in operator algebra has been open for several years. Problem 1.1. For some number field K (other than Q) exhibit an explicit quantum statistical mechanical system (A, σt) with the following properties: (1) The partition function Z(β) is the Dedekind zeta function of K.
Elliptic Curves with Complex Multiplication and the Conjecture of Birch and SwinnertonDyer
 Birch and SwinnertonDyer, Invent. Math
, 1981
"... A stronger version of (ii) (with no assumption that E have good reduction above p) was proved in [Ru2]. The program to prove (ii) was also begun by Coates and Wiles; it can ? Partially supported by the National Science Foundation. The author also gratefully acknowledges the CIME for its hospitality ..."
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Cited by 22 (0 self)
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A stronger version of (ii) (with no assumption that E have good reduction above p) was proved in [Ru2]. The program to prove (ii) was also begun by Coates and Wiles; it can ? Partially supported by the National Science Foundation. The author also gratefully acknowledges the CIME for its hospitality. ?? current address: Department of Mathematics, Stanford University, Stanford, CA 94305 USA, rubin@math.stanford.edu now be completed thanks to the recent Euler system machinery of Kolyvagin [Ko]. This proof will be given in x12, Corollary 12.13 and Theorem 12.19. The material through x4 is background which was not in the Cetraro lectures but is included here for completeness. In those sections we summarize, generally with references to [Si] instead of proofs, the basic properties of elliptic curves that will be needed later. For more details, including proofs, see Silverman's b
A CRT algorithm for constructing genus 2 curves over finite fields
, 2007
"... Abstract. — We present a new method for constructing genus 2 curves over a finite field Fn with a given number of points on its Jacobian. This method has important applications in cryptography, where groups of prime order are used as the basis for discretelog based cryptosystems. Our algorithm prov ..."
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Cited by 19 (7 self)
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Abstract. — We present a new method for constructing genus 2 curves over a finite field Fn with a given number of points on its Jacobian. This method has important applications in cryptography, where groups of prime order are used as the basis for discretelog based cryptosystems. Our algorithm provides an alternative to the traditional CM method for constructing genus 2 curves. For a quartic CM field K with primitive CM type, we compute the Igusa class polynomials modulo p for certain small primes p and then use the Chinese remainder theorem (CRT) and a bound on the denominators to construct the class polynomials. We also provide an algorithm for determining endomorphism rings of ordinary Jacobians of genus 2 curves over finite fields, generalizing the work of Kohel for elliptic curves. Résumé (Un algorithme fondé sur le théorème chinois pour construire des courbes de genre 2 sur des corps finis) Nous présentons une nouvelle méthode pour construire des courbes de genre 2 sur un corps fini Fn avec un nombre donné de points sur sa jacobienne. Cette méthode a des applications importantes en cryptographie, où des groupes d’ordre premier sont employés pour former des cryptosystèmes fondés sur le logarithme discret. Notre algorithme fournit une alternative à la méthode traditionnelle de multiplication complexe pour construire des courbes de genre 2. Pour un corps quartique K à multiplication complexe de type primitif, nous calculons les polynômes de classe d’Igusa modulo p pour certain petit premiers p et employons le théorème chinois et une borne sur les dénominateurs pour construire les polynômes de classe. Nous fournissons également un algorithme pour déterminer les anneaux d’endomorphismes des jacobiennes de courbes ordinaires de genre 2 sur des corps finis, généralisant le travail de Kohel pour les courbes elliptiques.