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34
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.
Faster Point Multiplication on Elliptic Curves with Efficient Endomorphisms
, 2001
"... The fundamental operation in elliptic curve cryptographic schemes is that of point multiplication of an elliptic curve point by an integer. This paper describes a new method for accelerating this operation on classes of elliptic curves that have efficientlycomputable endomorphisms. One advantage of ..."
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Cited by 68 (0 self)
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The fundamental operation in elliptic curve cryptographic schemes is that of point multiplication of an elliptic curve point by an integer. This paper describes a new method for accelerating this operation on classes of elliptic curves that have efficientlycomputable endomorphisms. One advantage of the new method is that it is applicable to a larger class of curves than previous such methods.
Mazur’s conjecture on higher Heegner points
 Invent. Math
"... In this article, we establish a nontriviality statement for Heegner points which was conjectured by B. Mazur [10], and has subsequently been used as a working hypothesis by a few authors in the study of the arithmetic of elliptic curves. ..."
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Cited by 37 (4 self)
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In this article, we establish a nontriviality statement for Heegner points which was conjectured by B. Mazur [10], and has subsequently been used as a working hypothesis by a few authors in the study of the arithmetic of elliptic curves.
Building curves with arbitrary small MOV degree over finite prime fields
 J. Cryptology
, 2002
"... We present a fast algorithm for building ordinary elliptic curves over finite prime fields having arbitrary small MOV degree. The elliptic curves are obtained using complex multiplication by any desired discriminant. ..."
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Cited by 31 (2 self)
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We present a fast algorithm for building ordinary elliptic curves over finite prime fields having arbitrary small MOV degree. The elliptic curves are obtained using complex multiplication by any desired discriminant.
Asymmetric multiple description lattice vector quantizers
 IEEE Trans. Inf. Theory
, 2002
"... Abstract—We consider the design of asymmetric multiple description lattice quantizers that cover the entire spectrum of the distortion profile, ranging from symmetric or balanced to successively refinable. We present a solution to a labeling problem, which is an important part of the construction, a ..."
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Cited by 24 (2 self)
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Abstract—We consider the design of asymmetric multiple description lattice quantizers that cover the entire spectrum of the distortion profile, ranging from symmetric or balanced to successively refinable. We present a solution to a labeling problem, which is an important part of the construction, along with a general design procedure. The highrate asymptotic performance of the quantizer is also studied. We evaluate the ratedistortion performance of the quantizer and compare it to known informationtheoretic bounds. The highrate asymptotic analysis is compared to the performance of the quantizer. Index Terms—Cubic lattice, highrate quantization, lattice quantization, multiple descriptions, quantization, source coding, successive refinement, vector quantization. I.
Building Cyclic Elliptic Curves Modulo Large Primes
 Advances in Cryptology  EUROCRYPT '91, Lecture Notes in Computer Science
, 1987
"... Elliptic curves play an important role in many areas of modern cryptology such as integer factorization and primality proving. Moreover, they can be used in cryptosystems based on discrete logarithms for building oneway permutations. For the latter purpose, it is required to have cyclic elliptic cu ..."
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Cited by 18 (2 self)
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Elliptic curves play an important role in many areas of modern cryptology such as integer factorization and primality proving. Moreover, they can be used in cryptosystems based on discrete logarithms for building oneway permutations. For the latter purpose, it is required to have cyclic elliptic curves over finite fields. The aim of this note is to explain how to construct such curves over a finite field of large prime cardinality, using the ECPP primality proving test of Atkin and Morain. 1 Introduction Elliptic curves prove to be a powerful tool in modern cryptology. Following the original work of H. W. Lenstra, Jr. [18] concerning integer factorization, many researchers have used this new idea to work out primality proving algorithms [8, 14, 2, 4, 22] as well as cryptosystems [21, 16] generalizing those of [12, 1, 9]. Recent work on these topics can be found in [20, 19]. More recently, Kaliski [15] has used elliptic curves in the design of oneway permutations. For this, the autho...
On the singular values of Weber modular functions
 Department of Mathematics National University of Singapore Kent Ridge, Singapore
, 1997
"... Abstract. The minimal polynomials of the singular values of the classical Weber modular functions give far simpler defining polynomials for the class fields of imaginary quadratic fields than the minimal polynomials of singular moduli of level 1. We describe computations of these polynomials and giv ..."
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Cited by 11 (1 self)
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Abstract. The minimal polynomials of the singular values of the classical Weber modular functions give far simpler defining polynomials for the class fields of imaginary quadratic fields than the minimal polynomials of singular moduli of level 1. We describe computations of these polynomials and give conjectural formulas describing the prime decomposition of their resultants and discriminants, extending the formulas of GrossZagier for the level 1 case.
Efficient Undeniable Signature Schemes Based on Ideal Arithmetic in Quadratic Orders
 Ideal Arithmetic in Quadratic Orders, Conference on The Mathematics of PublicKey Cryptography
, 1999
"... this paper we present new undeniable signature schemes which are constructed over an imaginary quadratic field. The basic scheme contains zeroknowledge confirmation and disavowal protocols which require operations of cubic bit complexity by the signer. In case one omits the part of the protocols wh ..."
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Cited by 10 (3 self)
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this paper we present new undeniable signature schemes which are constructed over an imaginary quadratic field. The basic scheme contains zeroknowledge confirmation and disavowal protocols which require operations of cubic bit complexity by the signer. In case one omits the part of the protocols which is costly the confirmation and disavowal protocol are not zeroknowledge but honestverifier zeroknowledge; the remaining operations for the signer have quadratic bit complexity. Additionally, the information which can be learned by a dishonest verifier can be characterized but will not be helpful to fake new signatures. Even tracing the operations done in this part leaks no information. In our basic scheme, the secret key of the signer is not needed to perform the additional operations for the zeroknowledge property; one can delegate this part to be performed by a certified software running on a terminal or PC to which the chip card is connected. Tracing the computations done by the certified software is allowed. One only has to be guaranteed that the results computed by this program are not manipulated. So, either in the basic protocol or in applications in which one knows the verifier to be trustworthy the tasks of the signer using the secret information can be performed in quadratic bit complexity, e.g. on a smart card. Buchmann and Williams proposed the first algorithm which achieves the DiffieHellman key distribution scheme using the class group in an imaginary quadratic field [5]. Later, Hafner and McCurley discovered the subexponential algorithm against the discrete logarithm problem of the class group [20]. Since then, cryptosystems over class groups have not gained much attention in practice. Recently, Huhnlein et. al. proposed an ElGamaltype public key crypt...
Computing the cardinality of CM elliptic curves using torsion points
, 2008
"... Let E be an elliptic curve having complex multiplication by a given quadratic order of an imaginary quadratic field K. The field of definition of E is the ring class field Ω of the order. If the prime p splits completely in Ω, then we can reduce E modulo one the factors of p and get a curve E define ..."
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Cited by 10 (1 self)
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Let E be an elliptic curve having complex multiplication by a given quadratic order of an imaginary quadratic field K. The field of definition of E is the ring class field Ω of the order. If the prime p splits completely in Ω, then we can reduce E modulo one the factors of p and get a curve E defined over Fp. The trace of the Frobenius of E is known up to sign and we need a fast way to find this sign. For this, we propose to use the action of the Frobenius on torsion points of small order built with class invariants à la Weber, in a manner reminiscent of the SchoofElkiesAtkin algorithm for computing the cardinality of a given elliptic curve modulo p. We apply our results to the Elliptic Curve Primality Proving algorithm (ECPP).