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13
On the construction of prime order elliptic curves
 Progress in cryptology—INDOCRYPT 2003, Springer Lecture Notes in Computer Science
"... Abstract. We consider a variant of the Complex Multiplication (CM) method for constructing elliptic curves (ECs) of prime order with additional security properties. Our variant uses Weber polynomials whose discriminant D is congruent to 3 (mod 8), and is based on a new transformation for converting ..."
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Abstract. We consider a variant of the Complex Multiplication (CM) method for constructing elliptic curves (ECs) of prime order with additional security properties. Our variant uses Weber polynomials whose discriminant D is congruent to 3 (mod 8), and is based on a new transformation for converting roots of Weber polynomials to their Hilbert counterparts. We also present a new theoretical estimate of the bit precision required for the construction of the Weber polynomials for these values of D. We conduct a comparative experimental study investigating the time and bit precision of using Weber polynomials against the (typical) use of Hilbert polynomials. We further investigate the time efficiency of the new CM variant under four different implementations of a crucial step of the variant and demonstrate the superiority of two of them. 1
Generating Prime Order Elliptic Curves: Difficulties and Efficiency
 Considerations, in International Conference on Information Security and Cryptology – ICISC 2004, Lecture Notes in Computer Science
, 2005
"... Abstract. We consider the generation of prime order elliptic curves (ECs) over a prime field Fp using the Complex Multiplication (CM) method. A crucial step of this method is to compute the roots of a special type of class field polynomials with the most commonly used being the Hilbert and Weber one ..."
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Cited by 4 (4 self)
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Abstract. We consider the generation of prime order elliptic curves (ECs) over a prime field Fp using the Complex Multiplication (CM) method. A crucial step of this method is to compute the roots of a special type of class field polynomials with the most commonly used being the Hilbert and Weber ones, uniquely determined by the CM discriminant D. In attempting to construct prime order ECs using Weber polynomials two difficulties arise (in addition to the necessary transformations of the roots of such polynomials to those of their Hilbert counterparts). The first one is that the requirement of prime order necessitates that D ≡ 3 (mod 8), which gives Weber polynomials with degree three times larger than the degree of their corresponding Hilbert polynomials (a fact that could affect efficiency). The second difficulty is that these Weber polynomials do not have roots in Fp. In this paper we show how to overcome the above difficulties and provide efficient methods for generating ECs of prime order supported by a thorough experimental study. In particular,
On the Use of Weber Polynomials in Elliptic Curve Cryptography
 In Proc. European PKI Workshop 2004, LNCS 3093
, 2004
"... Abstract. In many cryptographic applications it is necessary to generate elliptic curves (ECs) with certain security properties. These curves are commonly constructed using the Complex Multiplication method which typically uses the roots of Hilbert or Weber polynomials. The former generate the EC di ..."
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Cited by 2 (1 self)
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Abstract. In many cryptographic applications it is necessary to generate elliptic curves (ECs) with certain security properties. These curves are commonly constructed using the Complex Multiplication method which typically uses the roots of Hilbert or Weber polynomials. The former generate the EC directly, but have high computational demands, while the latter are faster to construct but they do not lead, directly, to the desired EC. In this paper we present in a simple and unifying manner a complete set of transformations of the roots of a Weber polynomial to the roots of its corresponding Hilbert polynomial for all discriminant values on which they are defined. Moreover, we prove a theoretical estimate of the precision required for the computation of Weber polynomials. Finally, we experimentally assess the computational efficiency of the Weber polynomials along with their precision requirements for various discriminant values and compare the results with the theoretical estimates. Our experimental results may be used as a guide for the selection of the most efficient curves in applications residing in resource limited devices such as smart cards that support secure and efficient Public Key Infrastructure (PKI) services. 1
How to Compute the Coefficients of the Elliptic Modular Function j(z)
, 2003
"... We discuss various methods to compute the Fourier coefficients of the elliptic modular function j(z). We present run times to compute the coefficients in practice. If possible we discuss... ..."
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Cited by 1 (0 self)
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We discuss various methods to compute the Fourier coefficients of the elliptic modular function j(z). We present run times to compute the coefficients in practice. If possible we discuss...
On the Efficient Generation of PrimeOrder Elliptic Curves∗
, 2009
"... Abstract. We consider the generation of primeorder elliptic curves (ECs) over a prime field Fp using the Complex Multiplication (CM) method. A crucial step of this method is to compute the roots of a special type of class field polynomials with the most commonly used being the Hilbert and Weber one ..."
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Abstract. We consider the generation of primeorder elliptic curves (ECs) over a prime field Fp using the Complex Multiplication (CM) method. A crucial step of this method is to compute the roots of a special type of class field polynomials with the most commonly used being the Hilbert and Weber ones. These polynomials are uniquely determined by the CM discriminant D. In this paper, we consider a variant of the CM method for constructing elliptic curves (ECs) of prime order using Weber polynomials. In attempting to construct primeorder ECs using Weber polynomials, two difficulties arise (in addition to the necessary transformations of the roots of such polynomials to those of their Hilbert counterparts). The first one is that the requirement of prime order necessitates that D ≡ 3 (mod 8), which gives Weber polynomials with degree ∗ This work was partially supported by the IST Programme of EU under contracts no. IST200133116 (FLAGS), and by the Action IRAKLITOS (Fellowships for Research in the University of Patras) with match
Introducing Ramanujan’s Class Polynomials in the Generation of Prime Order Elliptic Curves
, 804
"... Complex Multiplication (CM) method is a frequently used method for the generation of prime order elliptic curves (ECs) over a prime field Fp. The most demanding and complex step of this method is the computation of the roots of a special type of class polynomials, called Hilbert polynomials. These p ..."
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Complex Multiplication (CM) method is a frequently used method for the generation of prime order elliptic curves (ECs) over a prime field Fp. The most demanding and complex step of this method is the computation of the roots of a special type of class polynomials, called Hilbert polynomials. These polynonials are uniquely determined by the CM discriminant D. The disadvantage of these polynomials is that they have huge coefficients and thus they need high precision arithmetic for their construction. Alternatively, Weber polynomials can be used in the CM method. These polynomials have much smaller coefficients and their roots can be easily transformed to the roots of the corresponding Hilbert polynomials. However, in the case of prime order elliptic curves, the degree of Weber polynomials is three times larger than the degree of the corresponding Hilbert polynomials and for this reason the calculation of their roots involves computations in the extension field F p 3. Recently, two other classes of polynomials, denoted by MD,l(x) and MD,p1,p2(x) respectively, were introduced which can also be used in the generation of prime order elliptic curves. The advantage of these polynomials is that their degree is equal to the degree of the Hilbert polynomials and thus computations over the extension field can be avoided. In this paper, we propose the use of a new class of polynomials. We will call them Ramanujan polynomials named after Srinivasa Ramanujan who was the first to compute them for few values of D. We explicitly describe the algorithm for the construction of the new polynomials, show that their degree is equal to the degree of the corresponding Hilbert polynomials and give the necessary transformation of their roots (to the roots of the corresponding Hilbert polynomials). Moreover, we compare (theoretically and experimentally) the efficiency of using this new class against the use of the aforementioned Weber, MD,l(x) and MD,p1,p2(x) polynomials and show that they clearly outweigh all of them in the generation of prime order elliptic curves.
Constructing Twisted Anomalous Elliptic Curves
"... Abstract—Huhnlein et al showed that for totally nonaximal imaginary quadratic orders, the discrete logarithm problem can be reduced to the discrete logarithm problem in some finite fields. In previous work we showed that for twisted anomalous elliptic curves, the logarithm problem can also be reduc ..."
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Abstract—Huhnlein et al showed that for totally nonaximal imaginary quadratic orders, the discrete logarithm problem can be reduced to the discrete logarithm problem in some finite fields. In previous work we showed that for twisted anomalous elliptic curves, the logarithm problem can also be reduced to the logarithm problem in some finite fields. In this work we present an algorithm to construct this class of elliptic curves. Index Terms—discrete logarithm problem, imaginary quadratic order, elliptic curve, anomalous curve I I.
REGULAR CONTRIBUTION Efficient generation of secure elliptic curves
, 2006
"... Abstract In many cryptographic applications it is necessary to generate elliptic curves (ECs) whose order possesses certain properties. The method that is usually employed for the generation of such ECs is the socalled Complex Multiplication method. This method requires the use of the roots of cert ..."
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Abstract In many cryptographic applications it is necessary to generate elliptic curves (ECs) whose order possesses certain properties. The method that is usually employed for the generation of such ECs is the socalled Complex Multiplication method. This method requires the use of the roots of certain class field polynomials defined on a specific parameter called the discriminant. The most commonly used polynomials are the Hilbert and Weber ones. The former can be used to generate directly the EC, but they are characterized by high computational demands. The latter have usually much lower computational requirements, but they do not directly construct the desired EC. This can be achieved if transformations of their roots to the roots of This work was partially supported by the IST Programme of EC under contract no. IST200133116 (FLAGS), and by the Action IRAKLITOS (Fellowships for Research in the University of Patras) with matching funds from ESF (European Social Fund)
J. Cryptol. DOI: 10.1007/s0014500990372 On the Efficient Generation of PrimeOrder Elliptic Curves ∗
, 2008
"... Abstract. We consider the generation of primeorder elliptic curves (ECs) over a prime field Fp using the Complex Multiplication (CM) method. A crucial step of this method is to compute the roots of a special type of class field polynomials with the most commonly used being the Hilbert and Weber one ..."
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Abstract. We consider the generation of primeorder elliptic curves (ECs) over a prime field Fp using the Complex Multiplication (CM) method. A crucial step of this method is to compute the roots of a special type of class field polynomials with the most commonly used being the Hilbert and Weber ones. These polynomials are uniquely determined by the CM discriminant D. In this paper, we consider a variant of the CM method for constructing elliptic curves (ECs) of prime order using Weber polynomials. In attempting to construct primeorder ECs using Weber polynomials, two difficulties arise (in addition to the necessary transformations of the roots of such polynomials to those of their Hilbert counterparts). The first one is that the requirement of prime order necessitates that D ≡ 3 (mod 8), which gives Weber polynomials with degree ∗ This work was partially supported by the IST Programme of EU under contracts no. IST200133116 (FLAGS), and by the Action IRAKLITOS (Fellowships for Research in the University of Patras) with matching
Index Term — Discrete logarithm problem
"... Abstract — Huhnlein et al showed that for totally nonaximal imaginary quadratic orders, the discrete logarithm problem can be reduced to the discrete logarithm problem in some finite fields. In previous work we showed that for twisted anomalous elliptic curves, the logarithm problem can also be re ..."
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Abstract — Huhnlein et al showed that for totally nonaximal imaginary quadratic orders, the discrete logarithm problem can be reduced to the discrete logarithm problem in some finite fields. In previous work we showed that for twisted anomalous elliptic curves, the logarithm problem can also be reduced to the logarithm problem in some finite fields. In this work we present an algorithm to construct this class of elliptic curves.