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Evaluating large degree isogenies and applications to pairing based cryptography
"... Abstract. We present a new method to evaluate large degree isogenies between elliptic curves over finite fields. Previous approaches all have exponential running time in the logarithm of the degree. If the endomorphism ring of the elliptic curve is ‘small ’ we can do much better, and we present an a ..."
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Abstract. We present a new method to evaluate large degree isogenies between elliptic curves over finite fields. Previous approaches all have exponential running time in the logarithm of the degree. If the endomorphism ring of the elliptic curve is ‘small ’ we can do much better, and we present an algorithm with a running time that is polynomial in the logarithm of the degree. We give several applications of our techniques to pairing based cryptography. 1
COMPUTATIONAL CLASS FIELD THEORY
, 802
"... Abstract. Class field theory furnishes an intrinsic description of the abelian extensions of a number field that is in many cases not of an immediate algorithmic nature. We outline the algorithms available for the explicit computation of such extensions. 1. ..."
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Abstract. Class field theory furnishes an intrinsic description of the abelian extensions of a number field that is in many cases not of an immediate algorithmic nature. We outline the algorithms available for the explicit computation of such extensions. 1.
pADIC CLASS INVARIANTS
"... We develop a new padic algorithm to compute the minimal polynomial of a class invariant. Our approach works for virtually any modular function yielding class invariants. The main algorithmic tool is modular polynomials, a concept which we generalize to functions of higher level. ..."
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We develop a new padic algorithm to compute the minimal polynomial of a class invariant. Our approach works for virtually any modular function yielding class invariants. The main algorithmic tool is modular polynomials, a concept which we generalize to functions of higher level.
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.
Contents lists available at ScienceDirect Computers and Mathematics with Applications
"... journal homepage: www.elsevier.com/locate/camwa On nonoscillation of mixed advanceddelay differential equations with ..."
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journal homepage: www.elsevier.com/locate/camwa On nonoscillation of mixed advanceddelay differential equations with
unknown title
, 2008
"... Solutions by radicals at singular values kN from new class invariants for N ≡ 3 mod 8 ..."
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Solutions by radicals at singular values kN from new class invariants for N ≡ 3 mod 8
unknown title
, 2008
"... Solutions by radicals at singular values kN from new class invariants for N ≡ 3 mod 8 ..."
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Solutions by radicals at singular values kN from new class invariants for N ≡ 3 mod 8
Finite Fields and Their Applications
"... This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal noncommercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or sel ..."
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This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal noncommercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier’s archiving and manuscript policies are encouraged to visit:
ELLIPTIC RECIPROCITY
"... Abstract. The paper introduces the notions of an elliptic pair, an elliptic cycle and an elliptic list over a square free positive integer d. These concepts are related to the notions of amicable pairs of primes and aliquot cycles that were introduced by Silverman and Stange in [5]. Settling a matte ..."
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Abstract. The paper introduces the notions of an elliptic pair, an elliptic cycle and an elliptic list over a square free positive integer d. These concepts are related to the notions of amicable pairs of primes and aliquot cycles that were introduced by Silverman and Stange in [5]. Settling a matter left open by Silverman and Stange it is shown that for d = 3 there are elliptic cycles of length 6. For d 6 = 3 the question of the existence of proper elliptic lists of length n over d is reduced to the the theory of prime producing quadratic polynomials. For d = 163 a proper elliptic list of length 40 is exhibited. It is shown that for each d there is an upper bound on the length of a proper elliptic list over d. The final section of the paper contains heuristic arguments supporting conjectured asymptotics for the number of elliptic pairs below integer X. Finally, for d ≡8 3 the existence of infinitely many anomalous prime numbers is derived from Bounyakowski’s Conjecture for quadratic polynomials. 1.
Aliquot Cycles for Elliptic Curves with Complex Multiplication
, 2013
"... Undergraduate Theses—Unrestricted by an authorized administrator of Washington University Open Scholarship. For more information, please ..."
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Undergraduate Theses—Unrestricted by an authorized administrator of Washington University Open Scholarship. For more information, please