Results 1 
6 of
6
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 ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
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
pADIC CLASS INVARIANTS
"... Abstract. 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. 1. ..."
Abstract
 Add to MetaCart
Abstract. 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. 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. ..."
Abstract
 Add to MetaCart
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.
unknown title
, 2008
"... Solutions by radicals at singular values kN from new class invariants for N ≡ 3 mod 8 ..."
Abstract
 Add to MetaCart
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 ..."
Abstract
 Add to MetaCart
Solutions by radicals at singular values kN from new class invariants for N ≡ 3 mod 8
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 ..."
Abstract
 Add to MetaCart
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.