Nowadays, elliptic curve cryptosystems receive attention and much effort is being dedicated to make it more and more practical. It is worthwhile to construct discrete logarithm based cryptosystems using more general algebraic curves, because it supplies more security sources for public key cryptosystems. The presented paper introduces C ab curves. Roughly speaking, a curve is C ab if it is non-singular in its affine part and if its singularity at infinity is "nice". C ab curves compose a large family of algebraic curves, including elliptic, hyperelliptic and superelliptic curves. The paper shows an addition algorithm in Jacobian group of C ab curves in three steps: firstly with a geometrical point of view, which is impractical, secondly by translating the algorithm in the language of ideals, and finally, the final algorithm in which some costly steps are removed. The paper also gives experiments that prove that the algorithm behaves well in practice.
Abstract. The paper shows that some of elliptic curves over finite fields of characteristic three of composite degree are attacked by a more effective algorithm than Pollard's ρ method. For such an elliptic curve E, we construct a C ab curve D on its Weil restriction in order to reduce the discrete logarithm problem on E to that on D. And we show that the genus of D is small enough so that D is attacked by a modified form of Gaudry's variant for a suitable E. We also see such a weak elliptic curve is easily constructed.
This paper gives an efficient algorithm to compute addition in Jacobian of C 34 curves. The paper modifies the addition algorithm of [1], by classifying the forms of Groebner bases of all ideals involved in the addition in Jacobian, and by computing Groebner bases of ideals without using Buchberger algorithm. The algorithm computes the addition in Jacobian of C 34 curves in about 3 times amount of computation of the one in elliptic curves, when the sizes of groups are set to be the same.
Abstract. Gaudry has described a new algorithm (Gaudry's variant) for the discrete logarithm problem (DLP) in hyperelliptic curves. For hyperelliptic curves of small genus on finite field GF(q), Gaudry's variant solves for the DLP in O(q 2 log γ (q)) time. This paper shows that C ab curves can be attacked with a modified form of Gaudry's variant and presents the timing results of such attack. However, Gaudry's variant cannot be effective in all of the C ab curve cryptosystems, this paper provides an example of a C ab curve that is unassailable by Gaudry's variant.
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