It is proved that the rank of an elliptic curve one less arithmetic complexity corresponding non-commutative torus. As illustration, we consider a family curves with complex multiplication.

We show that the elliptic curve cryptosystems based on the Montgomery-form E : BY 2 = X+AX+X are immune to the timingattacks by using our technique of randomized projective coordinates, while Montgomery originally introduced this type of curves for speeding up the Pollard and Elliptic Curve Methods of integer factorization [Math. Comp. Vol.48, No.177, (1987) pp.243-264]. However, it should be n...

In this paper we look at three families of elliptic curves with rational 3-torsion over a finite field. These families include Hessian curves, twisted Hessian curves, and a new family we call generalized DIK curves. We find the number of Fq-isogeny classes of each family, as well as the number of Fq-isomorphism classes of the generalized DIK curves. We also include some formulas for efficient c...

Edwards curves are a particular form of elliptic curves that admit a fast, unified and complete addition law. Relations between Edwards curves and Montgomery curves have already been described. Our work takes the view of parameterizing elliptic curves given by their j-invariant, a problematic that arises from using curves with complex multiplication, for instance. We add to the catalogue the li...

To every non-singular elliptic curve (complex torus) we assign a C∗algebra Tθ = {u, v | vu = e uv} known as noncommutative torus. It is shown that morphisms of elliptic curves generate Morita equivalence of the corresponding noncommutative tori. Real number θ we call projective curvature attached to the elliptic curve. It is proved that projective curvatures of isomorphic elliptic curves are mo...

Signcryption is a new cryptographic approach which provides authentication and encryption simultaneously in a single logical step. The aim is to reduce the cost of signature-then-encryption approach. This cost includes computational cost and communication cost. Furthermore some signcryption schemes are based on RSA while some are based on elliptic curve. This paper provides a critical review of...

We collect various known results (about plane curves and the moduli space of stable maps) to derive new recursive formulas enumerating low genus plane curves of any degree with various behaviors. Recursive formulas are given for the characteristic numbers of rational plane curves, elliptic plane curves, and elliptic plane curves with fixed complex structure. Recursions are also given for the nu...

We study genus 2 covers of relative elliptic curves over an arbitrary base in which 2 is invertible. Particular emphasis lies on the case that the covering degree is 2. We show that the data in the " basic construction " of genus 2 covers of relative elliptic curves determine the cover in a unique way (up to isomorphism). A classical theorem says that a genus 2 cover of an elliptic curve of deg...

The GHS attack is known to solve discrete logarithm problems (DLP) in the Jacobian of a curve C0 defined over the d degree extension field kd of k := Fq by mapping it to the DLP in the Jacobian of a covering curve C of C0 over k. Recently, classifications for all elliptic curves and hyperelliptic curves C0/kd of genus 2,3 which possess (2, ..., 2)-covering C/k of P were shown under an isogeny c...