Computing the topology of a plane or space hyperelliptic curve

The Osculating Plane of a Space Curve - Synthetic Formulations

Hyperelliptic Curve Cryptography

The use of elliptic-curve groups in cryptography, suggested by Miller [1] and Koblitz [2] three decades ago,provides the same level of security for the Discrete Logarithm Problem as multiplicative groups, with much smallerkey sizes and parameters. The idea was refined two years later by Koblitz, who worked with the group formed bythe points of the Jacobian of hyperelliptic curve...

Hyperelliptic Curve Coprocessors on a FPGA

Cryptographic algorithms are used in a large variety of different applications to ensure security services. It is, thus, very interesting to investigate various implementation platforms. Hyperelliptic curve schemes are cryptographic primitives to which a lot of attention was recently given due to the short operand size compared to other algorithms. They are specifically interesting for special-...

Elliptic and Hyperelliptic Curve Cryptography

Definition 1. Let k be a field. An algebraic variety over k is a k-scheme X such that there exists a covering by a finite number of affine open subschemes Xi which are affine varieties over k, i.e. each Xi is the affine scheme associated to a finitely generated algebra over k. A projective variety over k is a projective scheme over k, i.e. a k-scheme isomorphic to Proj k[T0, . . . , Tn]/I for a...

Complete Linear Series on a Hyperelliptic Curve

In this paper we study complete linear series on a hyperelliptic curve C of arithmetic genus g. Let A be the unique line bundle on C such that |A| is a g 2 , and let L be a line bundle on C of degree d. Then L can be factorized as L = A ⊗ B where m is the largest integer satisfying H(C,L⊗A) 6= 0. Let b = deg(B). We say that the factorization type of L is (m, b). Our main results in this paper a...