Hyperelliptic curves

Algebraic Geometry

Published in Paul L. Goethals, Natalie M. Scala, Daniel T. Bennett, Mathematics in Cyber Research, 2022

The field of elliptic curve cryptography is a well-established field, so as we look to push the boundary of knowledge on this subject, we look to a field of mathematics that is very similar, hyperelliptic curve cryptography. Elliptic curves are known as genus one curves. Curves of higher order are higher genus curves. Hyperelliptic curves of genus two and three are known to be good for cryptography purposes, while there are known methods of attacks for curves of higher genus. Hence, for cryptographic purposes, we are interested in studying only curves of genus 2 and 3.

Hyperelliptic Curve Diffie–Hellman-Based Two-Server Password-Only Authenticated Key Exchange Protocol for Edge Computing Systems

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Journal Information

Published in IETE Journal of Research, 2021

K. Anitha Kumari, T. P. Kamatchi, R. Senthil Prabha, Bharath K. Samanthula

Group law on a Jacobian for a hyperelliptic curve was proposed by Koblitz in 1989 [27]. It is based on discrete logarithm problem and defined by Jacobian of genus g. It tends to possess qg points, where q denotes the number of elements of Jacobian. The HECC encryption scheme is unconditionally secure under Decisional Diffie–Hellman assumption. Hyperelliptic curves are special class of algebraic curves that are viewed as generalization of elliptic curves. A hyperelliptic curve of genus g = 1 is an elliptic curve. Hyperelliptic curve is given by, where h(x) is a polynomial of degree ≤ g over a finite field F, f(x) is a monic polynomial of degree 2g + 1 over F

Hyperelliptic curves

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Algebraic Geometry

Hyperelliptic Curve Diffie–Hellman-Based Two-Server Password-Only Authenticated Key Exchange Protocol for Edge Computing Systems