A Survey Report On Elliptic Curve Cryptography

The paper presents an extensive and careful study of elliptic curve cryptography (ECC) and its applications. This paper also discuss the arithmetic involved in elliptic curve  and how these curve operations is crucial in determining the performance of cryptographic systems. It also presents  different forms of elliptic curve in various coordinate system , specifying which is most widely used and why. It also explains how isogenenies between elliptic curve  provides the secure ECC. Exentended form of elliptic curve i.e hyperelliptic curve has been presented here with its pros and cons. Performance of ECC and HEC is also discussed based on scalar multiplication and DLP. Keywords: Elliptic curve cryptography (ECC), isogenies, hyperelliptic curve (HEC) , Discrete Logarithm Problem (DLP), Integer  Factorization , Binary Field, Prime FieldDOI:http://dx.doi.org/10.11591/ijece.v1i2.8

Bit Security of the Hyperelliptic Curves Diffie-Hellman Problem

The Diffie-Hellman problem as a cryptographic primitive plays an important role in modern cryptology. The Bit Security or Hard-Core Bits of Diffie-Hellman problem in arbitrary finite cyclic group is a long-standing open problem in cryptography. Until now, only few groups have been studied. Hyperelliptic curve cryptography is an alternative to elliptic curve cryptography. Due to the recent cryptanalytic results that the best known algorithms to attack hyperelliptic curve cryptosystems of genus $g<3$ are the generic methods and the recent implementation results that hyperelliptic curve cryptography in genus 2 has the potential to be competitive with its elliptic curve cryptography counterpart. In this paper, we generalize Boneh and Shparlinksi\u27s method and result about elliptic curve to the case of Jacobians of hyperelliptic curves. We prove that the least significant bit of each coordinate of hyperelliptic curves Diffie-Hellman secret value in genus 2 is hard as the entire Diffie-Hellman value, and then we also show that any bit is hard as the entire Diffie-Hellman value. Finally, we extend our techniques and results to hyperelliptic curves of any genus

Group law computations on Jacobians of hyperelliptic curves

We derive an explicit method of computing the composition step in Cantor’s algorithm for group operations on Jacobians of hyperelliptic curves. Our technique is inspired by the geometric description of the group law and applies to hyperelliptic curves of arbitrary genus. While Cantor’s general composition involves arithmetic in the polynomial ring F_q[x], the algorithm we propose solves a linear system over the base field which can be written down directly from the Mumford coordinates of the group elements. We apply this method to give more efficient formulas for group operations in both affine and projective coordinates for cryptographic systems based on Jacobians of genus 2 hyperelliptic curves in general form

Discrete logarithms in curves over finite fields

A survey on algorithms for computing discrete logarithms in Jacobians of curves over finite fields