An application of hyperelliptic curves in cryptography

Abstract

Cryptography is the science of securing private information. Cryptographers uses di erent approaches to secure important and personal data. Di erent aspects of data protection strategies are being introduced by cryptographers in order to con- vert secret data into an unreadable format using keys. Shanon gave the idea that a cryptosystem is secure if it creates uncertainty in data. In cryptography special types of curves that contain a group's con guration are fundamental and very useful resources. Elliptic curves are thought to be one of the most secure structures to minimize the risk of computational attacks. Elliptic curve cryptography (ECC) is a highly secured asymmetric encryption technique that uses the underlying mathe- matical structures involved in elliptic curve geometry. In cryptography, Koblitz and Miller independently gave the main example of EC over nite elds. When com- pared to other public key cryptosystems, the ECC has the same level of complexity while using a smaller key space. Substitution boxes (S-boxes) are the important non-linear component for security of cryptosystem. S-boxes are capable of creating confusion in the data that makes cryptosystem highly secured against cryptana- lytic attacks. Therefore many researchers introduced their own methodoligies for the construction of S-boxes to create confusion in the data. In this thesis we propose an e cient S-box generation scheme based on hyperel- liptic curves (HEC) over a prime eld. The rst chapter contains fundamentals of cryptography, some basic de nitions of hyperelliptic curve (HEC), and a detailed de- scription to the elliptic curves cryptography. In the second chapter we review some literature that proposed an e cient method to generate S-boxes that are based on a class of Mordell elliptic curves (MEC) over nite elds. In the third chapter newly developed technique uses the y-coordinates of hyperlliptic curve is explained whereas, a total ordering is applied on the points of an hyperelliptic curve to di use the y-coordinates. The proposed scheme o ers high level of security and generates a large number of distinct cryptographically secure S-boxes. Furthermore, to show the e ciency of the proposed method, the suggested S-boxes' security is analyzed and comparison is made with some already existing S-boxes generated by di erent mathematical methods.