On Ordered Mordell Elliptic Curves and Their Applications in Cryptography

Abstract

In this thesis, we consider ordered elliptic curves for cryptographic applications. The elliptic curves were appeared, when Diophantine Equations were being studied by the Greek mathematicians. The Diophantine Equation is in fact a polynomial equation, where someone is interested in integer or rational solutions. Algebraic number theory and algebraic varieties are two approaches used for the solutions of the polynomial equations. There are many kinds of polynomial equations such as, linear equations and equations of degree 2, which are not difficult to understand. The elliptic curves come into play, when the degree of polynomial equation is 3. So that a non-singular cubic curve E over a field K with a given point O is known as an elliptic curve, where the point O lies on each vertical line. The set of all points on an elliptic curve forms an abelian group. The points can have coordinates from any field such as real, complex, rational or prime field. However from application point of view elliptic curves over finite fields are preferred. Any point of finite order is called a torsion point. Moreover the point with order n is known as n-torsion point. Thus every point of an elliptic curve is a torsion point if and only if the elliptic curve is defined over a finite field. In general it is rather difficult to compute all points on an elliptic curve, however a theorem is developed by Hasse to compute the bounds on the number of points on any elliptic curve. In everyday life data is transferred electronically from one place to another. If the data is confidential then it is necessary to share the information safely and secretly. Cryptography plays a key role to protect the information form intruders. That is cryptography is in fact the science of hiding information from unauthorized users. The readable form of data is called plaintext. The process of hiding the information is known as encryption, whereas the encrypted data is said to be cipher-text. There are two main kinds of the cryptography namely symmetric key cryptography and asymmetric key cryptography. The main difference between the both types is iv that the symmetric key cryptosystem involves the same key for both sender and receiver, while in asymmetric key cryptography this is not the case. Symmetric key cryptography is also known as secret key cryptography. It is further categorized into stream ciphers and block ciphers. In stream ciphers a single bit is operated at a time, while in block ciphers blocks of data are used for encryption purpose. For further details about elliptic curves and cryptography, the readers are referred to [1, 2]. Many mathematician used elliptic curves for the solution of many problems. For example, in 1987 [3] Lenstra designed a new method for the factorization of positive integers. Basically Lenstra replaced the multiplication group in Pollard’s method by the group of the points on an elliptic curve. In 1995 [4] Wiles used elliptic curves to prove the Fermat’s Last Theorem. In 2015 [5] Star solved the congruent number problem using the ranks of elliptic curves. Miller and Neal Koblitz [6] proposed elliptic curve based cryptography which provides more security than other cryptosystems. In 2019 Hayat and Azam [7] developed a novel technique based on elliptic curves for the encryption of images. Researchers are still using elliptic curves for cryptographic and other applications. From literature review it follows that ordered elliptic curves are not yet utilized for information security. So this fact motivates us to use elliptic curves by defining new mathematical structures on their points for information security. We focus on a special kind of elliptic curves, that is Mordell elliptic curves to accomplish the following objectives. (i) To define new mathematical structures on elliptic curves that provide confusion and diffusion. (ii) To utilize the existing structures on elliptic curves using the new structures. (iii) Implementation of new and existing structures on elliptic curves. v (iv) Next to develop new schemes using above new structures for the construction of non-linear components of cryptosystems. (v) To count and statistically analyze the developed non-linear components on elliptic curves with new mathematical structures. (vi) To Design schemes based on the new structures for the generation of random numbers. (vii) To employ the designed non-linear components and random numbers for image encryption to measure their effectiveness. (viii) To establish the mathematical results associated with the proposed non-linear components and random numbers. Here we give a brief overview of the thesis: In Chapter 1, basic concepts and notions are discussed. In Chapter 2, a new method is proposed for the construction of substitution boxes based on Mordell elliptic curves. The proposed scheme is developed in such a way that for each input it outputs a substitution box in linear time and constant space. Due to this property, our method takes less time and space than the existing substitution box construction techniques over elliptic curves. Computational results show that the proposed method is capable of generating cryptographically strong substitution boxes with security comparable to some of the existing substitution boxes constructed via different mathematical structures. In Chapter 3, an efficient method based on ordered isomorphic elliptic curves for the generation of a large number of distinct, mutually uncorrelated and cryptographically strong injective substitution boxes is presented. The proposed scheme is characterized in terms of time complexity and the number of the distinct substitution boxes. Furthermore, rigorous analysis and comparison of the newly developed method with some of the existing methods are conducted. Experimental results reveal that the newly developed scheme can efficiently generate a large vi number of distinct, uncorrelated and secure substitution boxes when compared with some of the well-known existing schemes. In Chapter 4, secure generators of substitution boxes and pseudo random numbers are presented, which are essential for many well-known cryptosystems. These generators are based on a special class of ordered Mordell elliptic curves. The security strength of the proposed generators is tested via different tests. For a given prime, the experimental results reveal that the proposed generators are capable of generating a large number of distinct, cryptographically strong substitution boxes and sequences of random numbers in low time and space complexity. In Chapter 5, we propose a novel, fast and secure image encryption scheme based on Mordell elliptic curves. In this scheme, the receiver and sender agree on a public Mordell elliptic curve for data transmission which they generate by a simple search method instead of using complex arithmetic operations. Then the scheme performs pixel-masking and pixel-scrambling procedures by using random numbers and a dynamic substitution box to generate highly secure cipher-text. The random numbers and a substitution box are generated over Mordell elliptic curves that are isomorphic to the public Mordell elliptic curve such that for a fixed public Mordell elliptic curve the complexity of our scheme is essentially proportional to the size of the plain-text. In other words, the complexity of our scheme is independent of the point computation over Mordell elliptic curves. The random numbers and substitution boxes generated in our scheme are highly sensitive to the plain-text and hence our scheme is highly secure. We tested the security strength of our scheme against modern attacks including differential attacks, statistical attacks and key attacks by encrypting all standard images in USC-SIPI image database, and concluded that our scheme is fast and has high security. In Chapter 6, an image encryption scheme based on quasi-resonant Rossby/drift wave triads (related to elliptic surfaces) and Mordell elliptic curves is proposed. By defining a total order vii on quasi-resonant triads, at a first stage we construct quasi-resonant triads using auxiliary parameters of elliptic surfaces in order to generate pseudo random numbers. At a second stage, we employ a Mordell elliptic curve to construct a dynamic substitution box for the plain-image. The generated pseudo random numbers and substitution box are used respectively to provide diffusion and confusion in the tested image. We test the proposed scheme against well-known attacks by encrypting all gray images taken from the USC-SIPI image database. Our experimental results indicate the high security of the newly developed scheme. Finally, via extensive comparisons we show that the new scheme outperforms other popular schemes. In Chapter 7, the findings and future directions are discussed. The references are included at the end of the thesis.