Image Cryptosystems Using Elliptic Curve Cryptography

Abstract

This thesis is about applications of elliptic curves in cryptography. Cryptography is a technique for protecting data and communications. It is a purposeful effort to diffuse and confuse data so that the attackers cannot obtain confidential information. More specifically, cryptosystems involve the creation, evaluation, and use of mathematical defenses against adversarial attacks. An elliptic curve is a curve of the form y 2 = f(x), where f(x) is a cubic polynomial. The elliptic curve appears for the first time in the work of Diophantus in his text Arithmetica. To gain a general understanding of how operations on an elliptic curve actually work, we first prescribe the characteristics of an elliptic curve over the real numbers. Any field, including the complex, real, rational, or prime, can be used to compute the coordinates of an elliptic curve’s points. However, elliptic curves over finite fields are desirable from an application perspective. The set of points on an elliptic curve over a finite field, together with a special point “O” (“point at infinity”) form an additive abelian group. The group operation is a point addition that can be executed using arithmetic operations in the underlying finite field. Hasse’s theorem [1] gives an approximate bound about the points of an elliptic curve. Before the 1970's, encryption required two parties to physically meet to establish a shared secret key for secure communication, which made the procedure fairly difficult and time-consuming. Symmetric ciphers, which served as the foundation for private cryptosystems, were designed with this in mind. Miller [2] introduced the concept of Elliptic Curve Cryptography (ECC) and presented an elliptic curve-based cryptosystem that is 20 times faster than the Diffie-Hellmans algorithm. A cryptosystem based on an elliptic curve over a finite field is introduced by Koblitz in [3]. ECC-based algorithms are computationally efficient and provide greater security. For example, 256-bit ECC over a prime field and 2048-bit Rivest-Shamir-Adleman algorithm provide the same level of security [4]. In addition, ECC requires less memory on digital computers [5, 6]. iii Such features make ECC more suitable for devices with limited resources in power and network connectivity [7]. At present, ECC has currently gained commercial acceptance and has been embraced by numerous standard organizations, including NIST [8], ANSI [9], ISO [10] and IEEE [11]. Encryption techniques based on chaos maps are also used for image encryption. However, some chaotic systems with low dimensions have a short periodicity of orbits when implemented on digital computers that’s why they degrade quickly [12]. Nowadays, spatiotemporal chaotic systems such as coupled map lattices have been widely used in cryptography. The security of the coupled map lattices-based image encryption schemes is greatly improved because the coupled map lattices have larger keyspace, better randomness, longer cycle, and more parameters [13]. Amara [7] analyzed that ECC has high security than the Rivest-Shamir-Adleman algorithm. In [14] an elliptic curve-based random-number generator is used for diffusion while dynamic substitution boxes (S-box) for confusion. Hayat et al. proposed an S-box generator and an image cryptosystem based on elliptic curves over finite rings [15]. Azam et al., [16] designed a secure elliptic curve-based image cryptosystem. In [17], an asymmetric multiple-images encryption method based on an elliptic curve and a quick response code is given. Many researchers designed different schemes for image encryption and the construction of dy namic S-boxes. In [18], authors proposed an image encryption scheme using asymmetric key encryption. A genetic algorithm is used to generate a special key and then elliptic curves are utilized to encipher all pixels one by one. However, it increases the computational cost to en crypt all pixel one by one and search for the ideal keys. The scheme in [15] is highly secure but it is not possible to implement the scheme for elliptic curves of large size. A hybrid algorithm based on both Advanced Encryption Standard and ECC is introduced in [19]. iv The main motivation behind this thesis is to design a dynamic S-boxes generator and highly secure encryption schemes with lesser encryption time. The main objectives are listed below: 1- Defining new mathematical algorithms on elliptic curves to design new cryptosystems. 2- Construction of new structures on elliptic curves with the help of existing structures. 3- Developing encryption schemes that can provide both confidentiality and integrity. 4- Designing new structures and methods to construct a random-number generator based on elliptic curves. 5- Stating and proving theoretical results. The thesis is organized as follows: The used concepts and corresponding notations are described in Chapter 1. In Chapter 2, we introduced an S-box generator that is suitable for lightweight cryptography and outperforms previously designed S-box generators in terms of computation time and security analysis. We produce particular sequences of integers using ordered elliptic curves of short size and binary sequences, which are subsequently utilized to generate S-boxes. We conducted numerous conventional tests to find out the efficiency of the proposed generator. Comparisons indicate that the new generator requires less operating time and has more security against modern attacks than numerous existing well-known generators. In Chapter 3, a new parametrization of resonant discrete triads to develop a new algorithm for the generation of all resonant triads in a grid of size L. We define a new transformation to map the resonant triads on a conic. We provide a full list of discrete Rossby wave triads in a given grid by solving Diophantine equations which appear in the context of the Charney-Hasegawa Mima equation. Further, we extend the algorithm for the enumeration of quasi-resonant triads v and experimentally demonstrate the robustness the algorithm to design the network of quasi resonant triads. As an application, we apply a total order on generated triads to design an S-box generator. Finally, via extensive analysis, we show that the newly developed S-box outperforms the S-boxes generated by some of the existing algorithms. In Chapter 4, a novel and secure cryptosystem for the real-time transmission of digital images is presented. The proposed work is based on elliptic curves and couple map lattices. The encryption technique is divided into two steps. The plain-image is initially diffused using an elliptic curve-based pseudo-random-number generator (PRNG). Then an S-box generator based on elliptic curves and couple map lattices is designed. The proposed encryption scheme has a larger keyspace and is robust against modern attacks. The use of couple map lattice-system makes the proposed cryptosystem secure against the known-plaintext attack. In Chapter 5, image encryption is used to transform digital images into an unreadable form on open Networks. We have designed an image cryptosystem to tackle the issues related to small key sizes and plaintext attacks. The proposed cryptosystem is divided into three parts. In the first part, we have developed a PRNG to diffuse the pixels of the plain image and in the second step, an S-box generator is constructed to generate permutation S-boxes with high nonlinearity. In the final step, an image encryption technique is presented to encrypt grayscale images. Furthermore, the cryptographic properties of our pseudo-random-number generator are tested using NIST tests. Furthermore, pseudo-random-numbers are generated using the elliptic curves to create diffusion in the data of plain-images. An S-box is used as a permutation to scramble the diffused image. In Chapter 6, the research results and suggested future directions are discussed. The thesis concludes with a list of references