Color Image Encryption through Projective Coordinates of Elliptic Curve

Abstract

In recent decades, rapid advancements in science and digital technologies have significantly increased the prominence of digital data in various aspects of life, including education, business, banking, engineering, mathematics, art, advertisement, military, medicine, and scientific research. The growing influence of digital data in information technologies has heightened the significance of tools for processing and documenting digital information. For example, in the field of medicine, images serve as vital tools for visualizing various analyses and diagnoses. These analyses are transmitted in the form of images, enabling patients to seek consultations from medical specialists located anywhere around the world. In this scenario, the violation of integrity and confidentiality poses significant risks to the patient's well-being. Consequently, this surge has led to increased distribution of digital data over the internet. Given the internet's widespread accessibility, it introduces potential risks to the integrity and confidentiality of digital data during distribution. Consequently, diverse approaches have been advanced to safeguard their privacy and security. Foremost among these is the field of cryptography, which revolves around the science of encryption. Encryption techniques play a pivotal role in preserving the confidentiality of data. Numerous encryption methods have been employed to guarantee the privacy and integrity of digital images. Furthermore, there exist concealment techniques such as steganography and watermarking designed for safeguarding digital images. In steganography, a digital image can be covertly embedded within data, messages, or even within a distinct image. Likewise, watermarking is extensively employed to enhance the security of digital images, notably in the prevention of forgery and counterfeiting in contexts like currency protection. In contrast, encryption entails the complete concealment of the digital image by applying a transformation function to the original image's pixels. These methods can serve various purposes, with digital image encryption being a technique that converts an image into an encrypted form through pixel transformations. The foremost objective of this encryption process is to render the original image entirely unrecognizable and resistant to easy restoration. In cryptography, various methods, including chaotic and DNA-based approaches, are employed by researchers for image encryption. However, this thesis focuses on the implementation of Elliptic Curve Cryptography (ECC) for image encryption, recognizing its paramount importance in ensuring security and privacy. Originating in 1985 through 1 independent developments by Koblitz and Miller, ECC gained widespread acceptance around 2004. It stands out as a robust public key encryption method, offering heightened security for a given key size, distinguishing itself from encryption techniques relying on integer factorization or the discrete logarithmic problem. ECC widely adopted encryption technique is renowned for delivering enhanced security while requiring smaller key sizes, making it an optimal choice, especially for devices with limited computational capabilities. Similarly, cryptography, utilizing discrete mathematical structures, prominently employs the binary field 𝐺𝐹(2) and its extensions, such as 𝐺𝐹(2𝑚), in computer software and hardware. These algebraic structures, known for their soothing hardware implementation, play a crucial role in cryptography and computer science. The focus is on enhancing performance and reducing costs in finite field applications within cryptology. This thesis strategically leverages the Galois field extension 𝐺𝐹(2𝑚) and capitalizes on its distinctive structural attributes to attain optimal results. By harnessing the unique features inherent in the Galois field extension, our research aims to unlock its full potential and derive significant advantages. The utilization of 𝐺𝐹(2𝑚) serves as a cornerstone in our approach, allowing us to explore novel insights and exploit the unparalleled characteristics of this mathematical construct to enhance the overall effectiveness of our study. This thesis encompasses six chapters, with the first chapter providing a concise exploration of fundamental concepts in cryptography. This includes the classification of cryptography, key concepts employed, and basic algebraic definitions utilized throughout the entire thesis. Furthermore, the first chapter also discusses the definition of Elliptic Curves (ECs), elucidating their properties and structure, and establishing a foundational understanding. These defined terms and concepts play a pivotal role throughout the entirety of this thesis, forming the basis for subsequent discussions and analyses. The second chapter of this thesis presents a comprehensive review of cryptographic methods, conducting a comparative analysis to illuminate their strengths and weaknesses. it presents an overview of two research papers. In the first manuscript, the authors present a cryptosystem centered around ECs for the encryption of digital images. The scheme employs isomorphic ECs over a prime field to shuffle the pixel positions of the plain image. In the subsequent step, it generates multiple S-boxes using isomorphic ECs. The encryption procedure generates pseudo-random numbers (PRNs) through arithmetic operations on ECs, deviating from the EC group law. In the second manuscript, the authors introduce a technique for 2 crafting 16, 8 × 8 S-boxes across the elements of 16 Galois fields. To achieve this, they define 16 linear fractional transformations on these Galois fields. In chapter three, we propose novel nonlinear components that employ the concept of the Projective Coordinates (PCs) of ECs over the set of integers modulo 𝑝𝑟, where 𝑝 is prime and 𝑟 ≥ 1. We employ an approach that harnesses the potential of PCs of ECs in such a way that we commence by applying a set of conversion mappings, to facilitate its transformation from 𝑍𝑝𝑟 to 𝐺𝐹(𝑝𝑟). To enhance the intricacy of nonlinear mappings, numerous implementations with intriguing properties have been put forth in the academic literature. Specifically, an affine-power-affine structure has been devised for the Advanced Encryption Standard (AES), augmenting the complexity of its algebraic expression through the inclusion of additional terms. Leveraging the characteristics of the affine-power-affine structure, we incorporate a dual mapping strategy, beginning with an inverse mapping over the 𝐺𝐹(𝑝𝑟) and then seamlessly transitioning to an affine mapping, effectively yielding an expansive Keyspace for the generation of S-boxes. The culmination of these sequential procedures results in the creation of distinct 8 × 8 S-boxes, showcasing optimal statistical outcomes. These S-boxes are subsequently harnessed to introduce substantial distortion to the original image, while also orchestrating the permutation of pixels within the concealed image. The ultimate stride in achieving image diffusion and the attainment of a cipher image revolves around the application of PRNs. These PRNs are ingeniously generated by leveraging the properties of the EC over the set of integers modulo 𝑝𝑟, using various mappings defined on the PCs of EC. This intricate step significantly contributes to heightening the security and intricacy of the cipher image, effectively establishing it as a dependable and robust solution for image encryption. Through comprehensive computational experiments, our cryptosystem's superiority in safeguarding against linear, differential, and statistical attacks has been unequivocally demonstrated when compared to existing cryptographic systems. In chapter four, we present a novel and efficient cryptosystem that utilizes PCs of ECs for the encryption of digital images. Our scheme involves a well-structured sequence of steps. Initially, we leverage the power of the PCs of EC over the set of integers modulo 𝑝𝑟, where � � is prime and 𝑟 ≥ 1. Further, by implementing various bijective mappings on the PCs of EC to get the elements in the required range. Moreover, to preserve the algebraic operations and structure of the obtained PCs after the bijective mappings, we employed an automorphism to each coordinate of the EC. The culmination of these steps yields the creation of different 3 8 ×8 S-boxes with optimal statistical results, which are then employed to cause significant distortion in the original image and further used to permute the pixels of the masked image. Lastly, to achieve image diffusion and acquire a cipher image, we employ PRNs generated through the utilization of the EC over the set of integers modulo 𝑝𝑟 and considering various combinations of 𝑎 and 𝑏 on the defined PCs of EC. This step contributes significantly to enhancing the security and complexity of the cipher image, making it a reliable and robust solution for image encryption. In our computational experiments, we have shown that our proposed cryptosystem offers superior security against linear, differential, and statistical attacks when compared to existing cryptosystems. Recent research has shown that incorporating ECs within cryptosystems using finite rings can enhance safety by reducing the computational challenges of solving factorization and discrete logarithm problems. Inspired by this, in chapter five we have introduced an image encryption scheme that uses ECs over finite rings. This scheme has been developed to enhance the security and robustness of image encryption. To ensure sufficient complexity, we select a set of ECs, each chosen from distinct primes 𝑝 and 𝑞, such that their product surpasses the encrypted image dimensions. Initially, we employed permutation mapping on the points of an EC for shuffling the pixel arrangement of the original image. Further, we used the S-box derived from the permutation mapping relying on modular arithmetic operations over the affine coordinates of EC to shuffle the pixels of the diffused image, creating a substituted image. In the culmination of our encryption technique, a sequence of PRNs is generated. This is achieved through a series of mappings involving affine coordinates of ECs and the properties of the Euler phi function. Thus, this complex operation introduces a substantial level of randomness, thereby significantly enhancing the overall security of our scheme. Moreover, extensive computational experiments and the heightened level of security render the scheme remarkably resilient against classical attack methods. The final chapter summarizes key findings and presents concise conclusions. Additionally, it offers valuable suggestions for future research directions, encapsulating the essence of our work and providing a roadmap for further contributions in the field of cryptography.