Designing Cryptosystems with Optimal Security

Abstract

With the widespread use of digital devices and the internet, implementing advanced security systems is essential to protect confidential information from cy berattacks and unauthorized access. Cryptography, the art of encoding and decod ing information, plays a vital role in safeguarding data from unauthorized access and potential threats. Over time, cryptographic systems have evolved, striving to develop more powerful and efficient algorithms to ensure data confidentiality, integrity, and authenticity. The study of elliptic curves has emerged as a promising path for creating robust and secure cryptosystems. The inherent properties of el liptic curves offer a fertile ground for designing cryptographic systems that provide enhanced security with smaller key sizes compared to traditional methods. This research work explores the captivating domain of developing advanced cryptosystems by utilizing the potential of elliptic curves over finite fields in combi nation with optimization algorithms. Elliptic curves over finite fields have emerged as a highly promising basis for cryptographic schemes due to their inherent com plexity and resilience against computational challenges. The fusion of their math ematical properties with optimization techniques holds the potential to create ro bust, secure, and high-performing cryptographic solutions. The primary emphasis is on employing elliptic curves along with optimization algorithms for substitution box generator, pseudo-random number generator, and image encryption. Each of these elements is fundamental in constructing a robust and secure cryptographic system capable of withstanding adversarial attacks. The substitution box (S-box) is an integral part of security systems that con fuses confidential data. Randomized or optimized S-box generators are commonly used in these systems. The former aims to generate highly key-dependent dynam ical S-boxes, while the latter generates static S-boxes with optimal cryptographic properties. We utilize the high resistance of elliptic curves against modern crypt analysis to propose a novel S-box generator capable of generating both randomized and optimized S-boxes in minimal computation time. The dynamic behavior of the proposed generator is analyzed by proving the necessary conditions for out putting distinct S-boxes and ensuring that the resultant S-boxes have a uniform probability distribution. Pseudo-random number generators (PRNGs) play an important role to ensure the security and confidentiality of image cryptographic algorithms. Their primary function is to generate a sequence of numbers that possesses unpredictability and randomness, which is crucial for the algorithms to work effectively and provide the desired level of security. However, traditional PRNGs frequently encounter limita v tions like insufficient randomness, predictability, and vulnerability to cryptanalysis attacks. To overcome these limitations, we propose a novel method namely an el liptic curve genetic algorithm (ECGA) for the construction of an image-dependent pseudo-random number generator (IDPRNG) that merges elliptic curves (ECs) and a genetic algorithm (GA). The ECGA consists of two primary stages. First, we generate an EC-based initial sequence of random numbers using pixels of a plain-image and parameters of an EC, that differ from traditional methods of pop ulation initialization. In our proposed approach, the image itself serves as the seed for the initial population in the genetic algorithm optimization, taking into account the image-dependent nature of cryptographic applications. This allows the PRNG to adapt its behavior to the unique characteristics of the input image, leading to enhanced security and improved resistance against differential attacks. Furthermore, the use of a good initial population reduces the number of genera tions required by a genetic algorithm, which results in decreased computational cost. In the second stage, we use well-known operations of a genetic algorithm to optimize the generated sequence by maximizing a multivariable fitness function that is based on both the information entropy and the period of the PRNG. By combining elliptic curves and genetic algorithms, we enhance the randomness and security of the ECGA. Image encryption is the process of securing digital images by transforming their content into an encrypted, unreadable form. The purpose is to protect them from unauthorized access and ensure confidentiality. The main objective of image encryption is to make the image unreadable to anyone lacking the correct decryp tion key. Several encryption schemes have been proposed, but either they do not achieve high security or they are computationally costly for any image, and thus may not be suitable for real-time applications. We design an image encryption scheme by formulating an optimization problem with an objective function that captures both entropy and correlation. We select a core of the plain-image to compute a parameter that optimizes the objection function over the core using the artificial bee colony (ABC) algorithm. Furthermore, we propose a new efficient random number generator over isomorphic elliptic curves. This generator together with the parameter is then used to generate cipher-images. Our intention is that this work will inspire further exploration and advance ments in the field of cryptography. May the knowledge shared herein contribute to the broader landscape of digital security and inspire future researchers to unlock the full potential of elliptic curves and optimization algorithms in designing secure cryptosystems.