In this article, first we deliberate the theory of the Gaussian field and extension field of the Gaussian field. However, in the second phase, we provide a comprehensive construction scheme for BCH codes over the Gaussian field. The decoding of newly designed BCH codes is handled through a slightly amended modified Berlekamp-Massey algorithm. The coding gain is obtained by BCH codes over the Gaussian field. Accordingly, a better code rate and the number of code words are obtained as compared to the BCH codes over finite fields. Thus, this makes them a promising candidate for use in communication systems.

In block ciphers, the nonlinear components, also known as substitution boxes (S-boxes), are used with the purpose of inducing confusion in cryptosystems. For the last decade, most of the work on designing S-boxes over the points of elliptic curves has been published. The main purpose of these studies is to hide data and improve the security levels of crypto algorithms. In this work, we design pair of nonlinear components of a block cipher over the residue class of Gaussian integers (GI). The fascinating features of this structure provide S-boxes pair at a time by fixing three parameters. But the prime field dependent on the Elliptic curve (EC) provides one S-box at a time by fixing three parameters and . The newly designed pair of S-boxes are assessed by various tests like nonlinearity, bit independence criterion, strict avalanche criterion, linear approximation probability, and differential approximation probability.

In this paper, the authors present the design of a new pair of 8 × 8 S-boxes over the residue classes of Gaussian integers \(\:Z{\left[i\right]}_{h}\) and their usage for the encryption of multiple RGB images using a three-stage Substitution-Permutation Network (SPN). In the present work, the S-boxes are constructed using the properties of Gaussian Integers and inverse of affine then affine maps, where first S-box holds the roles of substitution layer and the second S-box looks after the permutation layer. The modulation is nonlinear by means of operating the Gaussian integer residue classes through modular arithmetic, which will greatly benefit from strong diffusion and confusion properties that are critical for cryptographic security. For even greater complexity an exclusive XOR operation on the two S-boxes is made to produce a third S-box. The third S-box is deployed in the encryption process in such manner which offers superior diffusion over all RGB image channels. When the proposed SPN framework is utilized for a set of multiple RGB images, the mappings employed involves substitution-permutation-XOR operations that are severe nonlinear functions for differential or linear attacks. The incorporation of these Gaussian-integer-based S-boxes into the SPN scheme makes it possible for each image channel to be transformed independently but mutually connected to produce a secure and efficient encryption function. The analysis of outcomes proves the efficiency of the developed method; the viability of its application for multi-image encryption with high entropy, low correlation and increased resistance to a variety of attacks. This ability to consider it as an effective tool in the sphere of secure multimedia communication.

This research paper proposes a novel approach for constructing substitution boxes (S-boxes) over Gaussian integers, which are complex numbers with integer coefficients. The proposed method is based on the properties of the Gaussian integers and their arithmetic operations and ensures the S-boxes exhibit strong cryptographic properties. Furthermore, the paper demonstrates how these S-boxes can be utilized for image encryption through a substitution-permutation network (SPN) over Gaussian integers. The SPN involves iteratively applying the S-box and a permutation layer to the input image, which effectively scrambles the image data. Experimental results show that the proposed method achieves high security and robustness against various attacks while providing efficient encryption and decryption performance. This research thus provides a promising avenue for developing secure image encryption schemes based on Gaussian integers.

In the field of cryptography, block ciphers are widely used to provide confidentiality and integrity of data. One of the key components of a block cipher is its nonlinear substitution function. In this paper, we propose a new design methodology for the nonlinear substitution function of a block cipher, based on the use of Quaternion integers (QI). Quaternions are an extension of complex numbers that allow for more complex arithmetic operations, which can enhance the security of the cipher. We demonstrate the effectiveness of our proposed design by implementing it in a block cipher and conducting extensive security analysis. Quaternion integers give pair of substitution boxes (S-boxes) after fixing parameters but other structures give only one S-box after fixing parameters. Our results show that the proposed design provides superior security compared to existing designs, two making on a promising approach for future cryptographic applications.