Galois Ring $GR\left( {2^{3} ,8} \right)$ Dependent $24 \times 24$ S-Box Design: An RGB Image Encryption Application

Abstract

An S-box is based on Boolean functions which are essentially the foundation of symmetric cryptographic systems. The Boolean functions are used for S-box designing in block ciphers and exploited as nonlinear components. Boolean functions with optimal nonlinearity and upright cryptographic stuffs play a significant role in the design of block ciphers. Traditionally $8 \times 8$ S-box is a $16 \times 16$ look up table over Galois field $GF\left( {2^{8} } \right)$ and has 112 feasible upper bonds for nonlinearity. A $24 \times 24$ S-box over Galois field $GF\left( {2^{24} } \right)$ is not viable as the computer memory does not support it. In this paper for the construction of $24 \times 24$ S-box a rout is adopted via maximal cyclic subgroup of the multiplicative group of units of Galois ring $GR\left( {2^{3} ,8} \right).$ The newly constructed S-box has much higher confusion capability than any of $8 \times 8$ S-box. To judge the impact of this new $24 \times 24$ S-box an RGB color image encryption application is demonstrated. Initially, in the proposed encryption scheme we use $24 \times 24$ S-box for confusion in RGB channels of plain image, however for diffusion linear permutation ${\text{P}} = \left( {{\text{i}} \times 32} \right) {\text{mod}}257$ is operated and then by the use of exclusive-or an encrypted image is obtained. Thus, we introduce a novel technique by which $24$ binary bits are divided into 3 bytes and each one deals R, G and B channel of the color image separately. A comparison with chaos and DNA based image encryption schemes shows the performance results of this novel RGB image encryption and observed as meeting the standard optimal level. Hence this $24 \times 24$ S-box dependent encryption method replaces $8 \times 8$ S-box based RGB color image encryption scheme.