Image Encryption Based on Elliptic Curve Points and Linear Fractional Transformation

Abstract

The elliptic curve (EC) offers a well-organized and secure method for implementing cryptographic protocols in computer systems. Its broad utilization in various applications, such as ensuring secure communications, digital signatures, and key agreement protocols, underscores its significance in contemporary computing. In this article, we employed EC over a binary extension field (BEF), a choice that not only maintains security but also reduces computational overhead when compared to the use of large prime numbers. Our encryption scheme encompasses three primary phases. Firstly, we employ EC points over a BEF in conjunction with a piecewise function to hide the original image. Subsequently, to inject a high degree of confusion into the plaintext, we constructed a substitution box (S-box) grounded in EC principles and applied linear fractional transformation (LFT) to the generated EC points. This S-box is then used to rearrange the pixels of the previously masked image. Lastly, to introduce diffusion throughout the image and generate a ciphertext, we derive pseudo-random numbers (PRNs) using the coordinates of the EC and the characteristics of the BEF. Moreover, the proposed scheme has nonlinearity (NL) 112 for each S-box which is optimal value. Furthermore, we conducted computational experiments that substantiate the robust security of our proposed cryptosystem against linear, differential, and statistical attacks when compared to existing cryptographic systems.

Keywords

• Mordell elliptic curve
• Galois field
• substitution box
• pseudo-random numbers Generator
• image encryption