Color image encryption by piecewise function and elliptic curve over the Galois field $GF\left(2^n\right)$

Elliptic curve (EC) cryptography supplies an efficient, secure, and lightweight method for executing computer cryptographic protocols. Its widespread use in various applications, including secure communications, digital signatures, and key agreement protocols, highlights its importance in modern computing. Moreover, EC-based image encryption is gaining popularity in cryptography as it offers strong protection with a relatively smaller key size than other famous cryptosystems. Inspired by this, we proposed a novel image encryption scheme that leverages ECs over a binary extension field (BEF). This approach also reduces computational workload using EC over BEF instead of large primes. Also, BEF can represent large numbers in a compact form, which is helpful in applications that require efficient data storage and transmission. Our scheme involves three main steps. Initially, we utilize points of an EC over a BEF and a piecewise function to mask the plain image. Next, to introduce a high level of confusion in the plain text, we create a substitution box (S-box) based on the EC and operation of BEF of order 256, which is then used to permute the pixels of the masked image. Finally, we generate pseudo-random numbers (PRNs) using EC coordinates and BEF characteristics to create diffusion in the image and obtain a cipher image. In addition, we accomplished computational experiments demonstrating that our proposed cryptosystem provides excellent security against linear, differential, and statistical attacks compared to existing cryptosystems.