Elliptic Curves (EC) provide a reliable and systematic framework for implementing cryptographic protocols in computer systems. This work presents a general algorithm using EC over the Galois field \(\:GF\left({2}^{n}\right)\) of order \(\:n\) , rather than building one around a specific Binary Extension Field (BEF), which directly reduces computational overhead compared to large-prime-number schemes while maintaining strong security. The approach begins by concealing the original image using the symmetric group \(\:S₁₆\) and EC points over a BEF, after which a Substitution-box (S-box) is constructed based on EC principles to introduce significant confusion into the plaintext. The Affine Inverse Map (AIM) is then applied over \(\:GF\left({2}^{n}\right)\) using an Irreducible Polynomial (IP) and the symmetric group \(\:S₁₆\) to the generated EC points. This combination substantially expands the key space and yields excellent S-box generation results. In most cases, this combination produces \(\:16!\times\:30\times\:255\times\:255\) possible configurations for S-boxes with Nonlinearity (NL) of \(\:112\) , which is the optimal value. For \(\:n=8\) , the method achieves a minimum key space of \(\:{2}^{479}\) , considerably larger than the conventional \(\:{2}^{128}\) threshold, and this security strengthens proportionally as n increases. The use of BEF ensures both computational efficiency and robustness. At the same time, the expanded key space significantly strengthens defences against exhaustive search attacks, establishing the proposed scheme as a strong and scalable image encryption framework.
