Dynamic S-Box Construction Using Mordell Elliptic Curves over Galois Field and Its Applications in Image Encryption

Dynamic S-Box Construction Using Mordell Elliptic Curves over Galois Field and Its Applications in Image Encryption Amal S. Alali Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia http://orcid.org/0000-0001-7856-2861 Rashad Ali Department of Mathematics, Riphah International University, Lahore 54000, Pakistan Muhammad Kamran Jamil Department of Mathematics, Riphah International University, Lahore 54000, Pakistan http://orcid.org/0000-0001-5715-9299 Javed Ali Department of Mathematics, Riphah International University, Lahore 54000, Pakistan Gulraiz Department of Mathematics, Riphah International University, Lahore 54000, Pakistan

Elliptic curve cryptography has gained attention due to its strong resilience against current cryptanalysis methods. Inspired by the increasing demand for reliable and secure cryptographic methods, our research investigates the relationship between complex mathematical structures and image encryption. A substitution box (S-box) is the single non-linear component of several well-known security systems. Mordell elliptic curves are used because of their special characteristics and the immense computational capacity of Galois fields. These S-boxes are dynamic, which adds a layer of complexity that raises the encryption process’s security considerably. We suggest an effective technique for creating S-boxes based on a class of elliptic curves over GF(2n),n≥8. We demonstrate our approach’s robustness against a range of cryptographic threats through thorough examination, highlighting its practical applicability. The assessment of resistance of the newly generated S-box to common attack methods including linear, differential, and algebraic attacks involves a thorough analysis. This analysis is conducted by quantifying various metrics such as non-linearity, linear approximation, strict avalanche, bit independence, and differential approximation to gauge the S-box’s robustness against these attacks. A recommended method for image encryption involves the use of built-in S-boxes to quickly perform pixel replacement and shuffling. To evaluate the efficiency of the proposed strategy, we employed various tests. The research holds relevance as it can provide alternative guidelines for image encryption, which could have wider consequences for the area of cryptography as a whole. We believe that our findings will contribute to the development of secure communication and data protection, as digital security is becoming increasingly important.
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