Combined Matrix-Block Encryption Algorithm Using Elliptic Curves

Abstract

The article examines a block cryptographic algorithm using a two-component shared secret key obtained according to the Diffie-Hellman key exchange principle on elliptic curve points over the field Z_p. The goal is to eliminate shortcom­ings of individual classical algorithms and, through their combination, increase overall system strength. Key generation and exchange between users are carried out using elliptic curve cryptographic systems with public key. Two methods are proposed for generating shared secret keys for interacting users: applying the Diffie-Hellman cryptographic protocol on multiple elliptic curve points or additionally using a recurrence formula. Encryption elements are represented by blocks as square matrices constructed on elliptic curve point coordinates. Encryption proceeds in two stages: the first uses stream cipher with scalar multiplication of elliptic curve points, and the second involves forming matrix blocks and performing Hill matrix transformation with feedback. Each encryption stage utilizes its corresponding component of the users’ shared secret key: a numerical gamma sequence or a square key matrix. The cryptographic strength is based on the computational complexity of solving the discrete logarithm problem on elliptic curves and the security of the sharing service with secure authentication of interacting users. The block implementation of the second encryption stage ensures the system’s resistance to frequency analysis. As an illustration of the presented algorithm’s operation, the article provides a step-by-step example of encrypting/decrypting a text message.