Cryptosystems based on Error correcting codes

In the world of networking and communication, cryptography and coding theory play a pivotal role. However, research combining both these concepts have not been explored to its full potential, and is a crucial topic of interest that forms this thesis.

A new class of codes based on hexadecimal representation is introduced in this dissertation to support the Advanced Encryption Standard (AES), to provide error detection and correction in addition to providing security. In a hexi field (= GF(24)), the 4 bit binary number is represented by hexadecimal numbers, with the irreducible polynomial m(x) = x4 + x + 1 in Z2[x]. The codes defined over a hexi field are called hexi codes, they find applications in code-based cryptosystems.

In this thesis, hexi block codes, hexi polynomial codes, quasi cyclic hexi codes and quasi cyclic partial hexi codes are introduced. Decoding these codes using syndrome decoding algorithm, coset leader method, Euclidean division decoding algorithm and majority logic decoding are discussed extensively. Their respective error correcting capacities are defined and studied. Hexi polynomial codes can be generalized over GF(2q).