Development of Matrix Cipher Modifications and Key Exchange Protocol

Abstract

In modern cryptographic methods, keys are the basis for secure communication channels and the establishment of secret keys is a challenging problem for the large-scale deployment of symmetric cryptography to control encryption and decryption. Key establishment protocols provide exchanging secret information between two or more parties, typically for subsequent use as symmetric keys for a variety of information security services including encryption, message authentication, and entity authentication. They may be broadly subdivided into key transport and key exchange. Notably, key exchange is one of the difficulties when using symmetric algorithms, the key exchange particularly useful from a security viewpoint, for each of the key-sharing parties can have its own control and a high confidence on the quality of the key output. Beside encryption, key exchange is one of the most basic problems in cryptography; it becomes another challenge in cryptography. This thesis is concerned with the modifications of the Hill cipher (HC), extension of Diffie Hellman and ElGamal key exchange protocols. The HC is one of the most popular symmetric key algorithms; it is resistant to brute-force and statistical attacks, but it can be broken with a known plaintext-ciphertext attack (KPCA). To overcome this vulnerability, several researchers tried to propose modifications of the Hill cipher and make it secure. However in the literature, most of these modifications are found to be either insecure or ineffective for image encryption. The Diffie-Hellman Key Exchange (DH) is known as one of the public key algorithms, its aim is to distribute the keys over insecure channels. It is based on the iv complexity of discrete logarithm problem (DLP) solving over a finite fieldGF(p) , where p is prime which considered as an advantage from the security viewpoint due to the challenging and difficulties for solving the discrete logarithm. But DH has drawbacks including the fact that there are heavy and expensive exponential operations in both sides (sender and receiver) which affect its efficiency; it can be used for exchanging secret keys. To overcome this drawback, DH protocol matrix oriented modifications based on DLP are proposed by several researchers. Moreover, in the literatures, most of the modifications still rely on the DLP. The ElGamal Public Key Cryptosystem and Signature (EPKCS) also rely on the computational complexity of finding discrete logarithms based on some publicly known primitive root (base element), GF(p) , where p is a large prime. Similar to DH protocol, the EPKCS has a drawback; it has a slow speed especially for signing in addition to the ciphertext is twice as long as the plaintext. In this thesis, we proposed two modifications of the Hill cipher, HCM-EE and HCM-PRE. A matrix-based Diffie-Hellman-like key exchange protocol is also proposed. ElGamal public key cryptosystem and signature scheme is extended to the group GU(m, p, n) of numbers co-prime to mpn.