𝟖×𝟖 𝑺−𝒃𝒐𝒙 𝑫𝒆𝒔𝒊𝒈𝒏 𝒐𝒗𝒆𝒓 𝑶𝒓𝒅𝒆𝒓 𝟔𝟒 𝑺𝒖𝒃𝒈𝒓𝒐𝒖𝒑 𝒐𝒇 𝑮𝒓𝒐𝒖𝒑 𝒐𝒇 𝑼𝒏𝒊𝒕𝒔 𝒐𝒇 𝑭𝒊𝒏𝒊𝒕𝒆 𝑪𝒉𝒂𝒊𝒏 𝑹𝒊𝒏𝒈 𝔽𝟐[𝛉]<𝛉𝟖>

Abstract

Increasing amounts of data are moved over the Internet and networks. Cryptography provides the security that these applications need. Cryptography is a deliberate attempt to conceal data so that unauthorized parties cannot access it. In symmetric cryptography, private-key encryption is the primary focus, and the key distribution and key-management issues make it useless for today's world. A new approach was needed to properly address these issues. However, asymmetric cryptography does not only present a solution, but it also marks the beginning of a new era in cryptography. The idea of key exchange protocol was first initiated by Merkle, Differ, and Hellman in the mid of 1970s. One of the earliest asymmetric cryptosystems is the famous RSA. Later, many more asymmetric algorithms were introduced, such as ElGamal, ECC which are based on the complexity of the integer factorization problem. This is still being further modified by different cryptologists. Data confidentially, integrity and authenticity are the fundamental protection goals of cryptography. The Hash function and digital signature improve message integrity and make it more authentic. Now a day, a critical problem that classical and modern cryptography fails to address is long-term security. Quantum cryptography seems to overcome this issue because it is based on the law of quantum physics which is believed to be valid forever. The hardness of the quantum algorithm makes it difficult to implement in many applications. In this respect, asymmetric cryptosystems based on matrix algebra over the residue ring have been studied from the last decade. The focus of this work is to ensure the improvement in [36], a proposed scheme based on a commutative subgroup of the finite chain ring. our main goal is to increase the security of the algorithm by using a unique algebraic structure of the finite chain ring 𝑅8=𝔽2[θ]<θ8> . however, the local ring ℤ𝑛 of the integers modulo 𝑛 makes both cryptosystems insecure in the sense that attacker, who is efficient in solving linear equations in ℤ𝑛 can easily break both schemes in a very limited period. In 2016, Jianwai Jia et al., they conduct a detailed analysis of the structure attack and deduced that both cryptosystems were breakable. In this thesis, we propose a new cryptosystem which are based on the subgroup of the finite chain ring in a similar way as done for cryptosystem 1 over finite chain ring [36]. The chain ring 𝑅8 has a special structure of polynomials, the coefficient of a polynomial is from 𝔽2 which make its calculation easy but unfeasible for the attackers to decrypt it.