Non-commutative cryptography: Diffie-Hellman and CCA secure cryptosystems using matrices over group rings and digital signatures

As computing speed has been following Moore's law without any inclination of tapering out, the need for ever more secure cryptographic protocols is becoming more and more relevant. During the past one and a half decades the field of non-commutative (or on-abelian) group based cryptography has seen a surge in interest.;Through this work we will present the classical Diffie-Hellman public key exchange protocol (DH PKE) and discuss two important notions related to it, the Computational Diffie-Hellman assumption and the Decision Diffie-Hellman assumption. We then proceed to look at a new platform group based on matrices over group rings and present work done by myself in collaboration with Delaram Kahrobaei and Vladimir Shpilrain. We discuss the viability of the new platform group and point out its benefits.;Additionally, I in collaboration with Delaram Kahrobaei and Vladimir Shpilrain propose to use the new platform group in the Cramer-Shoup cryptosystem. We demonstrate how one can implement the system using our platform and prove that the system is still CCA-2 secure.;Finally, we discuss the notion of classical digital signatures following the work of Goldwasser and Bellare and Schnorr. We then discuss some non-commutative digital signatures including those proposed by Ko, Choi, Cho and Lee, Wang and Hu Anjaneyulu, Reddy and Reddy and Chaum and van Antwerpen. We conclude by presenting work done my myself in conjunction with Delaram Kahrobaei which discusses a new non-commutative digital signature. We propose using groups for which the Conjugacy Search Problem is hard, or any group which is secure against length based attacks, such as polycyclic groups, as the platform for this signature.