Commutative Subgroup Pairs of B n with Applications for Cryptography.

Abstract

Cryptography is the science of writing and deciphering code. In the last few years Cyber-Cryptography, or the science of writing and deciphering digital code has become central in our society. Many methods of Cyber-Cryptography are known and used today by some of our most important institutions. One of the most popular methods of Cyber-Cryptography is known as Public Key Cryptography. Many variants of Public Key Cryptography are used today, such as the popular Diffie–Hellman Protocol and the RSA Algorithm. To combat any potential weaknesses alternative Public Key Cryptography methods have been proposed. One of these Methods, known as The Korean Protocol relies heavily on a mathematical structure known as a Braid Group. Underlying the Korean Protocol is the division of a braid group into a pair of commutative subgroups. There is a trivial division of B n into two subgroups, which is what has been traditionally used. A list of all such commutative subgroup pairs would be an invaluable addition to cryptography. We set out to categorize all commutative pairs of subgroups. In the end we were able to create an algorithm that can generate any arbitrary commutative pair of subgroups.