New Results in Group Theoretic Cryptology

With the publication of Shor's quantum algorithm for solving discrete logarithms in
finite cyclic groups, a need for new cryptographic primitives arose; namely, for more secure
primitives that would prevail in the post-quantum era.
The aim of this dissertation is to exploit some hard problems arising from group theory
for use in cryptography. Over the years, there have been many such proposals. We first look
at two recently proposed schemes based on some form of a generalization of the discrete
logari thm problem (DLP), identify their weaknesses, and cryptanalyze them.
By applying the exper tise gained from the above cryptanalyses, we define our own
generalization of the DLP to arbitrary finite groups. We show that such a definition leads
to the design of signature schemes and pseudo-random number generators with provable
security under a security assumption based on a group theoretic problem. In particular,
our security assumption is based on the hardness of factorizing elements of the projective
special linear group over a finite field in some representations. We construct a one-way
function based on this group theoretic assumption and provide a security proof.