This monograph deals with contributions to certain areas of contemporary cryptography based on recent developments in mathematics and computer science, including: (1) public-key cryptography based on combinatorial group theory, with an introduction to the exciting new area of braid group cryptography; (2) construction of one-way functions and pseudorandom number generators from a very general class of zeta functions, namely the feasible Selberg class. A focused survey of the underlying methods is presented together with careful computational constructions, allowing the reader to pursue independent investigations.