This talk is a survey of the appearance and uses of "ultimate periodicity" in the various branches of mathematics. This includes such widely differing areas as: generalized-inverses, non-negative matrices, FT semi-groups and Random Number Generation.
Applications to Cryptography can be found in the DES system, hyper-orbits in RSA, hash functions and the Birthday Problems as well as in prime factorization (Pollard's rho method, and the Rabin-Miller test)