.
The applications of orders, discrete logarithms, and the primitive root theorem are numerous. This chapter explores some contemporary applications such as the ElGamal encryption and ElGamal signature schemes. There is also an extensive discussion of public-key signature schemes including the Chaum-van Antwerpen undeniable signature scheme, blind signatures, and fail-stop signatures.
The multiplicative properties of congruences can be used to
analyze the period of the linear congruential number generator
of D. H. Lehmer. This popular pseudo-random number generator
is widely used in simulation and other statistical
applications. The section of pseudo-random sequences includes detailed proofs
of the period length of the sequence .