The Ubiquity of Prime Numbers

Abstract

It is often remarked that prime numbers finally found a legitimate practical application in the domain of cryptography. The cryptographic relevance is not disputed, but there are many other applications of the majestic primes. Some applications are industrial—such as applications in numerical analysis, applied mathematics, and other applied sciences—while some are of the " conceptual feedback " variety, in which primes and their surrounding concepts are used in theoretical work outside of, say, pure number theory. In this lucrative research mode, primes are used within algorithms that might appear a priori independent of primes, and so on. It seems fair to regard the prime number concept as ubiquitous, since the primes appear in so very many disparate domains of thought. 8.1 Cryptography On the face of it, the prime numbers apply to cryptography by virtue of the extreme difficulty of certain computations. Two such problems are factoring and the discrete logarithm problem. We shall discuss practical instances of these problems in the field of cryptography, and also discuss elliptic curve generalizations.