Large Random Numbers

Abstract

Sequences of “random” numerical values are used in many statistical procedures, in numerical mathematics, in physics, and also in number-theoretic applications to replace statistical observations or to automate the input of variable quantities. Random numbers are used:

• to select random samples from a larger set,

• in cryptography to generate keys and in running security protocols,

• as initial values in procedures to generate prime numbers,

• to test computer programs (a topic to which we shall return),

• for fun,

as well as in many additional applications. In computer simulations of natural phenomena random numbers can be used to represent measured values, thereby representing a natural process (Monte Carlo methods). Random numbers are useful even when numbers are required that can be selected arbitrarily. Before we set out in this chapter to produce some functions for the generation of large random numbers, which will be required, in particular, for cryptographic applications, we should take care of some methodological preparations.

Mathematics is full of pseudorandomness—plenty enough to supply all would-be creators for all time.

—D. R. Hofstadter, Gödel, Escher, Bach

Anyone who considers arithmetical methods of producing random digits is, of course, in a state of sin.

—John von Neumann