Cycles in random k-ary maps and the poor performance of random number generation

Knuth [The Art of Computer Programming, vol. 2: Semi-Numerical Algorithms, 3rd ed., Addison-Wesley, Reading, MA, 1997] shows that iterations of a random function perform poorly on average as a random number generator. He proposes a generalization in which the next value depends on two or more previous values. This note demonstrates, via an analysis of the cycle length of a random k-ary map, the equally poor performance of a random instance in Knuth's more general model.