Expected $\pi$ -Adic Security Measures of Sequences

Various measures of security of stream ciphers have been studied that are based on the problem of finding a minimum size generator for the keystream in some special class of generators. These include linear and p-adic spans, as well as π-adic span, which is based on a choice of an element π in a finite extension of the integers. The corresponding sequence generators are known as linear feedback shift registers, feedback with carry shift registers, and the more general algebraic feedback shift registers, respectively. In this paper, the average behavior of such security measures when π^d = p ≫ 0 or π² = -p ≪ 0 is studied. In these cases, if Z [π] is the ring of integers in its fraction field and is a UFD, it is shown that the average π-adic span is n - O(log(n)) for sequences with period n.