ON SECURE BINARY SEQUENCES GENERATED BY A FUNCTION f(x) = x + (g(x)2 _ C) mod 2n

Invertible transformations over n-bit words are essential ingredients in many cryptographic constructions. When n is large (e.g., n = 64) such invertible transformations are usually represented as a composition of simpler operations such as linear functions, S-P networks, Feistel structures and T-functions. Among them we will study T-functions which are probably invertible transformation and are very useful in stream ciphers. In this paper we will show that f(x) = x + (g(x)2 _ C) mod 2n is a permutation with a single cycle of length 2n if both the least significant bit and the third significant bit in the constant C are 1, where g(x) is a T-function.