Abstract: T-functions is a mapping from rrbits to rrbits words in which cach(0≤i≤n一1) bit i of the output depends only on bits 0,1，…,i of the input. Invertible T-functions is essential ingredients in many cryptographic applications. By using the cycle structure and parameter, we proved that every cycle has length of powers of 2. Then we studied on the cycle structure of the invertible T-functions intensively, and gave two different methods for retrieving the cycle structure of the invertible T-functions f(x)mod 2k+1 on basis of f(x)mod 2k. Moreover, based on the retriving cycle structure method, we presented a determinant condition of the cycle structure characteristic 2n-tX 2t.

Key words: Cryptography, T-functions, State transfer graph, Cycle structure, Paremeter