Structures and cryptographic properties of single cycle T-functions

In this paper, the structures and properties of single cycle T-function, which has been lately proposed by Klimov and Shamir, are studied. A new symbol cⁱ _j is introduced, which is called a carrier, to representing the carrier of the word x on jth position at ith time. By using the carriers, we reveal that the sequence composed of the values of the word x on jth position at all time, whose period is 2^j+1, is determined uniquely by a carriers' sequence with period only 2^j. The structures of single cycle T-functions can be represented by an arbitrary state and their carriers. And we need only 2^j successive values of the word x to recover all the data including the least significant bit on jth position. Under some conditions, we need less data to recover the word x on jth position, and there are more regular characters. We also find some linear equations holding for single cycle T-functions.