On the Periods of Certain Pseudorandom Sequences

In [1], Rader et al. gave a fast method for generating pseudorandom sequences. Concerning these sequences, Moriyama et al. [2] made a research including the computational results by computers. In this paper we shall study the periods of these sequences, and give an affirmative answer to the following conjecture presented in [2]: "Let k(n) be the maximum period of n-bit pseudorandom sequences generated by the Rader's method. Then k(2n} = 2k(n) for all ?z." We shall also prove a number of algebraic properties of the periods, and give an efficient algorithm for computing k(ri). We remark here that in this paper we are interested only in the algebraic properties of these sequences and not in the randomness of these sequences.