Trace Representations and Multi-rate Constructions of Two Classes of Generalized Cyclotomic Sequences

In this paper, two classes of generalized cyclotomic sequences of period pq are reconstructed by means of multi-rate parallel combinations of binary Legendre sequences which are clocked at different rates. Then these generalized cyclotomic sequences can be generated by combinations of short and cheap LFSR’s. From the multi-rate constructions and the trace representation of binary Legendre sequences, we present trace representations of these generalized cyclotomic sequences, which is important to the investigation of cryptographic properties of these sequences. keywords: stream cipher, cyclotomic sequence, linear complexity, minimal polynomial Mathematics Subject Classification 2000: 94A55 Introduction and Preliminaries This paper investigates the trace representations and generations of a binary Ding generalized cyclotomic sequence(DGCS) and a binary Whiteman generalized cyclotomic sequence(WGCS) by means of multi-rate parallel combinations of constituent binary Legendre sequences which are clocked at different rates. These combinations, proposed initially by M. G. Parker([1]), demonstrate that sequences of large linear complexity can be generated without resorting to linear feedback shift registers (LFSR) of large length. Trace representation is an important tool in the investigation of sequences, by which we can yield some properties such as linear complexity, correlation and distribution of runs. DGCS and WGCS, introduced by C. Ding in ∗Project supported by the National 973 High Technology Projects (No. G1999035804) and the National Natural Science Foundations of China (No. 90104005)