A New Construction Of Quaternary Sequence Family With Period Of 2(2~n-1)

Families of pseudorandom sequences have good properties, good randomness, good correlation, long period, large linear complexity and confirmability and repeatability. Good randomness and low correlation functions make it easy to separate from the signals. Large linear complexity make it can resist attacks, and ensure data security in stream cipher. Confirmability and repeatability make it possible to attain it. These characteristics make pseudorandom sequences be widely used in cryptogrphy systems, spread spectrum communication systems and so on. Binary sequences and quaternary sequences are the most widely used in communication systems because they are easier to implement in modulator than other types of sequences. The most common binary sequences are as follows : The first type is the sequences that can be realized by LFSR(linear feedback shift register), such as m sequences, GMW sequences, Gold sequences. The second type is the sequences based on number theory, such as Legendre sequences, Jacobi sequences and Cyclotomic sequence. The third type is the sequences based on the trace function over finite fields, such as Kasami sequences, No sequences and so on. The research on quaternary sequences is lately. In the late 1980 s, the quaternary sequences over ring were discovered. These sequences have many good properties,such as long period, good correlation, large linear complexity, good balance. Therefore, it has important significance of constructing new quaternary sequences with good properties.This paper first describes the research Background and Significance of cryptography, as well as the development history and present situation of pseudorandom sequences. And then describes the algebraic structure of groups, rings, finite, the knowledge of trace function, Galois and pseudorandom sequences. Next, we analyze the correlation distributions of quaternary sequences family A in the literature [28] in 1992. Then we use the Gray map and the inverse Gray map to present a new quaternary sequences family P based on the optimal quaternary sequences family A, which have even period 2(2n-1), family size 2n+1, and the maximum nontrivial correlation value is Rmax=( 1) 22n++2, which is optimal with repect to the Welch and Sidelnikow bounds, where n is an odd integer. Then, we give a detailed analysis of correlation distributions and a simple result of the maximum linear complexity of the new quaternary sequences family P. Finally, we have a comparison of the new quaternary sequences family P andother optimal quaternary sequences. In contrast to the known optimal quaternary sequences family A, the new quaternary sequences family P has the same family size, the same linear complexity, but has different period, different maximum nontrivial correlation value and different distributions. Also, the new quaternary sequences family P is simple to construct. Therefore, the new quaternary sequences family P can be used as a new key sequence to encrypt the plaintext information.