A Study On The Measure Of Pseudorandom Binary Sequences

The construction and randomness properties analysis of pseudorandom se-quences are the core problems for cryptography.Many scholars have constructed a series of binary sequences based on Fermat quotients and generalized cycloto-my.In this paper,We construct a large family of binary sequences and study their pseudorandomness by using the estimates for character sums of Fermat quotients.In addition,we study the autocorrelation of binary sequences and the correlation measure of order 3 and 4 respectively.Letbe a prime,the main contributions could be summarized as follows:1.Based on the properties of multiplicative characters modulo p²,we ob-tain the autocorrelation value of binary sequence (?) in the special case.Then,we construct a large family of binary sequences of length p² by using the esti-mates for character sums of Fermat quotients,and study the well-distribution,correlation,collision and avalanche effect.2.We study the correlation measure of binary sequence E_p²by using the estimates for exponential sums of Fermat quotients.Our results show that the correlation measure of order 3 is quite good,but the 4-order correlation measure of the sequence is very large.3.By the properties of multiplicative character sums modulo p^m,we study the correlation of order 4 of these sequences constructed by the generalized cy-clotomy.The results show the correlation of order 4 of these sequences can be very large.Therefore,they may not be recommended for cryptography.