Zhixiong Chen

Trace representation and linear complexity of binary sequences derived from Fermat quotients

We describe the trace representations of two families of binary sequences derived from the Fermat quotients modulo an odd prime p (one is the binary threshold sequences and the other is the Legendre Fermat quotient sequences) by determining the defining pairs of all binary characteristic sequences of cosets, which coincide with the sets of pre-images modulo p2 of each fixed value of Fermat quotients. From the defining pairs, we can obtain an earlier result of linear complexity for the binary threshold sequences and a new result of linear complexity for the Legendre Fermat quotient sequences under the assumption of 2p−1 ≢ 1 mod p2.

Structure of pseudorandom numbers derived from fermat quotients

We study the distribution of s-dimensional points of Fermat quotients modulo p with arbitrary lags. If no lags coincide modulo p the same technique as in [21] works. However, there are some interesting twists in the other case. We prove a discrepancy bound which is unconditional for s = 2 and needs restrictions on the lags for s > 2.We apply this bound to derive results on the pseudorandomness of the binary threshold sequence derived from Fermat quotients in terms of bounds on the well-distribution measure and the correlation measure of order 2, both introduced by Mauduit and Sarkozy. We also prove a lower bound on its linear complexity profile. The proofs are based on bounds on exponential sums and earlier relations between discrepancy and both measures above shown by Mauduit, Niederreiter and Sarkozy. Moreover, we analyze the lattice structure of Fermat quotients modulo p with arbitrary lags.

On the k -error linear complexity of binary sequences derived from polynomial quotients

The k-error linear complexity is an important cryptographic measure of pseudorandom sequences in stream ciphers. In this paper, we investigate the k-error linear complexity of p2-periodic binary sequences defined from the polynomial quotients modulo p, which are the generalizations of the well-studied Fermat quotients. Indeed, first we determine exact values of the k-error linear complexity over the finite field

Some Notes on Generalized Cyclotomic Sequences of Length pq

\mathbb{F}_2

Finite binary sequences constructed by explicit inversive methods

for these binary sequences under the assumption of 2 being a primitive root modulo p2, and then we determine their k-error linear complexity over the finite field

ON THE DISTRIBUTION OF PSEUDORANDOM NUMBERS AND VECTORS DERIVED FROM EULER–FERMAT QUOTIENTS

\mathbb{F}_p