We describe the trace representations of two families of binary sequences derived from Fermat quotients modulo an odd prime p (one is the binary threshold sequences, the other is the Legendre-Fermat quotient sequences) via determining the defining pairs of all binary characteristic sequences of cosets, which coincide with the sets of preimages modulo p 2 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 2 p−1 ≡ 1 mod p 2 .
Abstract.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 Sárközy. 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 Sárközy. Moreover, we analyze the lattice structure of Fermat quotients modulo p with arbitrary lags.
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