Linear complexity of pseudorandom sequences generated by Fermat quotients and their generalizations

We use polynomial quotients modulo an odd prime p  , which are generalized from the Fermat quotients, to define two families of d(⩾2)-ary sequences of period p2p2. If d   is a primitive element modulo p2p2, we determine the minimal characteristic polynomials of the sequences and hence their linear complexities, which depend on whether p≡1p≡1 or 3 (mod 4). Moreover, we generalize the result to the polynomial quotients modulo a power of p.

► Use polynomial quotients to define binary threshold sequences. ► Combine polynomial quotients and multiplicative characters to define sequences. ► Determine the minimal polynomials and linear complexities of both sequences. ► Generalize the result to the polynomial quotients modulo a power of p.