New Sequences of Period p n and p n + 1 via Projective Linear Groups

Two pseudorandom number generators are devised based on the projective linear group over \(\mathbb{F}_{p^n}\), outputting balanced sequences on \(\mathbb{F}_{p}\) meeting some statistical randomness properties. Sequences generated by the first generator have least period p n + 1 if p n ≥ 7, and linear complexity at least p n − p n − 1. Furthermore, autocorrelation of such sequences oscillates within a low amplitude except for the trivial peaks. If p n ∉ {2,4,8,16}, sequences generated by the second generator have least period p n , linear complexity at least p n − 1 + 1, and good k-error linear complexity when p = 2. If p = 2 and 2 n is large enough, then for a binary sequence generated by either generator, a randomly chosen 2-tuple is almost uniformly distributed in {00,01,10,11}, the probability that a randomly chosen 3-tuple is a run of length one is approximately 1/4. For such a binary sequence \(\vec{s}\) and integers 0 < i 1 < i 2 < ⋯ < i k ≤ m, s(t) + s(t + i 1) + s(t + i 2) + ⋯ + s(t + i k ) is equal to 0 or 1 at almost the same probability when m is far less than 2 n/2.