Binary Sequences Derived From Differences of Consecutive Primitive Roots

Let 1 <; g
₁
<; ⋯ <; g
_φ(p-1)
<; p-1 be the ordered primitive roots modulo p. We study the pseudorandomness of the binary sequence (s
_n
) defined by s
_n
≡ g
_n+1
+g
_n+2
mod 2, n=0,1,⋯ In particular, we study the balance, linear complexity and 2-adic complexity of (s
_n
). We show that for a typical p the sequence (s
_n
) is quite unbalanced. However, there are still infinitely many p such that (s
_n
) is very balanced. We also prove similar results for the distribution of longer patterns. Moreover, we give general lower bounds on the linear complexity and 2-adic complexity of (s
_n
) and state sufficient conditions for attaining their maximums. Hence, for carefully chosen p, these sequences are attractive candidates for cryptographic applications.