  Let $1<g_1<\ldots<g_{\varphi(p-1)}<p-1$ be the ordered primitive roots
modulo~$p$. We study the pseudorandomness of the binary sequence $(s_n)$
defined by $s_n\equiv g_{n+1}+g_{n+2}\bmod 2$, $n=0,1,\ldots$. 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.
