The Lucas-Pratt primality tree

In 1876, E. Lucas showed that a quick proof of primality for a prime $p$ could be attained through the prime factorization of $p-1$ and a primitive root for $p$. V. Pratt's proof that PRIMES is in NP, done via Lucas's theorem, showed that a certificate of primality for a prime $p$ could be obtained in $O(\log^2 p)$ modular multiplications with integers at most $p$. We show that for all constants $C \in \mathbb{R}$, the number of modular multiplications necessary to obtain this certificate is greater than $C \log p$ for a set of primes $p$ with relative asymptotic density 1.