Pseudo-primality test for Mersenne numbers faster than Lucas-Lehmer test?

Question

Definition

Let $M_p=2^p-1$ with $p$ prime and $p>2$ .

Lucas-Lehmer Test

$M_p$ is prime if and only if $S_{p-2} \equiv 0 \pmod {M_p}$

where $S_{k+1}=S^2_{k}-2$ and $S_0=4$ .

Pseudo-Primality Test

If $M_p$ is prime then $3^{\frac{M_p-1}{2}} \equiv -1 \pmod {M_p}$

For proof see this question .

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Question

Is this pseudo-primality test faster than Lucas-Lehmer test ?

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My PARI/GP implementation of PPT (see below) is approximately $1.5$ times faster than PARI/GP implementation of LLT .

LLT(p)=
{
my(s=Mod(4,2^p-1)); 
for(i=1,p-2, s=s^2-2); 
if(s==0,print("prime"),print("composite"))
}

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PPT(p)=
{
M=2^p-1;
if(lift(Mod(3,M)^((M-1)/2))==M-1,print("probably prime"),print("composite"))
}

Answer 1

Both tests need about $p$ multiplications modulo $M_p$, so they should be equally fast.

The lift-command seems to improve the calculation of large powers modulo large numbers, probably the reason PARI/GP is faster using this method.

The disadvantage of the pseudoprime-test is, of course, that the result is not $100$% correct, in contrary to the Lucas-Lehmer-Test.