A peaceful night for you,

you are right, but this is not really new for me.

You can transform the polynom for pollard rho in a 2x2 matrix

and calculate the 2x2 matrix with fast exponentation for the

primes < 10^9 for example.

You can use a subgroup either a vektor consisting of a pythagoraic triple

either a vektor base on the pell equation.

Nevertheless it is either a p-1 or a p+1 test,

I think with 40 digits it will go.

But elliptic curves are better because of its various group structure.

Nice greetings from the factoring part

Bernhard