I consider polnomial $T(x)$ defined over a finite field $\mathbb{F}_p$ where $p$ is a large prime number (256-bit). I want to factorize the polynomials over the finit field. The dgree of $T(x)$ is $c\cdot d$. I have $c$ number of polynomials $G_i(x)$ each of which has degree $d$ and they are also defined over the field.
Question: Which factorization is faster? factorizing polynomial $T(x)$ or factorizing all $c$ number of polynomials $G_i(x)$ ?
My Goal: As degree of the original polynomial $T(x)$ is very large $c \cdot d$, and factoring it may be too slow, I want to see if I break it up into smaller degree $d$ polynomals and factorize the polynomials I can speed up the factorizing the original polynomial $T(x)$.
Factoring $c$ degree $d$ polynomials is clearly faster. But your idea doesn't make a lot of sense, you would have to do considerable amount of work to factor a degree $c \cdot d$ polynomial into $c$ degree $d$ polynomials.
In the case that you have a polynomial of degree $cd$, and a method of factoring it into $c$ degree $d$ polynomials, then you already have your polynomial of degree $cd$ partially factored. You definitely don't want to multiply these factors together to get the big polynomial, then factor. If you have the small polynomials, a significant piece of the work you need to do is already done.