# How is the 'Algebraic Degree' calculated in the paper about analysing the white-box AES(Chow et al. 2002) by exploiting internal collisions?

This paper proposed a new attack on the initial white-box AES implementation of Chow et al.

In order to determine the good solution, we use the particular structure of the function $$S_{0}$$.

$$S^{-1} \circ S_{0} \left( \cdot \right) = P_{0}\left( \cdot \right)\oplus k_{0}$$

By definition of $$P_{0}$$, the above function has algebraic degree at most 4.

According to this paper, $$P_{0}$$ denotes bijective mapping on the vector space $$\mathbb{F}_{2}^{8}$$, which is the combination of two 4-bit input encodings and one 8 $$\times$$ 8-bit mixing bijection.

Given the next proposed Lemma 2 in this paper, I know the algebraic degree is about boolean function, then I study the boolean function and high-order derivations of Lai 1994.

But I still can't understand why the algebraic degree at most 4 by definition of $$P_{0}$$?