I've thought up a way to represent the transformation of S-Box in DES by ANF. Let $x_i\;(1\le i\le 6)$ be the input of an S-Box, $y_i\;(1\le i\le 4)$ be the output, for example, then $$y_1= 1\oplus x_1\oplus\dots\oplus x_1x_5\oplus\dots\oplus x_3x_4x_5x_6\oplus\cdots$$ Here the "$\oplus$" is "xor" operation and the operation between two continuous inputs is "and". I've omitted many terms because it's too long.
(This paragraph is in the old version)I think this is a very basic problem so it must be researched by so many people. But I can't find related papers. Can someone explain the ANF researches on S-Box of DES or provide some resources for me? Thanks a lot.
This is an explict expression because the left side is a single output and the right side is a polynomial of inputs with degree at most 6.
In algebraic cryptanalysis, reaserchers always represent the relations between inputs and outputs in an implicit way, like $$f(x_1,x_2,x_3,x_4,x_5,x_6,y_1,y_2,y_3,y_4)=0$$ What I want to know is that are there any advantages or applications of explicit form compared with the implict form?