Correlation Immunity of AES SBox

Question

How do I calculate the correlation immunity of the standard AES Sbox. All material ive read gives out calculation of correlation immunity of boolean functions with single bit outputs. How can it be translated to the 8bit AES Sbox.

Answer 1

Not unexpectedly, the correlation immunity of a vectorial boolean function $$S:\mathbb{F}_2^n\rightarrow \mathbb{F}_2^m,$$ is defined as $$ CI(S)=\min \left\{ CI(a^{\top} \cdot S(x)): a \in \mathbb{F}_2^m, a\neq 0_m\right\}, $$ where $0_m$ is the all zero vector of length $m.$ Thus, it is the minimum correlation immunity of all the possible nonzero XOR sums of the output bits of $S$. In the absence of further knowledge about the Sbox, you need to compute a boolean correlation immunity $2^{m}-1,$ times and take the minimum.

In the Rijndael specification, as well as in the book \textit{The Design of Rijndael}, there is a nice chapter entitled Propagation and Correlation, where this is explored from an algebraic point of view. This helps in analysing compositions of functions, when one looks at multiple rounds or cascaded functions.