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\section{Boolean Functions}  In this section, we will introduce some functions that are very important in information theory and in cryptography, i.e. \emph{Boolean functions}.\\  First we consider some examples. For \begin{Example}For  a given set of bits $S$ we define the "parity bit" a new bit that is $1$ if the number of bits $1$ is odd, $0$ otherwise. For example if $S$ has $3$ bits bits, that is $S=\{x,y,z\}$,  we have the function $f:(\Fb)^3 \rightarrow \Fb$, that is $f(x,y,z)=x+y+z$ where $x+y+z\in \Fb$. \end{Example}  Verify that the new bit $f(x,y,z)$ is the parity bit.  the occurrences of bits whose value is 1 is countedwe want to   Let $f:(\Fb)^3 \rightarrow \Fb$ be a polynomial function such that