deletions | additions       diff --git a/bits4.tex b/bits4.tex  index 01f99e8..8c62586 100644  --- a/bits4.tex  +++ b/bits4.tex 
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from $(\Fb)^3$ to $\Fb$ can be written as a polynomial $h \in \Fb[x,y,z]$ where   $0 \leq \deg_x(h),\deg_y(h),\deg_z(h) \leq 1$, for example,  $x^3 y^2 + z^{10} xy^2 = xy + zxy$, as a Boolean function.  \begin{Definition}\label{ANF} \begin{Definition}\label{Sqfree3}  A \emph{squarefree} monomial in $\Fb[x,y,z]$  is a monomial where each variable appears with exponent at most one. \end{Definition}  Even more generally, it is possible to prove that \textbf{every function} from   $(\Fb)^3$ to $\Fb$ can be written in a polynomial form, called the 
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The following function, often used in cryptography, is called the \emph{majority function}:  $$f: (\Fb)^3 \rightarrow \Fb $$  $$(\bar{x},\bar{y},\bar{z})\mapsto \bar{x}\bar{y}+\bar{x}\bar{z}+\bar{y}\bar{z}.$$  This function returns value $1$ if and only if the majority at least two  among the input bits $\{\bar{x},\bar{y},\bar{z}\}$ holds hold  value $1$. Obviously, it returns $0$ otherwise, i.e. if and only if the majority at least two  among the three input bits holds value $0$. \begin{Exercise}  Verify the above statement, by evaluating $f$ at all $3$-tuples of bits.  \end{Exercise}  We need to consider $\Fb[x_1,...,x_n]$ and so we can generalize definition \ref{Sqfree3} to the following  \begin{Definition}\label{Sqfreen}  A \emph{squarefree} monomial $\mathbf{t}$ in $\Fb[x_1,...,x_n]$ is a monomial where each variable appears with exponent at most one, that is, $0 \leq \deg_{x_1}(\mathbf{t}),\deg_{x_2}(\mathbf{t}),...,\deg_{x_n}(\mathbf{t}) \leq 1$.  \end{Definition}  Until now we have only given results and examples where a function has at most $3$ bits in input.  We would like to consider the most general case, where the number of bits in input is an arbitrary integer $n\geq 1$.