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\section{Multivariate polynomials on bits}  In many cryptographic applications, polynomials in only one variable are not enough and we need to deal with polynomials in two or more variables. In this section, we define such polynomials and we see how to sum and multiply them.  As for section \ref{Sec:Polynomials}, we start by defining terms (or monomials).  \\  We begin with two variables, $x$ and $y$.\\  A \emph{term} or \emph{monomial} in the $2$ variables $x$ and $y$ is a product of powers, i.e.   $x^{i} y^j$, for some $i,j$ in $\NN$.\\  For example, we can consider the following monomials $x^2y^3$ ($i=2,j=3$), $x^4$ ($i=4,j=0$), $y^7$ ($i=0,j=7$), $1$ ($i=j=0$).  Be careful that for example $x^(-4)$ is \textbf{not} a monomial.  \\  With monomials, we can define polynomials in the $2$ variables $x$ and $y$ and coefficients in $\Fb$ (i.e. multivariate polynomials in the field of bits) as sums of monomials (without coefficients). For example, $x^2y^3+x+x^10+y$, $xy+x^11y^2$ and $x^3y^5$ (a monomial is also a polynomial). \\  Formally, a polynomial in the $2$ variables $x$ and $y$ and coefficients in $\Fb$ is any   expression of the form  $$  f(x,y)=\sum_{i,j \in \NN} a_{i,j} x^{i}\cdots y^{j}   $$   such that  \begin{itemize}  \item for each $i,j$ in $\NN$, $ a_{i,j} \in \Fb$,  \item only a \textbf{finite} number of coefficients is nonzero.   \end{itemize}  The set containing all polynomials in $x,y$ with bits as coefficient is denoted by $\Fb[x,y]$.\\  We can notice that the definition of multivariate polynomial over the field of bits is the analogous of that over $\RR$.  \  A \emph{term} or \emph{monomial} in the  $n$ variables $x_1,...x_n$ is a product of powers of $x_1,...x_n$, i.e. $x_1^{i_1}\cdots x_n^{i_n}$ for some $i_1,...,i_n \in \NN$.\\  For example, for three variables $x_1,x_2,x_3$, it is clear that $x_1^3x_2^8x_3^6$, $x_2x_3^4=x_1^0x_2x_3^4$ ,$x_1^4=x_1^4x_2^0x_3^0$ are all terms.  \\