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We define an \( (m,n) \)-Boolean function to be an \( (m,n,2) \)-function . We also call \( (m,n) \)-Boolean functions Substitution boxes (or S-boxes).

See \( (m,n) \)-Boolean function .

We define a Boolean function to be an \( (m,n) \)-Boolean function in which \(m=n\).

Let \( p,m,n \in \Z^+\), where \(p\) is prime. An \( (m,n,p) \)-function is just a function : \[ f:\mathbb{F}_{p^m} \rightarrow \mathbb{F}_{p^n} \]

The graph of a function \( f:X \rightarrow Y \) is defined to be the cartesian product \( X \times f(X) \), and is denoted by \(\mathcal{G}_{f}\). More precisely stated: \[ \mathcal{G}_f=\{ (x, f(x)) : x \in X \} \]

Let \(a\) and \(b\), \(c\) be arbitrary objects. We define the ordered triple \((a,b,c)\) to be the ordered pair \(((a,b),c)\). remark (sidenote: We remark that the term _object_ is taken to be undefined. )

The preimage of a function \(f:X \rightarrow Y\) is defined to be the set \(f^{-1}(Y)\) of all \(x \in X\) such that \(f(x) \in Y\). That is: \[ f^{-1}(Y)=\{ x \in X : f(x) \in Y \} \] remark (sidenote: We remark that the preimage of a function is always a subset of the domain of \(f\). That is: \(f^{-1}(Y) \subseteq X\) )

The image of a function \(f:X \rightarrow Y\) is defined to be the set \(f(X)\) of all \(f(x) \in Y\), given \(x \in X\). That is: \[ f(X)=\{ f(x) : x \in X \} \] remark (sidenote: We remark that the image of a function is always a subset of the codomain of \(f\). That is: \(f(X) \subseteq Y\) )

Let \(a\) and \(b\) be arbitrary objects. We define the ordered pair \((a,b)\) of \(a\) and \(b\) to be the set defined by: \[ (a,b)=\{\{a\}, \{a,b\}\} \] remark (sidenote: We remark that the term _object_ is taken to be undefined. )

Let \(A\) and \(B\) be sets . We define the cartesian product of \(A\) and \(B\) to be the set \(A \times B\) of all ordered pairs of elements from \(A\) and elements of \(B\). That is: \[ A \times B = \{(a,b) : a \in A \text{ and } b \in B\} \]

Let \(A\) and \(B\) be sets . We say that \(A\) is a subset of \(B\) provided every element of \(A\) is also a element of \(A\): that is \(A\) is a subset of \(B\) if, and only if for every \(a \in A\), we also have \(a \in B\). We write \(A \subseteq B\) to denote that \(A\) is a subset of \(B\).

A function is a triple \((X,Y,f)\) such that \(f \subseteq X \times Y\) and whenever \((x,y) \in f\) and \((x,y') \in f\), then we have \(y=y'\). We denote the triple \((X,Y,f)\) simply by \(f\), or by the notation: \(f:X \rightarrow Y\). We call \(X\) the domain of \(f\) and we call \(Y\) the codomain of \(f\). Given \((x,y) \in f\) we, write \(f(x)=y\).

See function

See function

A set is a an arbitrary collection of objects. We call the objects of a set elements. If \(A\) is a set, and \(a\) is a element of \(A\), then we write \(a \in A\). We remark that that the terms object and collection are to be taken without definition.

An Abelian group is a group \(G\) whose binary operation is commutative remark (sidenote: We remark that Abelian groups are also sometimes called commutative groups. )

Let \(A\) be a set . We call a binary operation \(\ast: A \times A \rightarrow A\) commutative if for every \(a,b \in A\): \[ a \ast b=b \ast a \]

Let \(A\) be a set . We call a binary operation \(\ast: A \times A \rightarrow A\) associative if for every \(a,b,c \in A\): \[ (a \ast b) \ast c = a \ast (b \ast c) \]

Let \(A\) be a set , and \(\ast:A \times A \rightarrow A\) a binary operation on \(A\). For any element \(a\) in \(A\), we define an inverse element of \(a\) to be an element \(b \in A\) such that: \[ a \ast b=b \ast a=e \] where \(e\) is defined to be an identity element of \(A\). remark (sidenote: We remark that when \(A\) is a group , then the inverse elements are unique.

Let \(A\) be a set , and \(\ast:A \times A \rightarrow A\) be a binary operation on \(A\). We define an identity element of \(A\) to be an element \(e \in A\) such that: \[ e \ast a=a \ast e=a \] for every \(a \in A\). remark (sidenote: We remark that when \(A\) is a group , then the identity element is unique. )