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Let \(R\) be a ring , we define the polynomial ring \(R[x]\), in the indeterminate \(x\), over \(R\) to be the ring of all polyonmials in \(x\) over \(R\); that is: \[ R[x]=\{ f(x)=a_0+a_1x+\dots+a_nx^n : a_0, \dots, a_n \in R \text{ and } n \in \Z^+\} \] made into a ring under polynomial addition and polynomial multiplication .

Let \(R\) be a ring , and \(x\) an indeterminate. We define a polynomial in \(x\), over \(R\) to be an expression of the form: \[ f(x)=a_0+a+1x+\dots+a_nx^n \] for some integer \(n \in \Z^+\), and where \(a_0, \dots, a_n \in R\). remark (sidenote: In most cases, the indeterminate \(x\) is understood and so we omit any explicit mention to it most of the time. )

We define the group of units of a ring with identity \(1 \neq 0\) to be the group \(\mathcal{U}(R)\) of all units of \(R\), under the binary operation \(\cdot:R \times R \rightarrow R\). remark (sidenote: For a field \(F\), we remark that \(\mathcal{U}(F)={F \backslash \{0\}}\), so we just denote the group of units of \(F\) by \(F^\ast\). Indeed, we observe that \(F\) is an Abelian group under \(+:F \times F \rightarrow F\), and \(F^\ast\) is an Abelian group under \(\cdot:F \times F \rightarrow F\).

Let \(R\) be a ring with identity \(1 \neq 0\). We call a non-zero element \(a \in R\) a unit if there exists a non-zero element \(b \in R\) for which \[ab=1\] remark (sidenote: The identity element \(1\) of \(R\) is by definition, a unit. )

We call a commuatative ring with identity \(1 \neq 0\) an integral domain if it contains no zero-divisors .

Let \(R\) be a ring . We call a non-zero element \(a \in R\) a zero-divisor if there exists a non-zero element \(b \in R\) for which \[ab=0\]

We define the empty set to be the set having no elements, and we denote it by: \(\emptyset\).

We call a set non-empty if it is not the empty set . remark (sidenote: That is, the set \(A\) is non-empty if it contains at-least one element. )

Let \(A\) be a set , and let \(\ast:A \times A \rightarrow A\) and \(\star:A \times A \rightarrow A\) be binary operations . We say that \(\star\) distributes over \(\ast\) provided that it left distributes and right distributes .

Let \(A\) be a set , and let \(\ast:A \times A \rightarrow A\) and \(\star:A \times A \rightarrow A\) be binary operations . We say that \(\star\) right distributes over \(\ast\) provided for any \(a,b,c \in A\): \[ (b \ast c) \star a=(b \star a) \ast (c \star a) \]

Let \(A\) be a set , and let \(\ast:A \times A \rightarrow A\) and \(\star:A \times A \rightarrow A\) be binary operations . We say that \(\star\) left distributes over \(\ast\) provided for any \(a,b,c \in A\): \[ a \star (b \ast c)=(a \star b) \ast (a \star c) \]

A field is a commutative division ring . remark (sidenote: A field can also be defined as an integral domain in which every element is a unit . )

A ring with identity \(R\) is called a division ring if \(1 \neq 0\), and every element \(a \in R\) has an inverse element under the operation \( \cdot: R \times R \rightarrow R \).

A ring \(R\) is said to have identity if \(R\) has an identity element under the binary operation \( \cdot: R \times R \rightarrow R \). We denote the identity element of \(R\) under \( \cdot: R \times R \rightarrow R \) by \(1\).

A commmutative ring is a ring \(R\) whose operation \(\cdot:R \times R \rightarrow R\) is commutative .

A ring is a non-empty set \(R\) together with two binary operations \(+:R \times R \rightarrow R\) and \(\cdot:R \times R \rightarrow R\) such that: (1). \(R\) is an Abelian group under the operation \(+:R \times R \rightarrow R\). Moreover the identity is denoted \(0\) and the inverse elements are denoted \(-a\), for all \(a \in R\). (2). For every \(a,b \in R\), \( a \cdot b \in R\), and the operation \(\cdot:R \times R \rightarrow R\) is associative .

An \((m,n,p)\)-function is called almost perfect non-linear (or APN) if it is differentially \(2\)-uniform

An \((m,n,p)\)-function is called perfect non-linear (or PN) if it is differentially \(p^{m-n}\)-uniform .

Let \(f:\mathbb{F}_{p^m} \rightarrow \mathbb{F}_{p^n}\) be an \((m,n,p)\)-function , and take \(a \in \mathbb{F}_{p^m}\). The derivative of \(f\) in the direction of \(a\) is defined to be: \[ D_a{f(x)}=f(x+a)-f(x) \]

Given \( \delta \in \Z^+ \), we call an \( (m,n,p) \)-function differentially \(\delta\)-uniform if for every nonzero \( a \in \mathbb{F}_{p^m} \), and for every \( b \in \mathbb{F}_{p^n} \), the difference equation: \[ D_a{f(x)}=b \] has at most \(\delta\)-solutions.