Posts

A group is a groupoid which is also a monoid .

A monoid is a category having only one object.

A groupoid is a category in which every morphism is an isomorphism .

Let \(\mathscr{C}\) be a category , and let \(a\) and \(b\) be object in \(\mathscr{C}\). We call a morphism \(f:a \rightarrow b\) an isomorphism between \(a\) and \(b\) provided there exists a morphism \(g:b \rightarrow a\) such that: \[ g \circ f=id_a \text{ and } f \circ g=id_b \] If such an isomorphism exists between \(a\) and \(b\), we call \(a\) and \(b\) isomorphic.

A category \(\mathscr{C}\) is a collection of objects, denoted \(\text{obj}{\mathscr{C}}\) together with a collection of classes of morphisms between pairs of objects, each of which is denoted \(\text{Mor}{(a,b)}\) for any pair \(a,b \in \text{obj}{\mathscr{C}}\), such that the following data is given: (1). For any objects \(a,b,c \in \text{obj}{\mathscr{C}}\), there exist composition maps \[ \circ:\text{Mor}{(b,c)} \times \text{Mor}{(a,b)} \rightarrow \text{Mor}{(a,c)} \] associating to each pair of morphisms \((g,f)\) a morphism \(g \circ f\) called the composition of the morphism \(g\) with the morphism \(f\).

We define the characteristic of a ring \(R\) to be the smallest such positive integer \(m \in \Z^+\) for which: \[ m \cdot 1=0 \] in \(R\). We denote the characteristic of \(R\) by: \(\text{char}{R}=m\). remark (sidenote: When \(R\) is an integral domain , then the characteristic of \(R\) must be a [prime integer](). )

We call a polynomial \(A(x) \in \mathbb{F}_q\) a affine polynomial if it can be written in the form: \[ A(x)=L(x)+c \text{ for some } c \in \mathbb{F}_q \] where \(L(x) \in \mathbb{F}_q[x]\) is a linearized polynomial .

We call a polynomial \(L(x) \in \mathbb{F}_q\) a linearized polynomial if it can be written in the form: \[ L(x)=\sum_{c_i \in \mathbb{F}_q}{c_ix^{p^i}} \] where \(\text{char}{\mathbb{F}_q}=p\) .

We call a linearized polynomial \(A(x)\) on \(\mathbb{F}_q\) a affine permutation if the evaluation map \[ c \rightarrow A(c) \] permutes \(\mathbb{F}_q\).

We call a linearized polynomial \(L(x)\) on \(\mathbb{F}_q\) a linear permutation if the evaluation map \[ c \rightarrow L(c) \] permutes \(\mathbb{F}_q\).

We call two \((m,n,p)\)-functions \(f(x)\) and \(g(x)\) extended affine equivalent (or EA-equivalent) provided there exist affine permutations \(A_1(x)\) and \(A_2(x)\), and an affine polynomial \(A(x)\) for which: \[ g(x)=A_1 \circ f \circ A_2(x)+A(x) \] remark (sidenote: We have the following implication: EA-equivalence \(\implies\) affine equivalence )

Two \((m,n,p)\)-functions \(f(x)\) \(g(x)\) and are called linear equivalent if there exits linearized polynomials \(L_1(x)\) and \(L_2(x)\) such that: \[ g(x)=L_1 \circ f \circ L_2(x) \]

Two \((m,n,p)\)-functions \(f(x)\) \(g(x)\) and are called affine equivalent if there exits affine polynomials \(A_1(x)\) and \(A_2(x)\) such that: \[ g(x)=A_1 \circ f \circ A_2(x) \] remark (sidenote: We have the following implication: Affine equivalence \(\implies\) linear equivalence )

We call two \((m,n,p)\)-functions \(f(x)\) and \(g(x)\) Carlet-Charpin-Zimoviev equivalent (or CCZ-equivalent) provided for any Affine Permutation \(\mathcal{L}\), the image of the graph of \(f(x)\) under \(\mathcal{L}\) is equal to the graph of \(g(x)\). That is: \[ \mathcal{L}(\mathcal{G}_f)=\mathcal{G}_g \] remark (sidenote: In general, CCZ-equivalence of two \((m,n,p)\)-functions is hard to prove. However, we do have the following implication: (1). CCZ-equivalence \(\implies\) EA-equivalence )

Let \[ f(x)=a_0+a_1x+\dots+a_nx^n \] be a polynomial over a ring . We define the degree of \(f(x)\) to be the highest power in the expression of \(f(x)\). That is, provided \(a_n \neq 0\), we denote \(\deg{f}=n\).

Let \(R\) be a ring, and let \(f(x)=a_0+a_1x+\dots+a_mx^m\) and \(g(x)=b_0+b_1x+\dots+b_nx^n\) polynomials over \(R\). We define _polynomial multiplication over \(R\) to be the binary operation \(\cdot\) given by the rule: \[ \cdot:(f(x),g(x)) \rightarrow fg(x)=f(x)g(x)=c_0+c_1x+\dots+c_kx^k \] where \[ c_j=\sum_{i=0}^j{a_ib_{j-1}} \text{ and } k=m+n \]

Let \(R\) be a ring, and let \(f(x)=a_0+a_1x+\dots+a_mx^m\) and \(g(x)=b_0+b_1x+\dots+b_nx^n\) polynomials over \(R\), with \(m \leq n\). We define polynomial addition over \(R\) to be the binary operation \(+\) given by the rule: \[ +:(f(x),g(x)) \rightarrow f+g(x)=f(x)+g(x)=c_0+c_1x+\dots+c_nx^n \] where \[ c_j=a_j+b_j \text{ and } a_j=0 \text{ for all } m < j \leq n \]

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 \(\cdot:F \times F \rightarrow F\).