feat: make 信安数学基础2 paper
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---
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title: 2024Fall_期末
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category:
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- 软件工程学院
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- 课程资料
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tag:
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- 试卷
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author:
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- タクヤマ
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---
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## 说明
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本学期（2024年秋季学期）的期末试卷难度**相当**低，本人评价为有手就行，故没有特意记忆所有题目，而是挑出一些有代表性的题目以供参考
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我们这里再提供一道填空题，一道选择题，以显示这张卷子的**诚意**
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<ol>
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<li>
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设 $G$ 是一个群，对于 $\forall a,b \in G$， $a*b=b*a$， 则 $G$ 是 ____ 群
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</li>
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<li>
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设 $R$ 是一个环，对于 $\forall a,b \in R$， $a+b=b+a$， 则 $R$ 是？
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A. 整环 B. 交换环 C. 含幺环 D. 环
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</li>
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</ol>
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很大程度上，老师的出题和给分是十分宽容的，请大家好好学习这门课程，**数学并不是妖魔鬼怪！**
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## 题目
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> **1. 设 $G=\{a,b,c,e\}$ 是一个群， $H=\{a,e\}$ 是 $G$ 的子群，写出 $H$ 的所有左陪集**
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<details>
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<summary>解：</summary>
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将 $G$ 中元素 $g$ 各个代入，计算 $gH$
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- $g = e$：
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$eH = H$
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- $g = a$：
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$aH = \{ a \cdot a, a \cdot e \} = \{ a^2, a \}$，由于 $G$ 是群，且 $H$ 是子群，所以 $a^2$ 必须是 $G$ 中元素。
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故 $a^2 = e$，则： $aH = \{ e, a \} = H$
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- $g = b$：
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$bH = \{ b \cdot a, b \cdot e \} = \{ ba, b \}$，
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假设 $ba = c$ ，则：
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$bH = \{ c, b \}$
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- $g = c$：
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$cH = \{ c \cdot a, c \cdot e \} = \{ ca, c \}$，
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假设 $ca = b$ ，则：
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$cH = \{ b, c \} = bH$
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综上所述，H的所有子陪集是 $\{ e, a \}, \{ b, c \}$
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</details>
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***
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> **2. 写出 $R=Z/6Z$ 的所有零因子**
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<details>
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<summary>解：</summary>
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$R=Z/6Z \cong Z_6$，我们只要考虑 $Z_6$ 上的性质：
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显然有 $2\*3 \equiv 0 \pmod{6}, 4*3 \equiv 0 \pmod{6}$，所以零因子是2，3，4
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</details>
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***
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> **3. 定义在有限域 $F_{17}$ 上的椭圆曲线 $E: x^3 + 2x + 3 = y^2$ 上有点 $P(2, 7), Q(11, 8)$ , 计算 $P+Q$ , $2P$**
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<details>
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<summary>解：</summary>
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直接计算即可，我们这里直接给出答案： $P+Q = (8, 15)$ , $2P = (14, 15)$
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</details>
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***
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> **4. 求域 $F_{16}=F_2[x]/(x^4+x^3+1)$的一个生成元 $g(x)$，并用 $g(x)$ 的幂表示 $F_{16}$ 中的所有非零元**
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<details>
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<summary>解：</summary>
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这时候有同学要问了，生成元怎么找啊？其实很简单，直接验证就可以了，（出于强大的直觉和观察力）我们在这里直接验证 $x$ :
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| $x^n$ | $x^n\pmod{x^4+x^3+1}$ |
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| -------- | --------------------- |
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| $x^0$ | $1$ |
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| $x^1$ | $x$ |
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| $x^2$ | $x^2$ |
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| $x^3$ | $x^3$ |
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| $x^4$ | $x^3 + 1$ |
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| $x^5$ | $x^3 + x + 1$ |
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| $x^6$ | $x^3 + x^2 + x + 1$ |
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| $x^7$ | $1 + x^2 + x$ |
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| $x^8$ | $x^2 + x + 1$ |
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| $x^9$ | $x^3 + x^2 + x$ |
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| $x^{10}$ | $x^3 + x^2 + x + 1$ |
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| $x^{11}$ | $x^3 + x + 1$ |
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| $x^{12}$ | $x^3 + 1$ |
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| $x^{13}$ | $x^3$ |
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| $x^{14}$ | $x^2$ |
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如上表，由 $x$ 生成的15个非零元素互不相等，所以 $x$ 确实是生成元，非零元的表示如表中所示
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</details>
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***
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> **5. 证明：有限环的特征一定不为0**
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证明：请参考课件Chap7.pdf中27-28页
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@ -1,67 +0,0 @@
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# 答案
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##
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### 1.设 $G=\{a,b,c,e\}$ 是一个群， $H=\{a,e\}$ 是 $G$ 的子群，写出 $H$ 的所有左陪集
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解：
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将 $G$ 中元素 $g$ 各个代入，计算 $gH$
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- $g = e$：
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$eH = H$
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- $g = a$：
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$aH = \{ a \cdot a, a \cdot e \} = \{ a^2, a \}$，由于 $G$ 是群，且 $H$ 是子群，所以 $a^2$ 必须是 $G$ 中元素。
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故 $a^2 = e$，则： $aH = \{ e, a \} = H$
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- $g = b$：
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$bH = \{ b \cdot a, b \cdot e \} = \{ ba, b \}$，
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假设 $ba = c$ ，则：
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$bH = \{ c, b \}$
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- $g = c$：
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$cH = \{ c \cdot a, c \cdot e \} = \{ ca, c \}$，
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假设 $ca = b$ ，则：
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$cH = \{ b, c \} = bH$
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综上所述，H的所有子陪集是 $\{ e, a \}, \{ b, c \}$
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**2.写出 $R=Z/6Z$ 的所有零因子**
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解：
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$R=Z/6Z \cong Z_6$，我们只要考虑 $Z_6$ 上的性质：
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显然有 $2\*3 \equiv 0 \pmod{6}, 4*3 \equiv 0 \pmod{6}$，所以零因子是2，3，4
|
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|
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**3.定义在有限域 $F_{17}$ 上的椭圆曲线 $E: x^3 + 2x + 3 = y^2$ 上有点 $P(2, 7), Q(11, 8)$ , 计算 $P+Q$ , $2P$**
|
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解：
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直接计算即可，我们这里直接给出答案： $P+Q = (8, 15)$ , $2P = (14, 15)$
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**4.求域 $F_{16}=F_2[x]/(x^4+x^3+1)$的一个生成元 $g(x)$，并用 $g(x)$ 的幂表示 $F_{16}$ 中的所有非零元**
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解：
|
|
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这时候有同学要问了，生成元怎么找啊？其实很简单，直接验证就可以了，（出于强大的直觉和观察力）我们在这里直接验证 $x$ :
|
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| $x^n$ | $x^n\pmod{x^4+x^3+1}$ |
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|------|----------|
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| $x^0$ | $1$ |
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| $x^1$ | $x$ |
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| $x^2$ | $x^2$ |
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| $x^3$ | $x^3$ |
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| $x^4$ | $x^3 + 1$ |
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| $x^5$ | $x^3 + x + 1$ |
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| $x^6$ | $x^3 + x^2 + x + 1$ |
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| $x^7$ | $1 + x^2 + x$ |
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| $x^8$ | $x^2 + x + 1$ |
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| $x^9$ | $x^3 + x^2 + x$ |
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| $x^{10}$ | $x^3 + x^2 + x + 1$ |
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| $x^{11}$ | $x^3 + x + 1$ |
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| $x^{12}$ | $x^3 + 1$ |
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| $x^{13}$ | $x^3$ |
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| $x^{14}$ | $x^2$ |
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如上表，由 $x$ 生成的15个非零元素互不相等，所以 $x$ 确实是生成元，非零元的表示如表中所示
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### 5.证明：有限环的特征一定不为0
|
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证明：请参考课件Chap7.pdf中27-28页
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@ -1,19 +0,0 @@
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## 说明：
|
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||||||
本学期（2024年秋季学期）的期末试卷难度**相当**低，本人评价为有手就行，故没有特意记忆所有题目，而是挑出一些有代表性的题目以供参考
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|
|
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<ol>
|
|
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<li>设 $G=\{a,b,c,e\}$ 是一个群， $H=\{a,e\}$ 是 $G$ 的子群，写出 $H$ 的所有左陪集</li>
|
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<li>写出 $R=Z/6Z$ 的所有零因子</li>
|
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<li>定义在有限域 $F_{17}$ 上的椭圆曲线 $E: x^3 + 2x + 3 = y^2$ 上有点 $P(2, 7), Q(11, 8)$ , 计算 $P+Q$ , $2P$</li>
|
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<li>求域 $F_{16}=F_2[x]/(x^4+x^3+1)$的一个生成元 $g(x)$，并用 $g(x)$ 的幂表示 $F_{16}$ 中的所有非零元</li>
|
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<li>证明：有限环的特征一定不为0</li>
|
|
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</ol>
|
|
||||||
|
|
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我们这里再提供一道填空题，一道选择题，以显示这张卷子的**诚意**
|
|
||||||
<ol>
|
|
||||||
<li>设 $G$ 是一个群，对于 $\forall a,b \in G$， $a*b=b*a$， 则 $G$ 是 ____ 群</li>
|
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<li>设 $R$ 是一个环，对于 $\forall a,b \in R$， $a+b=b+a$， 则 $R$ 是？
|
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A. 整环 B.交换环 C.含幺环 D.环</li>
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</ol>
|
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||||||
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||||||
很大程度上，老师的出题和给分是十分宽容的，请大家好好学习这门课程，**数学并不是妖魔鬼怪！**
|
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@ -1,5 +1,5 @@
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---
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---
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title: 信息安全数学基础（二）
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title: 形式语言与自动机理论
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category:
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category:
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- 软件工程学院
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- 软件工程学院
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- 课程评价
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- 课程评价
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@ -9,5 +9,4 @@ tag:
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- 抽象代数
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- 抽象代数
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---
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---
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### 这门课是抽象代数，但是并没有涉及到抽象代数真正复杂的部分，较为浅尝辄止，所以课程本身的学习难度并不高。
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### 同样地，这门课实际考试不难，老师会捞，请大家不要害怕。
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@ -1846,76 +1846,76 @@
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"dev": true,
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"dev": true,
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"dev": true,
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|
||||||
"integrity": "sha512-HIca0daEySJ8zuy9bdrtcBPhcYBo8wR1dyHk1vKrOuwDsITtZuQeGhEkcEfWc6IDyTcom7LRFCH6P7ljGSCEiQ==",
|
"integrity": "sha512-Y1MbMfVO5baRz7Boo7EoD36TmzfUx/I5n8e+wZumx6SlUA81Zj1ZwNJL871iIuSHrdsheV4AxJtHQ9mlooklmg==",
|
||||||
"dev": true,
|
"dev": true,
|
||||||
"license": "MIT",
|
"license": "MIT",
|
||||||
"dependencies": {
|
"dependencies": {
|
||||||
"@shikijs/types": "3.12.0"
|
"@shikijs/types": "3.12.1"
|
||||||
}
|
}
|
||||||
},
|
},
|
||||||
"node_modules/@shikijs/themes": {
|
"node_modules/@shikijs/themes": {
|
||||||
"version": "3.12.0",
|
"version": "3.12.1",
|
||||||
"resolved": "https://registry.npmjs.org/@shikijs/themes/-/themes-3.12.0.tgz",
|
"resolved": "https://registry.npmjs.org/@shikijs/themes/-/themes-3.12.1.tgz",
|
||||||
"integrity": "sha512-/lxvQxSI5s4qZLV/AuFaA4Wt61t/0Oka/P9Lmpr1UV+HydNCczO3DMHOC/CsXCCpbv4Zq8sMD0cDa7mvaVoj0Q==",
|
"integrity": "sha512-9JrAm9cA5hqM/YXymA3oAAZdnCgQf1zyrNDtsnM105nNEoEpux4dyzdoOjc2KawEKj1iUs/WH2ota6Atp7GYkQ==",
|
||||||
"dev": true,
|
"dev": true,
|
||||||
"license": "MIT",
|
"license": "MIT",
|
||||||
"dependencies": {
|
"dependencies": {
|
||||||
"@shikijs/types": "3.12.0"
|
"@shikijs/types": "3.12.1"
|
||||||
}
|
}
|
||||||
},
|
},
|
||||||
"node_modules/@shikijs/transformers": {
|
"node_modules/@shikijs/transformers": {
|
||||||
"version": "3.12.0",
|
"version": "3.12.1",
|
||||||
"resolved": "https://registry.npmjs.org/@shikijs/transformers/-/transformers-3.12.0.tgz",
|
"resolved": "https://registry.npmjs.org/@shikijs/transformers/-/transformers-3.12.1.tgz",
|
||||||
"integrity": "sha512-HcJwlvMAyZzOY+ayEAGE891BdJ7Vtio+qdWUTF9ki4d0LIkDb6DBz8ynOWGAEglHv6eQs/WcAWf/h6ina6IgCw==",
|
"integrity": "sha512-crGh3cSZf6mwg3K2W8i79Ja+q4tVClRHdHLnUGi5arS58+cqdzsbkrEZBDMyevf9ehmjFUWDTEwCMEyp9I3z0g==",
|
||||||
"dev": true,
|
"dev": true,
|
||||||
"license": "MIT",
|
"license": "MIT",
|
||||||
"dependencies": {
|
"dependencies": {
|
||||||
"@shikijs/core": "3.12.0",
|
"@shikijs/core": "3.12.1",
|
||||||
"@shikijs/types": "3.12.0"
|
"@shikijs/types": "3.12.1"
|
||||||
}
|
}
|
||||||
},
|
},
|
||||||
"node_modules/@shikijs/types": {
|
"node_modules/@shikijs/types": {
|
||||||
"version": "3.12.0",
|
"version": "3.12.1",
|
||||||
"resolved": "https://registry.npmjs.org/@shikijs/types/-/types-3.12.0.tgz",
|
"resolved": "https://registry.npmjs.org/@shikijs/types/-/types-3.12.1.tgz",
|
||||||
"integrity": "sha512-jsFzm8hCeTINC3OCmTZdhR9DOl/foJWplH2Px0bTi4m8z59fnsueLsweX82oGcjRQ7mfQAluQYKGoH2VzsWY4A==",
|
"integrity": "sha512-Is/p+1vTss22LIsGCJTmGrxu7ZC1iBL9doJFYLaZ4aI8d0VDXb7Mn0kBzhkc7pdsRpmUbQLQ5HXwNpa3H6F8og==",
|
||||||
"dev": true,
|
"dev": true,
|
||||||
"license": "MIT",
|
"license": "MIT",
|
||||||
"dependencies": {
|
"dependencies": {
|
||||||
|
|
@ -2912,6 +2912,16 @@
|
||||||
"vuepress": "2.0.0-rc.24"
|
"vuepress": "2.0.0-rc.24"
|
||||||
}
|
}
|
||||||
},
|
},
|
||||||
|
"node_modules/@vuepress/plugin-redirect/node_modules/commander": {
|
||||||
|
"version": "14.0.0",
|
||||||
|
"resolved": "https://registry.npmjs.org/commander/-/commander-14.0.0.tgz",
|
||||||
|
"integrity": "sha512-2uM9rYjPvyq39NwLRqaiLtWHyDC1FvryJDa2ATTVims5YAS4PupsEQsDvP14FqhFr0P49CYDugi59xaxJlTXRA==",
|
||||||
|
"dev": true,
|
||||||
|
"license": "MIT",
|
||||||
|
"engines": {
|
||||||
|
"node": ">=20"
|
||||||
|
}
|
||||||
|
},
|
||||||
"node_modules/@vuepress/plugin-rtl": {
|
"node_modules/@vuepress/plugin-rtl": {
|
||||||
"version": "2.0.0-rc.112",
|
"version": "2.0.0-rc.112",
|
||||||
"resolved": "https://registry.npmjs.org/@vuepress/plugin-rtl/-/plugin-rtl-2.0.0-rc.112.tgz",
|
"resolved": "https://registry.npmjs.org/@vuepress/plugin-rtl/-/plugin-rtl-2.0.0-rc.112.tgz",
|
||||||
|
|
@ -3070,15 +3080,15 @@
|
||||||
}
|
}
|
||||||
},
|
},
|
||||||
"node_modules/@vueuse/core": {
|
"node_modules/@vueuse/core": {
|
||||||
"version": "13.8.0",
|
"version": "13.9.0",
|
||||||
"resolved": "https://registry.npmjs.org/@vueuse/core/-/core-13.8.0.tgz",
|
"resolved": "https://registry.npmjs.org/@vueuse/core/-/core-13.9.0.tgz",
|
||||||
"integrity": "sha512-rmBcgpEpxY0ZmyQQR94q1qkUcHREiLxQwNyWrtjMDipD0WTH/JBcAt0gdcn2PsH0SA76ec291cHFngmyaBhlxA==",
|
"integrity": "sha512-ts3regBQyURfCE2BcytLqzm8+MmLlo5Ln/KLoxDVcsZ2gzIwVNnQpQOL/UKV8alUqjSZOlpFZcRNsLRqj+OzyA==",
|
||||||
"dev": true,
|
"dev": true,
|
||||||
"license": "MIT",
|
"license": "MIT",
|
||||||
"dependencies": {
|
"dependencies": {
|
||||||
"@types/web-bluetooth": "^0.0.21",
|
"@types/web-bluetooth": "^0.0.21",
|
||||||
"@vueuse/metadata": "13.8.0",
|
"@vueuse/metadata": "13.9.0",
|
||||||
"@vueuse/shared": "13.8.0"
|
"@vueuse/shared": "13.9.0"
|
||||||
},
|
},
|
||||||
"funding": {
|
"funding": {
|
||||||
"url": "https://github.com/sponsors/antfu"
|
"url": "https://github.com/sponsors/antfu"
|
||||||
|
|
@ -3088,9 +3098,9 @@
|
||||||
}
|
}
|
||||||
},
|
},
|
||||||
"node_modules/@vueuse/metadata": {
|
"node_modules/@vueuse/metadata": {
|
||||||
"version": "13.8.0",
|
"version": "13.9.0",
|
||||||
"resolved": "https://registry.npmjs.org/@vueuse/metadata/-/metadata-13.8.0.tgz",
|
"resolved": "https://registry.npmjs.org/@vueuse/metadata/-/metadata-13.9.0.tgz",
|
||||||
"integrity": "sha512-BYMp3Gp1kBUPv7AfQnJYP96mkX7g7cKdTIgwv/Jgd+pfQhz678naoZOAcknRtPLP4cFblDDW7rF4e3KFa+PfIA==",
|
"integrity": "sha512-1AFRvuiGphfF7yWixZa0KwjYH8ulyjDCC0aFgrGRz8+P4kvDFSdXLVfTk5xAN9wEuD1J6z4/myMoYbnHoX07zg==",
|
||||||
"dev": true,
|
"dev": true,
|
||||||
"license": "MIT",
|
"license": "MIT",
|
||||||
"funding": {
|
"funding": {
|
||||||
|
|
@ -3098,9 +3108,9 @@
|
||||||
}
|
}
|
||||||
},
|
},
|
||||||
"node_modules/@vueuse/shared": {
|
"node_modules/@vueuse/shared": {
|
||||||
"version": "13.8.0",
|
"version": "13.9.0",
|
||||||
"resolved": "https://registry.npmjs.org/@vueuse/shared/-/shared-13.8.0.tgz",
|
"resolved": "https://registry.npmjs.org/@vueuse/shared/-/shared-13.9.0.tgz",
|
||||||
"integrity": "sha512-x4nfM0ykW+RmNJ4/1IzZsuLuWWrNTxlTWUiehTGI54wnOxIgI9EDdu/O5S77ac6hvQ3hk2KpOVFHaM0M796Kbw==",
|
"integrity": "sha512-e89uuTLMh0U5cZ9iDpEI2senqPGfbPRTHM/0AaQkcxnpqjkZqDYP8rpfm7edOz8s+pOCOROEy1PIveSW8+fL5g==",
|
||||||
"dev": true,
|
"dev": true,
|
||||||
"license": "MIT",
|
"license": "MIT",
|
||||||
"funding": {
|
"funding": {
|
||||||
|
|
@ -3591,13 +3601,13 @@
|
||||||
}
|
}
|
||||||
},
|
},
|
||||||
"node_modules/commander": {
|
"node_modules/commander": {
|
||||||
"version": "14.0.0",
|
"version": "8.3.0",
|
||||||
"resolved": "https://registry.npmjs.org/commander/-/commander-14.0.0.tgz",
|
"resolved": "https://registry.npmjs.org/commander/-/commander-8.3.0.tgz",
|
||||||
"integrity": "sha512-2uM9rYjPvyq39NwLRqaiLtWHyDC1FvryJDa2ATTVims5YAS4PupsEQsDvP14FqhFr0P49CYDugi59xaxJlTXRA==",
|
"integrity": "sha512-OkTL9umf+He2DZkUq8f8J9of7yL6RJKI24dVITBmNfZBmri9zYZQrKkuXiKhyfPSu8tUhnVBB1iKXevvnlR4Ww==",
|
||||||
"dev": true,
|
"dev": true,
|
||||||
"license": "MIT",
|
"license": "MIT",
|
||||||
"engines": {
|
"engines": {
|
||||||
"node": ">=20"
|
"node": ">= 12"
|
||||||
}
|
}
|
||||||
},
|
},
|
||||||
"node_modules/connect-history-api-fallback": {
|
"node_modules/connect-history-api-fallback": {
|
||||||
|
|
@ -4533,16 +4543,6 @@
|
||||||
"katex": "cli.js"
|
"katex": "cli.js"
|
||||||
}
|
}
|
||||||
},
|
},
|
||||||
"node_modules/katex/node_modules/commander": {
|
|
||||||
"version": "8.3.0",
|
|
||||||
"resolved": "https://registry.npmjs.org/commander/-/commander-8.3.0.tgz",
|
|
||||||
"integrity": "sha512-OkTL9umf+He2DZkUq8f8J9of7yL6RJKI24dVITBmNfZBmri9zYZQrKkuXiKhyfPSu8tUhnVBB1iKXevvnlR4Ww==",
|
|
||||||
"dev": true,
|
|
||||||
"license": "MIT",
|
|
||||||
"engines": {
|
|
||||||
"node": ">= 12"
|
|
||||||
}
|
|
||||||
},
|
|
||||||
"node_modules/kind-of": {
|
"node_modules/kind-of": {
|
||||||
"version": "6.0.3",
|
"version": "6.0.3",
|
||||||
"resolved": "https://registry.npmjs.org/kind-of/-/kind-of-6.0.3.tgz",
|
"resolved": "https://registry.npmjs.org/kind-of/-/kind-of-6.0.3.tgz",
|
||||||
|
|
@ -5929,18 +5929,18 @@
|
||||||
"license": "ISC"
|
"license": "ISC"
|
||||||
},
|
},
|
||||||
"node_modules/shiki": {
|
"node_modules/shiki": {
|
||||||
"version": "3.12.0",
|
"version": "3.12.1",
|
||||||
"resolved": "https://registry.npmjs.org/shiki/-/shiki-3.12.0.tgz",
|
"resolved": "https://registry.npmjs.org/shiki/-/shiki-3.12.1.tgz",
|
||||||
"integrity": "sha512-E+ke51tciraTHpaXYXfqnPZFSViKHhSQ3fiugThlfs/om/EonlQ0hSldcqgzOWWqX6PcjkKKzFgrjIaiPAXoaA==",
|
"integrity": "sha512-eMlxVaXyuNQAQCaMtDKQjKv0eVm+kA6fsZtv9UqKgspP+7lWCVi7SoN+cJq1dawvIDQY7TI3SixamztotM6R6Q==",
|
||||||
"dev": true,
|
"dev": true,
|
||||||
"license": "MIT",
|
"license": "MIT",
|
||||||
"dependencies": {
|
"dependencies": {
|
||||||
"@shikijs/core": "3.12.0",
|
"@shikijs/core": "3.12.1",
|
||||||
"@shikijs/engine-javascript": "3.12.0",
|
"@shikijs/engine-javascript": "3.12.1",
|
||||||
"@shikijs/engine-oniguruma": "3.12.0",
|
"@shikijs/engine-oniguruma": "3.12.1",
|
||||||
"@shikijs/langs": "3.12.0",
|
"@shikijs/langs": "3.12.1",
|
||||||
"@shikijs/themes": "3.12.0",
|
"@shikijs/themes": "3.12.1",
|
||||||
"@shikijs/types": "3.12.0",
|
"@shikijs/types": "3.12.1",
|
||||||
"@shikijs/vscode-textmate": "^10.0.2",
|
"@shikijs/vscode-textmate": "^10.0.2",
|
||||||
"@types/hast": "^3.0.4"
|
"@types/hast": "^3.0.4"
|
||||||
}
|
}
|
||||||
|
|
|
||||||
Loading…
Reference in New Issue