From ef97e28852771ffc63dd632780a55fce2bd043f4 Mon Sep 17 00:00:00 2001
From: KirisameVanilla <118162831+KirisameVanilla@users.noreply.github.com>
Date: Tue, 2 Sep 2025 13:51:07 +0800
Subject: [PATCH 1/5] =?UTF-8?q?feat:=20make=20=E4=BF=A1=E5=AE=89=E6=95=B0?=
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.../信息安全数学基础（二）/2024秋期末考试答案.md | 123 +++++++++++++++++
.../24FallFinalExam/2024秋期末考试答案.md | 67 ----------
.../24FallFinalExam/2024秋期末考试（部分）题目.md | 19 ---
.../形式语言与自动机理论/index.md | 5 +-
package-lock.json | 124 +++++++++---------
5 files changed, 187 insertions(+), 151 deletions(-)
create mode 100644 docs/courses/软件工程学院/信息安全数学基础（二）/2024秋期末考试答案.md
delete mode 100644 docs/courses/软件工程学院/信息安全数学基础（二）/24FallFinalExam/2024秋期末考试答案.md
delete mode 100644 docs/courses/软件工程学院/信息安全数学基础（二）/24FallFinalExam/2024秋期末考试（部分）题目.md
diff --git a/docs/courses/软件工程学院/信息安全数学基础（二）/2024秋期末考试答案.md b/docs/courses/软件工程学院/信息安全数学基础（二）/2024秋期末考试答案.md
new file mode 100644
index 0000000..8dba8c4
--- /dev/null
+++ b/docs/courses/软件工程学院/信息安全数学基础（二）/2024秋期末考试答案.md
@@ -0,0 +1,123 @@
+---
+title: 2024Fall_期末
+category:
+ - 软件工程学院
+ - 课程资料
+tag:
+ - 试卷
+author:
+ - タクヤマ
+---
+
+## 说明
+
+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
+
+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**
+
+ -
+
+ 设 $G$ 是一个群，对于 $\forall a,b \in G$， $a*b=b*a$， 则 $G$ 是 ____ 群
+
+ -
+
+ 设 $R$ 是一个环，对于 $\forall a,b \in R$， $a+b=b+a$， 则 $R$ 是？
+ A. 整环 B. 交换环 C. 含幺环 D. 环
+
+
+
+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！**
+
+## 题目
+
+> **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**
+
+
+
+解：
+
+将 $G$ 中元素 $g$ 各个代入，计算 $gH$
+
+- $g = e$：
+ $eH = H$
+
+- $g = a$：
+ $aH = \{ a \cdot a, a \cdot e \} = \{ a^2, a \}$，由于 $G$ 是群，且 $H$ 是子群，所以 $a^2$ 必须是 $G$ 中元素。
+ 故 $a^2 = e$，则： $aH = \{ e, a \} = H$
+
+- $g = b$：
+ $bH = \{ b \cdot a, b \cdot e \} = \{ ba, b \}$，
+ 假设 $ba = c$ ，则：
+ $bH = \{ c, b \}$
+
+- $g = c$：
+ $cH = \{ c \cdot a, c \cdot e \} = \{ ca, c \}$，
+ 假设 $ca = b$ ，则：
+ $cH = \{ b, c \} = bH$
+
+综上所述，H的所有子陪集是 $\{ e, a \}, \{ b, c \}$
+
+
+
+***
+
+> **2. 写出 $R=Z/6Z$ 的所有零因子**
+
+
+
+解：
+
+ $R=Z/6Z \cong Z_6$，我们只要考虑 $Z_6$ 上的性质：
+ 显然有 $2\*3 \equiv 0 \pmod{6}, 4*3 \equiv 0 \pmod{6}$，所以零因子是2，3，4
+
+
+
+***
+
+> **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 $P+Q$ , $2P$**
+
+
+
+解：
+
+ 直接计算即可，我们这里直接给出答案： $P+Q = (8, 15)$ , $2P = (14, 15)$
+
+
+
+***
+
+> **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**
+
+
+
+解：
+
+ 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 $x$ :
+
+ | $x^n$ | $x^n\pmod{x^4+x^3+1}$ |
+ | -------- | --------------------- |
+ | $x^0$ | $1$ |
+ | $x^1$ | $x$ |
+ | $x^2$ | $x^2$ |
+ | $x^3$ | $x^3$ |
+ | $x^4$ | $x^3 + 1$ |
+ | $x^5$ | $x^3 + x + 1$ |
+ | $x^6$ | $x^3 + x^2 + x + 1$ |
+ | $x^7$ | $1 + x^2 + x$ |
+ | $x^8$ | $x^2 + x + 1$ |
+ | $x^9$ | $x^3 + x^2 + x$ |
+ | $x^{10}$ | $x^3 + x^2 + x + 1$ |
+ | $x^{11}$ | $x^3 + x + 1$ |
+ | $x^{12}$ | $x^3 + 1$ |
+ | $x^{13}$ | $x^3$ |
+ | $x^{14}$ | $x^2$ |
+
+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
+
+
+
+***
+
+> **5. 证明：有限环的特征一定不为0**
+
+证明：请参考课件Chap7.pdf中27-28页
diff --git a/docs/courses/软件工程学院/信息安全数学基础（二）/24FallFinalExam/2024秋期末考试答案.md b/docs/courses/软件工程学院/信息安全数学基础（二）/24FallFinalExam/2024秋期末考试答案.md
deleted file mode 100644
index 5641ec1..0000000
--- a/docs/courses/软件工程学院/信息安全数学基础（二）/24FallFinalExam/2024秋期末考试答案.md
+++ /dev/null
@@ -1,67 +0,0 @@
-# 答案
-
-##
-
-### 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
-
-解：
- 将 $G$ 中元素 $g$ 各个代入，计算 $gH$
-
-- $g = e$：
- $eH = H$
-
-- $g = a$：
- $aH = \{ a \cdot a, a \cdot e \} = \{ a^2, a \}$，由于 $G$ 是群，且 $H$ 是子群，所以 $a^2$ 必须是 $G$ 中元素。
- 故 $a^2 = e$，则： $aH = \{ e, a \} = H$
-
-- $g = b$：
- $bH = \{ b \cdot a, b \cdot e \} = \{ ba, b \}$，
- 假设 $ba = c$ ，则：
- $bH = \{ c, b \}$
-
-- $g = c$：
- $cH = \{ c \cdot a, c \cdot e \} = \{ ca, c \}$，
- 假设 $ca = b$ ，则：
- $cH = \{ b, c \} = bH$
-
-综上所述，H的所有子陪集是 $\{ e, a \}, \{ b, c \}$
-
-**2.写出 $R=Z/6Z$ 的所有零因子**
-
-解：
- $R=Z/6Z \cong Z_6$，我们只要考虑 $Z_6$ 上的性质：
- 显然有 $2\*3 \equiv 0 \pmod{6}, 4*3 \equiv 0 \pmod{6}$，所以零因子是2，3，4
-
-**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 $P+Q$ , $2P$**
-
-解：
- 直接计算即可，我们这里直接给出答案： $P+Q = (8, 15)$ , $2P = (14, 15)$
-
-**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**
-
-解：
- 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 $x$ :
-
- | $x^n$ | $x^n\pmod{x^4+x^3+1}$ |
- |------|----------|
- | $x^0$ | $1$ |
- | $x^1$ | $x$ |
- | $x^2$ | $x^2$ |
- | $x^3$ | $x^3$ |
- | $x^4$ | $x^3 + 1$ |
- | $x^5$ | $x^3 + x + 1$ |
- | $x^6$ | $x^3 + x^2 + x + 1$ |
- | $x^7$ | $1 + x^2 + x$ |
- | $x^8$ | $x^2 + x + 1$ |
- | $x^9$ | $x^3 + x^2 + x$ |
- | $x^{10}$ | $x^3 + x^2 + x + 1$ |
- | $x^{11}$ | $x^3 + x + 1$ |
- | $x^{12}$ | $x^3 + 1$ |
- | $x^{13}$ | $x^3$ |
- | $x^{14}$ | $x^2$ |
-
-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
-
-### 5.证明：有限环的特征一定不为0
-
-证明：请参考课件Chap7.pdf中27-28页
diff --git a/docs/courses/软件工程学院/信息安全数学基础（二）/24FallFinalExam/2024秋期末考试（部分）题目.md b/docs/courses/软件工程学院/信息安全数学基础（二）/24FallFinalExam/2024秋期末考试（部分）题目.md
deleted file mode 100644
index 4499a66..0000000
--- a/docs/courses/软件工程学院/信息安全数学基础（二）/24FallFinalExam/2024秋期末考试（部分）题目.md
+++ /dev/null
@@ -1,19 +0,0 @@
-## 说明：
-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
-
-
- - 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
- - 写出 $R=Z/6Z$ 的所有零因子
- - 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 $P+Q$ , $2P$
- - 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
- - 证明：有限环的特征一定不为0
-
-
-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**
-
- - 设 $G$ 是一个群，对于 $\forall a,b \in G$， $a*b=b*a$， 则 $G$ 是 ____ 群
- - 设 $R$ 是一个环，对于 $\forall a,b \in R$， $a+b=b+a$， 则 $R$ 是？
- A. 整环 B.交换环 C.含幺环 D.环
-
-
-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！**
diff --git a/docs/courses/软件工程学院/形式语言与自动机理论/index.md b/docs/courses/软件工程学院/形式语言与自动机理论/index.md
index e248e6d..ccbb1c4 100644
--- a/docs/courses/软件工程学院/形式语言与自动机理论/index.md
+++ b/docs/courses/软件工程学院/形式语言与自动机理论/index.md
@@ -1,5 +1,5 @@
---
-title: 信息安全数学基础（二）
+title: 形式语言与自动机理论
category:
- 软件工程学院
- 课程评价
@@ -9,5 +9,4 @@ tag:
- 抽象代数
---
-### 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。
-### 同样地，这门课实际考试不难，老师会捞，请大家不要害怕。
+
diff --git a/package-lock.json b/package-lock.json
index b42a27d..bb8546b 100644
--- a/package-lock.json
+++ b/package-lock.json
@@ -1846,76 +1846,76 @@
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+ "@shikijs/types": "3.12.1",
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+ "integrity": "sha512-9JrAm9cA5hqM/YXymA3oAAZdnCgQf1zyrNDtsnM105nNEoEpux4dyzdoOjc2KawEKj1iUs/WH2ota6Atp7GYkQ==",
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"license": "MIT",
"dependencies": {
- "@shikijs/types": "3.12.0"
+ "@shikijs/types": "3.12.1"
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"dev": true,
"license": "MIT",
"dependencies": {
- "@shikijs/core": "3.12.0",
- "@shikijs/types": "3.12.0"
+ "@shikijs/core": "3.12.1",
+ "@shikijs/types": "3.12.1"
}
},
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- "resolved": "https://registry.npmjs.org/@shikijs/types/-/types-3.12.0.tgz",
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"license": "MIT",
"dependencies": {
@@ -2912,6 +2912,16 @@
"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": {
"version": "2.0.0-rc.112",
"resolved": "https://registry.npmjs.org/@vuepress/plugin-rtl/-/plugin-rtl-2.0.0-rc.112.tgz",
@@ -3070,15 +3080,15 @@
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- "resolved": "https://registry.npmjs.org/@vueuse/core/-/core-13.8.0.tgz",
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+ "version": "13.9.0",
+ "resolved": "https://registry.npmjs.org/@vueuse/core/-/core-13.9.0.tgz",
+ "integrity": "sha512-ts3regBQyURfCE2BcytLqzm8+MmLlo5Ln/KLoxDVcsZ2gzIwVNnQpQOL/UKV8alUqjSZOlpFZcRNsLRqj+OzyA==",
"dev": true,
"license": "MIT",
"dependencies": {
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- "@vueuse/shared": "13.8.0"
+ "@vueuse/metadata": "13.9.0",
+ "@vueuse/shared": "13.9.0"
},
"funding": {
"url": "https://github.com/sponsors/antfu"
@@ -3088,9 +3098,9 @@
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+ "version": "13.9.0",
+ "resolved": "https://registry.npmjs.org/@vueuse/metadata/-/metadata-13.9.0.tgz",
+ "integrity": "sha512-1AFRvuiGphfF7yWixZa0KwjYH8ulyjDCC0aFgrGRz8+P4kvDFSdXLVfTk5xAN9wEuD1J6z4/myMoYbnHoX07zg==",
"dev": true,
"license": "MIT",
"funding": {
@@ -3098,9 +3108,9 @@
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+ "version": "13.9.0",
+ "resolved": "https://registry.npmjs.org/@vueuse/shared/-/shared-13.9.0.tgz",
+ "integrity": "sha512-e89uuTLMh0U5cZ9iDpEI2senqPGfbPRTHM/0AaQkcxnpqjkZqDYP8rpfm7edOz8s+pOCOROEy1PIveSW8+fL5g==",
"dev": true,
"license": "MIT",
"funding": {
@@ -3591,13 +3601,13 @@
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+ "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": ">=20"
+ "node": ">= 12"
}
},
"node_modules/connect-history-api-fallback": {
@@ -4533,16 +4543,6 @@
"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": {
"version": "6.0.3",
"resolved": "https://registry.npmjs.org/kind-of/-/kind-of-6.0.3.tgz",
@@ -5929,18 +5929,18 @@
"license": "ISC"
},
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- "version": "3.12.0",
- "resolved": "https://registry.npmjs.org/shiki/-/shiki-3.12.0.tgz",
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+ "version": "3.12.1",
+ "resolved": "https://registry.npmjs.org/shiki/-/shiki-3.12.1.tgz",
+ "integrity": "sha512-eMlxVaXyuNQAQCaMtDKQjKv0eVm+kA6fsZtv9UqKgspP+7lWCVi7SoN+cJq1dawvIDQY7TI3SixamztotM6R6Q==",
"dev": true,
"license": "MIT",
"dependencies": {
- "@shikijs/core": "3.12.0",
- "@shikijs/engine-javascript": "3.12.0",
- "@shikijs/engine-oniguruma": "3.12.0",
- "@shikijs/langs": "3.12.0",
- "@shikijs/themes": "3.12.0",
- "@shikijs/types": "3.12.0",
+ "@shikijs/core": "3.12.1",
+ "@shikijs/engine-javascript": "3.12.1",
+ "@shikijs/engine-oniguruma": "3.12.1",
+ "@shikijs/langs": "3.12.1",
+ "@shikijs/themes": "3.12.1",
+ "@shikijs/types": "3.12.1",
"@shikijs/vscode-textmate": "^10.0.2",
"@types/hast": "^3.0.4"
}
From 95bcf297402e304b8bd791cc29f56c5ff346eb9c Mon Sep 17 00:00:00 2001
From: KirisameVanilla <118162831+KirisameVanilla@users.noreply.github.com>
Date: Tue, 2 Sep 2025 14:40:51 +0800
Subject: [PATCH 2/5] =?UTF-8?q?feat:=202024=20fall=20=E6=95=B0=E8=AE=BA1?=
=?UTF-8?q?=E6=9C=9F=E6=9C=AB=E8=AF=95=E5=8D=B7=EF=BC=88=E9=9C=80=E8=A6=81?=
=?UTF-8?q?=E6=A0=B9=E6=8D=AE=E5=9B=BE=E7=89=87=E6=A0=A1=E5=AF=B9=E5=86=85?=
=?UTF-8?q?=E5=AE=B9=EF=BC=89?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit
---
.../期末试卷/2024Fall_期末.md | 129 ++++++++++++++++++
1 file changed, 129 insertions(+)
create mode 100644 docs/courses/软件工程学院/信息安全数学基础（一）/期末试卷/2024Fall_期末.md
diff --git a/docs/courses/软件工程学院/信息安全数学基础（一）/期末试卷/2024Fall_期末.md b/docs/courses/软件工程学院/信息安全数学基础（一）/期末试卷/2024Fall_期末.md
new file mode 100644
index 0000000..d2ba070
--- /dev/null
+++ b/docs/courses/软件工程学院/信息安全数学基础（一）/期末试卷/2024Fall_期末.md
@@ -0,0 +1,129 @@
+---
+title: 2024Spring_期末
+category:
+ - 软件工程学院
+ - 课程资料
+tag:
+ - 试卷
+author:
+ - タクヤマ
+ - KirisameVanilla
+---
+
+## 华东师范大学期末试卷（A）
+
+### 一、选择题（共 10 小题，每小题 2 分，共 20 分）
+
+> **设 $m > 1$ 是整数，$(a, m) = 1$，则下列选项中不正确的是**
+
+A. 若 $b \equiv a \pmod{m}$，则 $\mathrm{ord}_m(b) = \mathrm{ord}_m(a)$.
+B. $a^d \equiv a^k \pmod{m}$ 成立的充要条件是 $d \equiv k \pmod{\mathrm{ord}_m(a)}$.
+C. 若 $a' \cdot a \equiv 1 \pmod{m}$，则 $\mathrm{ord}_m(a') = \mathrm{ord}_m(a)$.
+D. 若 $\mathrm{ord}_m(a) = st$，则 $\mathrm{ord}_m(a^s) = t$.
+
+> **下列哪个数不是模 11 的原根？**
+
+A. 7 B. 6 C. 4 D. 2
+
+> **9 模 14 的指数 $\mathrm{ord}_{14}(9)$ 是**
+
+A. 6 B. 3 C. 2 D. 1
+
+> **设 $a, b, c \in \mathbb{Z}, c \ne 0$，下列结论不正确的是**
+
+A. 若 $c \mid a, c \mid b$，则 $c \mid (a + b)$.
+B. 若 $c \mid a$，则 $c \mid ab$.
+C. 若 $b \mid c, c \mid a$，则 $b \mid a$.
+D. 若 $c \mid (a^2 - b^2)$，则 $c \mid (a - b)$ 或 $c \mid (a + b)$.
+
+> **模 40 的简化剩余系中元素的个数为**
+
+A. 16 B. 28 C. 39 D. 40
+
+> **已知 $\mathrm{ord}_{137}(47) = 136$, $\mathrm{ord}_{739}(47) = 82$，则 $\mathrm{ord}_{101243}(47) =$**
+
+A. 136 B. 82 C. 5576 D. 11152
+
+> **7. 设 $n$ 为整数，则下列选项中一定可以被 6 整除的是**
+
+A. $n(n^2 + 1)$
+B. $n(n^2 - 1)$
+C. $n(n + 1)$
+D. $n(n - 1)$
+
+> **8. 下列选项中正确的是**
+
+A. 若 $m = 1458$，则模 $m$ 的原根不存在。
+B. 1275 是 Carmichael 数。
+C. 2047 是对于基 2 的拟素数。
+
+> **9. 模 24 的一个简化剩余系为**
+
+A. $\{-1, 2, 3, 5, 7, 9, 19, 23\}$
+B. $\{-7, -1, 9, 13, 17, 23\}$
+C. $\{1, 5, 7, 11, 13, 17, 19, 23\}$
+D. $\{3, 7, 11, 13, 17, 19, 23\}$
+
+> ** 10. 以下哪个数不是模 71 的二次剩余？
+
+A. 35 B. 36 C. 37 D. 38
+
+---
+
+### 二、填空题（共 10 小题，每小题 3 分，共 30 分）
+
+1. 13 模 21 的指数 $\mathrm{ord}_{21}(13) =$ ________.
+
+2. $3865749 \mod 11 =$ ________.
+
+3. 同余方程 $17x \equiv 14 \pmod{21}$ 的解为 ________.
+
+4. 已知 $a = 123, b = 321$，则有 $s =$ ________, $t =$ ________，使得 $sa + tb = (a, b) =$ ________.
+
+5. 下面的方程组的解为 ________.
+
+$$ \begin{cases}
+3x + 5y \equiv 38 \pmod{47} \\
+x - y \equiv 10 \pmod{47}
+\end{cases} $$
+
+6. $\left( \frac{65}{103} \right) =$ ________.
+
+7. 同余方程 $6x \equiv 3 \pmod{9}$ 的解为 ________.
+
+8. 模 29 的最小正原根为 ________.
+
+9. $2^{2002}$ 被 7 除所得的余数为 ________.
+
+10. 已知 443 是素数，同余方程 $x^2 \equiv 26 \pmod{443}$ 有 ________ 个解。
+
+---
+
+### 三、计算题（共 25 分）
+
+> **1. 判断方程 $x^{15} \equiv 14 \pmod{41}$ 解的个数，并求出所有解**
+
+模 41 以 6 为原根的指数表如下，其中第一列表示十位数，第一行表示个位数，交叉位置表示该数的指数：
+
+| | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
+|-----|---|---|---|---|---|---|---|---|---|---|
+| 0 | | 40| 26| 15| 12| 22| 1 | 39| 16| 30|
+| 1 | 8 | 3 | 27| 31| 23| 37| 24| 33| 16| 9 |
+| 2 | 34| 14| 29| 36| 13| 4 | 17| 5 | 11| 7 |
+| 3 | 23| 28| 10| 18| 19| 21| 2 | 32| 35| 6 |
+| 4 | 20| | | | | | | | | |
+
+---
+
+> **2. 计算 Legendre 符号**
+
+1) $\left( \frac{33}{317} \right)$
+2) $\left( \frac{286}{563} \right)$ （10 分）
+
+---
+
+### 四、证明题（25 分）
+
+> **1. 证明：121 是对基 3 的拟素数。（10 分）**
+
+> **2. 设 $n$ 为偶数, $p$ 为素数, 且 $p \mid n^{4} + 1$, 证明 $p \equiv 1 \pmod 8$ (15 分)**
From 0f3776e23ae793a2bcd112996d15daeba551fc72 Mon Sep 17 00:00:00 2001
From: KirisameVanilla <118162831+KirisameVanilla@users.noreply.github.com>
Date: Tue, 2 Sep 2025 15:10:10 +0800
Subject: [PATCH 3/5] =?UTF-8?q?wip:=20=E6=95=B4=E7=90=86=E6=95=B0=E8=AE=BA?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit
---
.../{期末试卷/2024Fall_期末.md => 2024Spring_期末.md} | 0
.../信息安全数学基础（一）/index.md | 8 +++++++-
.../信息安全数学基础（一）/期末试卷/试卷p1.jpg | Bin 225013 -> 0 bytes
.../信息安全数学基础（一）/期末试卷/试卷p2.png | Bin 11656 -> 0 bytes
.../信息安全数学基础（一）/课件/Chap4-2.pdf | Bin 390535 -> 0 bytes
.../信息安全数学基础（一）/课件/Chap5.pdf | Bin 353677 -> 0 bytes
.../{2024秋期末考试答案.md => 2024Fall_期末_含答案.md} | 2 +-
.../信息安全数学基础（二）/Slides/Chap8.pdf | Bin 379590 -> 0 bytes
.../信息安全数学基础（二）/Slides/Chap9.pdf | Bin 374677 -> 0 bytes
.../信息安全数学基础（二）/index.md | 12 ++++++++++--
10 files changed, 18 insertions(+), 4 deletions(-)
rename docs/courses/软件工程学院/信息安全数学基础（一）/{期末试卷/2024Fall_期末.md => 2024Spring_期末.md} (100%)
delete mode 100644 docs/courses/软件工程学院/信息安全数学基础（一）/期末试卷/试卷p1.jpg
delete mode 100644 docs/courses/软件工程学院/信息安全数学基础（一）/期末试卷/试卷p2.png
delete mode 100644 docs/courses/软件工程学院/信息安全数学基础（一）/课件/Chap4-2.pdf
delete mode 100644 docs/courses/软件工程学院/信息安全数学基础（一）/课件/Chap5.pdf
rename docs/courses/软件工程学院/信息安全数学基础（二）/{2024秋期末考试答案.md => 2024Fall_期末_含答案.md} (99%)
delete mode 100644 docs/courses/软件工程学院/信息安全数学基础（二）/Slides/Chap8.pdf
delete mode 100644 docs/courses/软件工程学院/信息安全数学基础（二）/Slides/Chap9.pdf
diff --git a/docs/courses/软件工程学院/信息安全数学基础（一）/期末试卷/2024Fall_期末.md b/docs/courses/软件工程学院/信息安全数学基础（一）/2024Spring_期末.md
similarity index 100%
rename from docs/courses/软件工程学院/信息安全数学基础（一）/期末试卷/2024Fall_期末.md
rename to docs/courses/软件工程学院/信息安全数学基础（一）/2024Spring_期末.md
diff --git a/docs/courses/软件工程学院/信息安全数学基础（一）/index.md b/docs/courses/软件工程学院/信息安全数学基础（一）/index.md
index a3883c9..0aa0188 100644
--- a/docs/courses/软件工程学院/信息安全数学基础（一）/index.md
+++ b/docs/courses/软件工程学院/信息安全数学基础（一）/index.md
@@ -9,8 +9,14 @@ tag:
- 数论
---
-# 课程评价
+## 资源列表
+
+- [课件 2022级更新 (3-1 到 5)](https://drive.vanillaaaa.org/SharedCourses/%E8%BD%AF%E4%BB%B6%E5%B7%A5%E7%A8%8B%E5%AD%A6%E9%99%A2/%E4%BF%A1%E6%81%AF%E5%AE%89%E5%85%A8%E6%95%B0%E5%AD%A6%E5%9F%BA%E7%A1%80/2022%E7%BA%A7)
+- [2024春学期期末试卷](https://courses.ecnu.vanillaaaa.org/courses/软件工程学院/信息安全数学基础（一）/2024Spring_期末.html)
+
## 2024Spring
+
### 沈家辰
+
- 初等数论, 涉及知识较为广阔, 但是并不要求高妙的数学技巧, 也不需要惊人的注意力, 认真学习完全可以对付
- 考试不难, 老师会捞
diff --git a/docs/courses/软件工程学院/信息安全数学基础（一）/期末试卷/试卷p1.jpg b/docs/courses/软件工程学院/信息安全数学基础（一）/期末试卷/试卷p1.jpg
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