From 5a56b26e3e78badfea35cfcc1769e66c8c25e127 Mon Sep 17 00:00:00 2001
From: tyxben <20570617+tyxben@users.noreply.github.com>
Date: Tue, 30 Jul 2024 11:55:42 +0800
Subject: [PATCH 1/2] Update readme.md
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

优化中国剩余定理求解函数，确保模逆元非负
---
 08_Remainder/readme.md | 21 +++++++++++++++------
 1 file changed, 15 insertions(+), 6 deletions(-)

diff --git a/08_Remainder/readme.md b/08_Remainder/readme.md
index 113bd1b..d84e8d5 100644
--- a/08_Remainder/readme.md
+++ b/08_Remainder/readme.md
@@ -283,6 +283,12 @@ $a_3 = 2, b_3 = 15, b_3' \equiv 15^{-1} = 1\pmod{7}$
 
 ```python
 def extended_gcd(a, b):
+    """
+    扩展欧几里得算法，用于求解 ax + by = gcd(a, b) 的整数解
+    :param a: 整数
+    :param b: 整数
+    :return: (gcd(a, b), x, y)
+    """
     if b == 0:
         return a, 1, 0
     else:
@@ -292,7 +298,6 @@ def extended_gcd(a, b):
 def chinese_remainder_theorem(congruences):
     """
     中国剩余定理求解函数
-
     :param congruences: 模线性同余方程组，格式为 [(a1, m1), (a2, m2), ..., (an, mn)]，表示方程组为 x ≡ ai (mod mi)
     :return: 方程组的解 x
     """
@@ -303,18 +308,22 @@ def chinese_remainder_theorem(congruences):
 
     # 计算 Mi 和 Mi 的模逆元素
     M_values = [M // mi for _, mi in congruences]
-    Mi_values = [extended_gcd(Mi, mi)[1] for Mi, (_, mi) in zip(M_values, congruences)]
+    inverse_values = [extended_gcd(Mi, mi)[1] % mi for Mi, (_, mi) in zip(M_values, congruences)]
 
     # 计算解 x
-    x = sum(ai * Mi * mi for (ai, _), Mi, mi in zip(congruences, Mi_values, M_values)) % M
+    x = sum(ai * Mi * inverse for (ai, mi), Mi, inverse in zip(congruences, M_values, inverse_values)) % M
 
     return x
 
 # 示例：解 x ≡ 2 (mod 3), x ≡ 3 (mod 5), x ≡ 2 (mod 7)
 congruences = [(2, 3), (3, 5), (2, 7)]
 solution = chinese_remainder_theorem(congruences)
-print(f"同余方程组的解为 x ≡ {solution} (mod {congruences[0][1] * congruences[1][1] * congruences[2][1]})")
-# 同余方程组的解为 x ≡ 23 (mod 105)
+M_product = 1
+for _, mi in congruences:
+    M_product *= mi
+
+print(f"同余方程组的解为 x ≡ {solution} (mod {M_product})")
+# 输出: 同余方程组的解为 x ≡ 23 (mod 105)
 ```
 
 ### 4.5 逆向使用
@@ -337,4 +346,4 @@ $$
 
 ## 5. 总结
 
-这一讲，我们学习了剩余类，同余方程组，和中国剩余定理。其中中国剩余定理不仅可以解同余方程组，还可以逆向使用，将大问题分解为小问题，在零知识证明中非常重要。
\ No newline at end of file
+这一讲，我们学习了剩余类，同余方程组，和中国剩余定理。其中中国剩余定理不仅可以解同余方程组，还可以逆向使用，将大问题分解为小问题，在零知识证明中非常重要。

From 05135b0444a3bd0271a2dc2a389cc5fda1e62bd6 Mon Sep 17 00:00:00 2001
From: weitianyuan <15245012960@163.com>
Date: Wed, 31 Jul 2024 11:50:19 +0800
Subject: [PATCH 2/2] update jupyter code

---
 08_Remainder/Remainder.ipynb | 46 +++++++++++++++++++++++-------------
 1 file changed, 30 insertions(+), 16 deletions(-)

diff --git a/08_Remainder/Remainder.ipynb b/08_Remainder/Remainder.ipynb
index 5dfa37d..e9dcbc0 100644
--- a/08_Remainder/Remainder.ipynb
+++ b/08_Remainder/Remainder.ipynb
@@ -2,20 +2,21 @@
  "cells": [
   {
    "cell_type": "code",
-   "execution_count": 2,
    "id": "40f73521",
-   "metadata": {},
-   "outputs": [
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "同余方程组的解为 x ≡ 23 (mod 105)\n"
-     ]
+   "metadata": {
+    "ExecuteTime": {
+     "end_time": "2024-07-31T03:46:04.473502Z",
+     "start_time": "2024-07-31T03:46:04.468611Z"
     }
-   ],
+   },
    "source": [
     "def extended_gcd(a, b):\n",
+    "    \"\"\"\n",
+    "    扩展欧几里得算法，用于求解 ax + by = gcd(a, b) 的整数解\n",
+    "    :param a: 整数\n",
+    "    :param b: 整数\n",
+    "    :return: (gcd(a, b), x, y)\n",
+    "    \"\"\"\n",
     "    if b == 0:\n",
     "        return a, 1, 0\n",
     "    else:\n",
@@ -25,7 +26,6 @@
     "def chinese_remainder_theorem(congruences):\n",
     "    \"\"\"\n",
     "    中国剩余定理求解函数\n",
-    "\n",
     "    :param congruences: 模线性同余方程组，格式为 [(a1, m1), (a2, m2), ..., (an, mn)]，表示方程组为 x ≡ ai (mod mi)\n",
     "    :return: 方程组的解 x\n",
     "    \"\"\"\n",
@@ -36,19 +36,33 @@
     "\n",
     "    # 计算 Mi 和 Mi 的模逆元素\n",
     "    M_values = [M // mi for _, mi in congruences]\n",
-    "    Mi_values = [extended_gcd(Mi, mi)[1] for Mi, (_, mi) in zip(M_values, congruences)]\n",
+    "    inverse_values = [extended_gcd(Mi, mi)[1] % mi for Mi, (_, mi) in zip(M_values, congruences)]\n",
     "\n",
     "    # 计算解 x\n",
-    "    x = sum(ai * Mi * mi for (ai, _), Mi, mi in zip(congruences, Mi_values, M_values)) % M\n",
+    "    x = sum(ai * Mi * inverse for (ai, mi), Mi, inverse in zip(congruences, M_values, inverse_values)) % M\n",
     "\n",
     "    return x\n",
     "\n",
     "# 示例：解 x ≡ 2 (mod 3), x ≡ 3 (mod 5), x ≡ 2 (mod 7)\n",
     "congruences = [(2, 3), (3, 5), (2, 7)]\n",
     "solution = chinese_remainder_theorem(congruences)\n",
-    "print(f\"同余方程组的解为 x ≡ {solution} (mod {congruences[0][1] * congruences[1][1] * congruences[2][1]})\")\n",
-    "# 同余方程组的解为 x ≡ 23 (mod 105)"
-   ]
+    "M_product = 1\n",
+    "for _, mi in congruences:\n",
+    "    M_product *= mi\n",
+    "\n",
+    "print(f\"同余方程组的解为 x ≡ {solution} (mod {M_product})\")\n",
+    "# 输出: 同余方程组的解为 x ≡ 23 (mod 105)"
+   ],
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "同余方程组的解为 x ≡ 23 (mod 105)\n"
+     ]
+    }
+   ],
+   "execution_count": 1
   },
   {
    "cell_type": "code",