update en code, and fix tutorial

24_Polynomial/Polynomial.ipynb

@@ -67,7 +67,7 @@
     "        Y = list(data)\n",
     "\n",
     "# 使用 lagrange 函数构造拉格朗日插值多项式\n",
-    "interpolation_polynomial = interpolating_poly(len(data_points), x, X, Y)\n",
+    "interpolation_polynomial = interpolating_poly(len(data), x, X, Y)\n",
     "\n",
     "# 输出插值多项式\n",
     "print(\"拉格朗日多项式:\", interpolation_polynomial)\n",

27_GaloisField/GaloisField.ipynb

@@ -44,9 +44,9 @@
     "print(\"有限域 GF(4) 的元素\", GF4.elements)\n",
     "\n",
     "print(\"有限域 GF(4) 的加法表\")\n",
-    "print(GF7.arithmetic_table(\"+\"))\n",
+    "print(GF4.arithmetic_table(\"+\"))\n",
     "print(\"有限域 GF(4) 的乘法表\")\n",
-    "print(GF7.arithmetic_table(\"*\"))\n",
+    "print(GF4.arithmetic_table(\"*\"))\n",
     "\n",
     "## 输出示例\n",
     "# 有限域 GF(4) 的性质 Galois Field:\n",

27_GaloisField/readme.md

@@ -122,9 +122,9 @@ print("有限域 GF(4) 的性质", GF4.properties)
 print("有限域 GF(4) 的元素", GF4.elements)
 
 print("有限域 GF(4) 的加法表")
-print(GF7.arithmetic_table("+"))
+print(GF4.arithmetic_table("+"))
 print("有限域 GF(4) 的乘法表")
-print(GF7.arithmetic_table("*"))
+print(GF4.arithmetic_table("*"))
 
 ## 输出示例
 # 有限域 GF(4) 的性质 Galois Field:

32_ECC/ECC.ipynb

@@ -78,6 +78,7 @@
     "# EC ElGamal\n",
     "\n",
     "from py_ecc.secp256k1 import secp256k1\n",
+    "from random import randint\n",
     "\n",
     "def elgamal_encrypt(G, Y, M):\n",
     "    k = randint(1, secp256k1.N - 1)\n",

32_ECC/readme.md

@@ -132,6 +132,7 @@ Bob 收到密文 $(C_1, C_2)$ 后，需要使用私钥 $x$ 进行解密：
 
 ```python
 from py_ecc.secp256k1 import secp256k1
+from random import randint
 
 def elgamal_encrypt(G, Y, M):
     k = randint(1, secp256k1.N - 1)

Languages/en/24_Polynomial/Polynomial.ipynb

@@ -0,0 +1,115 @@
+{
+ "cells": [
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "id": "a20a2420",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Original polynomial: x**3 - 10*x**2 + 31*x - 30\n",
+      "Factorization of polynomial: (x - 5)*(x - 3)*(x - 2)\n"
+     ]
+    }
+   ],
+   "source": [
+    "# Prime Factorization of polynomials\n",
+    "from sympy import symbols, factor\n",
+    "from sympy.abc import x\n",
+    "\n",
+    "polynomial = x**3 -10*x**2 + 31*x - 30\n",
+    "\n",
+    "factored_polynomial = factor(polynomial)\n",
+    "\n",
+    "print(\"Original polynomial:\", polynomial)\n",
+    "print(\"Factorization of polynomial:\", factored_polynomial)\n",
+    "\n",
+    "# Output\n",
+    "# Original polynomial: x**3 - 10*x**2 + 31*x - 30\n",
+    "# Factorization of polynomial: (x - 5)*(x - 3)*(x - 2)\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "id": "89ff714b",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Lagrange Polynomial: (x - 3)*(x - 2) - 3*(x - 3)*(x - 1) + 4*(x - 2)*(x - 1)\n",
+      "Simplified Polynomial: 2*x**2 - 5*x + 5\n",
+      "Value at x=4: 8\n"
+     ]
+    }
+   ],
+   "source": [
+    "# Lagrange Interpolation Method\n",
+    "from sympy import symbols\n",
+    "from sympy.abc import x\n",
+    "from sympy.polys.polyfuncs import interpolating_poly\n",
+    "\n",
+    "# Given interpolation points and their corresponding function values\n",
+    "data = [(1, 2), (2, 3), (3, 8)]\n",
+    "\n",
+    "if isinstance(data, dict):\n",
+    "    X, Y = list(zip(*data.items()))\n",
+    "else:\n",
+    "    if isinstance(data[0], tuple):\n",
+    "        X, Y = list(zip(*data))\n",
+    "    else:\n",
+    "        X = list(range(1, n + 1))\n",
+    "        Y = list(data)\n",
+    "\n",
+    "# Use the interpolating_poly function to construct the Lagrange interpolation polynomial\n",
+    "interpolation_polynomial = interpolating_poly(len(data), x, X, Y)\n",
+    "\n",
+    "# Output the interpolation polynomial\n",
+    "print(\"Lagrange Polynomial:\", interpolation_polynomial)\n",
+    "print(\"Simplified Polynomial:\", interpolation_polynomial.expand())\n",
+    "# Calculate the value at x=4 using the interpolation polynomial\n",
+    "result = interpolation_polynomial.subs(x, 3)\n",
+    "print(\"Value at x=4:\", result)\n",
+    "\n",
+    "# Sample Output\n",
+    "# Lagrange Polynomial: (x - 3)*(x - 2) - 3*(x - 3)*(x - 1) + 4*(x - 2)*(x - 1)\n",
+    "# Simplified Polynomial: 2*x**2 - 5*x + 5\n",
+    "# Value at x=4: 8\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "6f1b05a6",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3 (ipykernel)",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.7.7"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}

Languages/en/25_PolyRing/PolyRing.ipynb

@@ -0,0 +1,111 @@
+{
+ "cells": [
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "id": "a20a2420",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Polynomial p1: Poly(2*x**3 - 2*x**2 + x + 1, x, modulus=5)\n",
+      "Polynomial p2: Poly(x**2 - x - 1, x, modulus=5)\n",
+      "Addition result: Poly(2*x**3 - x**2, x, modulus=5)\n",
+      "Subtraction result: Poly(2*x**3 + 2*x**2 + 2*x + 2, x, modulus=5)\n",
+      "Multiplication result: Poly(2*x**5 + x**4 + x**3 + 2*x**2 - 2*x - 1, x, modulus=5)\n",
+      "Modulo operation: Poly(-2*x + 1, x, modulus=5)\n",
+      "Quotient: Poly(2*x, x, modulus=5)\n",
+      "Greatest common divisor: Poly(x + 2, x, modulus=5)\n",
+      "Result of substituting x = 2 into polynomial p1: 1\n"
+     ]
+    }
+   ],
+   "source": [
+    "from sympy import symbols, Poly, GF, gcd\n",
+    "\n",
+    "# Define symbolic variables\n",
+    "x = symbols('x')\n",
+    "\n",
+    "# Define polynomials with coefficients in the ring of integers modulo 5\n",
+    "p1 = Poly(2*x**3 + 3*x**2 + x + 1, x, domain=GF(5))\n",
+    "p2 = Poly(x**2 + 4*x + 4, x, domain=GF(5))\n",
+    "\n",
+    "# Print the polynomials\n",
+    "print(\"Polynomial p1:\", p1)\n",
+    "print(\"Polynomial p2:\", p2)\n",
+    "\n",
+    "\n",
+    "# Polynomial addition\n",
+    "add_result = p1 + p2\n",
+    "print(\"Addition result:\", add_result)\n",
+    "\n",
+    "# Polynomial subtraction\n",
+    "sub_result = p1 - p2\n",
+    "print(\"Subtraction result:\", sub_result)\n",
+    "\n",
+    "# Polynomial multiplication\n",
+    "mul_result = p1 * p2\n",
+    "print(\"Multiplication result:\", mul_result)\n",
+    "\n",
+    "# Note: In the ring of integers modulo n, polynomial division is not always possible\n",
+    "# However, modulo operation can be performed\n",
+    "mod_result = p1 % p2\n",
+    "print(\"Modulo operation:\", mod_result)\n",
+    "\n",
+    "# Quotient\n",
+    "quotient_result = p1 // p2\n",
+    "print(\"Quotient:\", quotient_result)\n",
+    "\n",
+    "# Greatest common divisor\n",
+    "gcd_result = gcd(p1, p2)\n",
+    "print(\"Greatest common divisor:\", gcd_result)\n",
+    "\n",
+    "# Substituting values to solve\n",
+    "value_result = p1.subs(x, 2)\n",
+    "print(\"Result of substituting x = 2 into polynomial p1:\", value_result)\n",
+    "\n",
+    "## Output Example\n",
+    "# Polynomial p1: Poly(2*x**3 - 2*x**2 + x + 1, x, modulus=5)\n",
+    "# Polynomial p2: Poly(x**2 - x - 1, x, modulus=5)\n",
+    "# Addition result: Poly(2*x**3 - x**2, x, modulus=5)\n",
+    "# Subtraction result: Poly(2*x**3 + 2*x**2 + 2*x + 2, x, modulus=5)\n",
+    "# Multiplication result: Poly(2*x**5 + x**4 + x**3 + 2*x**2 - 2*x - 1, x, modulus=5)\n",
+    "# Modulo operation: Poly(-2*x + 1, x, modulus=5)\n",
+    "# Quotient: Poly(2*x, x, modulus=5)\n",
+    "# Greatest common divisor: Poly(x + 2, x, modulus=5)\n",
+    "# Result of substituting x = 2 into polynomial p1: 1\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "6f1b05a6",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3 (ipykernel)",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.7.7"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}

Languages/en/26_FieldExtension/FieldExtension.ipynb

@@ -0,0 +1,78 @@
+{
+ "cells": [
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "id": "04209408",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Extension Field:  QQ<sqrt(2)>\n",
+      "a = ANP([MPQ(1,2)], [MPQ(1,1), MPQ(0,1), MPQ(-2,1)], QQ)\n",
+      "b = ANP([MPQ(3,1)], [MPQ(1,1), MPQ(0,1), MPQ(-2,1)], QQ)\n",
+      "a + b = ANP([MPQ(7,2)], [MPQ(1,1), MPQ(0,1), MPQ(-2,1)], QQ)\n",
+      "a * b = ANP([MPQ(3,2)], [MPQ(1,1), MPQ(0,1), MPQ(-2,1)], QQ)\n"
+     ]
+    }
+   ],
+   "source": [
+    "from sympy import symbols, QQ, RootOf\n",
+    "\n",
+    "# Define symbolic variables\n",
+    "x = symbols('x')\n",
+    "\n",
+    "# Construct the algebraic extension QQ/(sqrt(2)) (which is a root of x^2 - 2 = 0) over the rational number field QQ\n",
+    "ext_field = QQ.algebraic_field(RootOf(x**2 - 2, 1))\n",
+    "\n",
+    "# Output the extension field and algebraic element\n",
+    "print(\"Extension Field: \", ext_field)\n",
+    "# QQ<sqrt(2)>\n",
+    "\n",
+    "\n",
+    "\n",
+    "# Calculation on extention field\n",
+    "a = ext_field(QQ(1, 2))\n",
+    "b = ext_field(3)\n",
+    "c = a + b\n",
+    "d = a * b\n",
+    "\n",
+    "print(\"a =\", a)\n",
+    "print(\"b =\", b)\n",
+    "print(\"a + b =\", c)\n",
+    "print(\"a * b =\", d)\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "4c678725",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3 (ipykernel)",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.7.7"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}