CSC15005/DLP-based/DLP-based.ipynb

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    "# DLP-based cryptosystem\n",
    "\n",
    "## 0. Lí thuyết nhóm\n",
    "Cho $G \\neq \\emptyset$, định nghĩa phép toán $\\cdot: G \\times G \\rightarrow G$\n",
    "\n",
    "$(G, \\cdot)$ là một nhóm (Group) nếu:\n",
    "\n",
    "0. Tiên đề đóng: $\\forall a, b \\in G: a\\cdot b \\in G$\n",
    "1. $G$ có phần tử đơn vị: $\\exists e \\in G: e\\cdot a = a\\cdot e = a, \\forall a \\in G$\n",
    "2. $G$ có phần tử nghịch đảo: $\\exists a, e \\in G: \\exists a^{-1} \\in G: a \\cdot a^{-1} = a^{-1} \\cdot a = e$\n",
    "3. $G$ có tính kết hợp: $\\forall a, b, c \\in G: (a \\cdot b) \\cdot c = a \\cdot (b \\cdot c)$\n",
    "\n",
    "$G$ là nhóm giao hoán (Nhóm Abel) nếu thỏa thêm điều kiện\n",
    "\n",
    "4. $\\forall a, b \\in G: a\\cdot b = b\\cdot a$\n"
   ]
  },
  {
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   "id": "b272b593",
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   "source": [
    "## 1. DLP - discrete logarithm problem\n",
    "Cho $g \\in G \\equiv \\mathbb{Z}^{*}_p, x \\in \\mathbb{Z}, y \\equiv g^x \\pmod{p}$. Khi đó bài toán tìm $x$ là một bài toán DLP."
   ]
  },
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   "cell_type": "markdown",
   "id": "16e248e6",
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   "source": [
    "## 2. Phần tử sinh của nhóm\n",
    "Ví dụ: Xét nhóm $\\left(\\mathbb{Z}^{*}_7, \\cdot_7\\right)$ (tạm hiểu $\\cdot_7$ là phép mũ $7^x\\ \\forall x \\in \\mathbb{Z}^{*}_7$)\n",
    "\n",
    "Với $2 \\in \\mathbb{Z}^{*}_7$, tập các giá trị có thể có là $\\{1, 2, 4\\}$:\n",
    "- $2^0 \\equiv 1 \\pmod 7$\n",
    "- $2^1 \\equiv 2 \\pmod 7$\n",
    "- $2^2 \\equiv 4 \\pmod 7$\n",
    "- $2^3 \\equiv 1 \\pmod 7$ (quay lại ban đầu)\n",
    "..\n",
    "\n",
    "Với 3 thì khác, tập giá trị có thể có là $\\{1, .., 6\\}$. \n",
    "\n",
    "Gọi $\\text{ord}\\left(g\\right)$ là bậc của $g \\in G$. Đặt $|G| = n$, nếu $\\text{ord}\\left(g\\right) = n - 1$ thì $g$ là một phần tử sinh của $G$.\n",
    "\n",
    "Thuật toán phía dưới tìm nhanh các phần tử sinh của $n$."
   ]
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     "text": [
      "Generators of Z_11:  [2, 6, 7, 8]\n",
      "Generators of Z_13:  [2, 6, 7, 11]\n",
      "Generators of Z_17:  [3, 5, 6, 7, 10, 11, 12, 14]\n"
     ]
    }
   ],
   "source": [
    "def naive_generator_list(n):\n",
    "    result = []\n",
    "    for i in range(2, n):\n",
    "        is_generator = True\n",
    "        power = 1\n",
    "        generated = []\n",
    "        while 1:\n",
    "            m = (i ** power) % n\n",
    "            if (m not in generated):\n",
    "                generated.append(m)\n",
    "            else: break\n",
    "            power += 1\n",
    "\n",
    "        if len(generated) != n - 1: is_generator = False\n",
    "\n",
    "        if is_generator: result.append(i)\n",
    "    return result\n",
    "\n",
    "print('Generators of Z_11: ', naive_generator_list(11))\n",
    "print('Generators of Z_13: ', naive_generator_list(13))\n",
    "print('Generators of Z_17: ', naive_generator_list(17))"
   ]
  },
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   "id": "1c8d6eb0",
   "metadata": {},
   "source": [
    "### Một thuật toán tìm nhanh phần tử sinh\n",
    "\n",
    "Cho $p$ là một số nguyên tố có dạng $p = 2q + 1$ (như ta đã tạo trong RSA) $\\Rightarrow (p - 1)$ không phải là một số nguyên tố.\n",
    "\n",
    "Ta có thể biểu diễn $(p - 1) = s_1^{e_1}s_2^{e_2}..s_m^{e_m} (e_i \\geq 1)$. Thuật toán tìm nhanh phần tử sinh được miêu tả như sau:\n",
    "\n",
    "1. Lấy random một số nguyên $x \\in (2, p)$\n",
    "2. $\\forall s_i \\in [1, m]$:\n",
    "\n",
    "   Nếu $x^{\\frac{(p - 1)}{s_i}} \\equiv 1 \\pmod p$ thì $x$ không phải là phần tử sinh của $p$, trở lại bước 1.\n",
    "3. $x$ là một phần tử sinh, return x.\n",
    "\n",
    "Thuật toán trình bày bên dưới."
   ]
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   "cell_type": "code",
   "execution_count": 2,
   "id": "eea5aa53",
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   "outputs": [
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     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2\n",
      "6\n",
      "5\n"
     ]
    }
   ],
   "source": [
    "import random\n",
    "from functools import reduce\n",
    "\n",
    "# For consistent output\n",
    "random.seed(42)\n",
    "\n",
    "def factors(n):\n",
    "    return set(reduce(list.__add__,\n",
    "                ([i, n//i] for i in range(1, int(n**0.5) + 1) if n % i == 0)))\n",
    "\n",
    "def find_random_generator(n):\n",
    "    while 1:\n",
    "        x = random.randrange(2, n)\n",
    "        is_generator = True\n",
    "        for i in factors(n - 1):\n",
    "            if i == 1: continue\n",
    "            k = (n - 1) // i\n",
    "            if (x ** k) % n == 1:\n",
    "                is_generator = False\n",
    "                break\n",
    "        if is_generator: return x\n",
    "        \n",
    "print(find_random_generator(11))\n",
    "print(find_random_generator(13))\n",
    "print(find_random_generator(17))"
   ]
  },
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   "id": "1e5b71cc",
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   "source": [
    "## Key Exchange Protocol (Diffie - Hellman)\n",
    "Điểm chính: thống nhất 1 bí mật chung giữa người gửi và người nhận.\n",
    "\n",
    "### Pha 0: Chuẩn bị\n",
    "1. Tạo ngẫu nhiên $p$ là một số nguyên tố có độ an toàn $\\lambda$.\n",
    "2. Chọn $g$ là một phần tử sinh của $p$.\n",
    "\n",
    "### Pha 1: Mã hóa & giải mã\n",
    "- Phía người gửi:\n",
    "  1. Chọn ngẫu nhiên $x \\in (2, p)$.\n",
    "  2. Tạo khóa $k_A \\equiv g^x \\pmod p$\n",
    "  3. Gửi khóa $k_A$ cho người nhận.\n",
    "\n",
    "- Phía người nhận:\n",
    "  1. Chọn ngẫu nhiên $y \\in (2, p)$.\n",
    "  2. Tạo khóa $k_B \\equiv g^y \\pmod p$\n",
    "  3. Gửi khóa $k_B$ cho người gửi.\n",
    "  \n",
    "Khi đó khóa $k$ để mã hóa & giải mã được tính như sau:\n",
    "$k \\equiv k_B^x \\equiv k_A^y \\pmod p$\n",
    "\n",
    "#### Ví dụ 1: Mã hóa đối xứng\n",
    "Phía người gửi tính $c \\equiv mk \\pmod p$\n",
    "\n",
    "Phía người nhận tính $m \\equiv ck^{-1} \\pmod p$\n",
    "\n",
    "#### Ví dụ 2: Mã hóa bất đối xứng (Elgamal)\n",
    "Giả sử Alice gửi mã và Bob giải mã, vậy **Bob sẽ là người tạo khóa**.\n",
    "\n",
    "- Phía người nhận:\n",
    "  0. Chọn $p, g$ như Pha 0 bên trên.\n",
    "  1. Chọn ngẫu nhiên $d \\in (2, p)$.\n",
    "  2. Tính khóa mã $e \\equiv g^d \\pmod p$\n",
    "  3. Gửi bộ $p, g, e$ cho người gửi.\n",
    " \n",
    "- Phía người gửi:\n",
    "  0. Nhận $p, g, e$.\n",
    "  1. Chọn ngẫu nhiên $x \\in (2, p)$.\n",
    "  2. Tính $c_1, c_2$:\n",
    "    - $c_1 \\equiv g^x \\pmod p$\n",
    "    - $c_2 \\equiv me^x = mg^{dx} \\pmod p$\n",
    "  3. Gửi $c = (c_1, c_2)$ cho người nhận.\n",
    "  \n",
    "  \n",
    "Người nhận giải mã bằng cách tính $m \\equiv (c_1^d)^{-1}c_2 \\pmod p$\n",
    "\n",
    "**Chú ý**: Chọn $p$ quá nhỏ sẽ dẫn đến trường hợp encrypt ra các kí tự bị collision."
   ]
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   "id": "e02c6932",
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   "source": [
    "# Giải bài toán logarithm rời rạc\n",
    "\n",
    "Các thuật DLP dựa vào độ khó của bài toán logarithm rời rạc (i.e cho $g, h, p$, tìm $x$ sao cho $g^x \\equiv h \\pmod p$). Giải bài toán logarithm rời rạc, thường dùng là thuật toán Index Calculus.\n",
    "\n",
    "## Thuật toán Index Calculus\n",
    "### 1. B-Smooth\n",
    "$n \\in \\mathbb{Z}$ là một số $B$-smooth khi và chỉ khi $\\displaystyle n = \\prod_{p_i \\in \\mathbb{P}} p_i (p_i \\leq B)$ (với $\\mathbb{P}$ là tập các số nguyên tố).\n",
    "\n",
    "### 2. Thuật toán\n",
    "Cho $g^x \\equiv h \\pmod p, g, p$. Tìm $x$?\n",
    "\n",
    "1. Giải $g^x \\equiv l \\pmod p\\ (l \\leq B)$\n",
    "2. Tìm $k$ sao cho $hg^{-k}$ là một số $B$-smooth\n",
    "\n",
    "$\\displaystyle \\iff hg^{-k} = \\prod_{p_i \\in \\mathbb{P}} p_i^{e_{p_i}} \\pmod p\\ (p_i \\leq B) \\\\ \\iff \\log_{g}h = k + \\sum_{p_i \\leq B} e_{p_i} \\log_{g} p_i \\pmod p $\n",
    "\n",
    "### 3. Ví dụ\n",
    "Giải phương trình đồng dư $6^x = 57 \\pmod{107}$\n",
    "\n",
    "1. Chọn $B = 7 \\Rightarrow \\mathbb{B} = \\{2, 3, 5, 7\\}$\n",
    "2. Thử-và-sai nhiều lần:\n",
    "- Chọn $k = 24 \\Rightarrow 6^k \\equiv 42 \\pmod{107} = 2\\times3\\times7$\n",
    "- Chọn $k = 6 \\Rightarrow 6^k \\equiv 4 \\pmod{107} = 2^2$\n",
    "- Chọn $k = 33 \\Rightarrow 6^k \\equiv 15 \\pmod{107} = 3\\times5$\n",
    "- Chọn $k = 34 \\Rightarrow 6^k \\equiv 90 \\pmod{107} = 2\\times3^2\\times5$\n",
    "\n",
    "Điều này tương đương với\n",
    "\n",
    "$$\\begin{cases}\n",
    "24 \\equiv \\log_{6}2 + \\log_{6}3 + \\log_{6}7 \\pmod{107}\\\\\n",
    "6 \\equiv 2\\log_{6}2 \\pmod{107}\\\\\n",
    "33 \\equiv \\log_{6}3 + \\log_{6}5 \\pmod{107}\\\\\n",
    "34 \\equiv \\log_{6}2 + 2\\log_{6}3 + \\log_{6}5 \\pmod{107}\n",
    "\\end{cases}$$\n",
    "\n",
    "Lần lượt đặt $\\log_{6}2 = a, \\log_{6}3 = b, \\log_{6}5 = c, \\log_{6}7 = d$, ta có hệ 4 phương trình 4 ẩn. Giải hệ phương trình, ta thu được các nghiệm\n",
    "\n",
    "$$\\begin{cases}\n",
    "a = \\log_{6}2 = 3\\\\\n",
    "b = \\log_{6}3 = 104\\\\\n",
    "c = \\log_{6}5 = 35\\\\\n",
    "d = \\log_{6}7 = 23\n",
    "\\end{cases}$$\n",
    "\n",
    "(Chú ý đây không phải phép logarit thông thường!)\n",
    "\n",
    "3. Tiếp tục thử-sai nhiều lần: chọn $u$ sao cho $hg^u$ là một số $B$-smooth\n",
    "\n",
    "Chọn $u = 38 \\Rightarrow hg^u \\equiv 35 = 5\\times7 \\pmod{107}$\n",
    "\n",
    "Lấy logarithm 2 vế cho $6$:\n",
    "\n",
    "$\\log_{6}(57\\times 6^{38}) = \\log_{6}5\\times \\log_{6}7 \\iff \\log_{6}57 + 38 = \\log_{6}5 + \\log_{6}7$. Mà $6^x \\equiv 57 \\pmod{107} \\Rightarrow x \\equiv \\log_{6}57 = \\log_{6}5 + \\log_{6}7 - 38 = 35 + 23 - 38 = 20$.\n"
   ]
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   "source": [
    "# Đường cong Elliptic\n",
    "\n",
    "## Định nghĩa:\n",
    "Nhóm $(\\mathbb{G}, +)$ là nhóm các điểm nằm trên đường cong Elliptic $(E): y^2 = x^3 + ax + b$ với:\n",
    "\n",
    "- $\\mathbb{G} = \\{P(x, y) \\in (E), \\forall x, y \\in \\mathbb{Z}_p\\}$\n",
    "- $+:\\mathbb{G} \\times \\mathbb{G}\\to \\mathbb G,\\; P + Q\\mapsto R $ (Chọn 2 điểm $P, Q \\in (E)$, đường thẳng đi qua $P, Q$ cắt $E$ tại điểm thứ 3 là $Z$. Lấy $R$ là đối xứng của $Z$ qua $Ox$).\n",
    "- Nếu $P$ và $Q$ không cắt $(E)$ ở điểm thứ 3 thì $P + Q = O$ với $O$ là điểm vô cực.\n",
    "\n",
    "Hình bên dưới là biểu diễn của đường cong Elliptic $(E): y^2 = x^3 - x + 1, P = (1, -1), Q = (0, -1), R = P + Q$."
   ]
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      "text/plain": [
       "<Figure size 1440x720 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "a = -1\n",
    "b = 1\n",
    "\n",
    "y, x = np.ogrid[-5:5:100j, -5:5:100j]\n",
    "\n",
    "plt.figure(figsize=(20, 10))\n",
    "plt.contour(x.ravel(), y.ravel(), pow(y, 2) - pow(x, 3) - x * a - b, [0])\n",
    "plt.grid()\n",
    "\n",
    "plt.scatter(1, -1)\n",
    "plt.annotate('P', (1, -1), size = 20)\n",
    "plt.scatter(0, -1)\n",
    "plt.annotate('Q', (0, -1), size = 20)\n",
    "plt.axhline(y = -1)\n",
    "\n",
    "plt.scatter(-1, -1)\n",
    "plt.annotate('Z', (-1, -1), size = 20)\n",
    "\n",
    "plt.scatter(-1, 1)\n",
    "plt.annotate('R', (-1, 1), size = 20)\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "982829dd",
   "metadata": {},
   "source": [
    "## Hệ mã ECC (Elliptic Curve Cryptosystem)\n",
    "\n",
    "0. Chọn $p \\in \\mathbb{P}$ và $(E)$ là một đường cong Elliptic; Chọn $G \\in (E)$.\n",
    "1. Chọn $d$ ngẫu nhiên, $e = dG = G + G + .. + G$ ($d$ lần).\n",
    "2. Pha mã hóa: chọn $y$ ngẫu nhiên, $c_1 = yG, c_2 = M + ye$ ($M$ là tin cần mã hóa).\n",
    "3. Pha giải mã: $M = c_2 - dc_1$."
   ]
  }
 ],
 "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": "3.9.7"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}