MLD_Example.ipynb

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   "source": [
    "## Multilinear Detection"
   ]
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   "source": [
    "import sys\n",
    "import math\n",
    "import warnings\n",
    "from random import Random\n",
    "import sympy as sp\n",
    "from IPython.display import display_latex\n",
    "\n",
    "warnings.filterwarnings('ignore')\n",
    "import galois"
   ]
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    "def solve(p, n, k, max_degree, fingerprint_size, rand: Random):\n",
    "    \"\"\"Implementation of Algorithm 1 in Koutis and Williams, 2016.\"\"\"\n",
    "\n",
    "    v = [rand.randint(0, 2 ** k - 1) for _ in range(n)]\n",
    "    ell = 3 + int(math.ceil(math.log2(max_degree)))\n",
    "    GF = galois.GF(2, ell)\n",
    "    A_tilde = [GF(rand.randint(0, 2 ** ell - 1)) for _ in range(fingerprint_size)]\n",
    "    S = GF(0)\n",
    "    \n",
    "    for j in range(2 ** k):\n",
    "        X_tilde = [(vi & j).bit_count() % 2 for vi in v]\n",
    "        S = S + p(X_tilde, A_tilde)\n",
    "\n",
    "    return bool(S)"
   ]
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    "class Degree:\n",
    "    def __init__(self, d): self.d = d\n",
    "    def __add__(self, other): return self if isinstance(other, int) else Degree(max(self.d, other.d))\n",
    "    def __mul__(self, other): return self if isinstance(other, int) else Degree(self.d + other.d)\n",
    "    __radd__ = __add__\n",
    "    __rmul__ = __mul__\n",
    "    def __int__(self): return self.d\n",
    "    def __repr__(self): return f'{self.d}'"
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    "class Circuit:\n",
    "    def __init__(self, p, n, fingerprint_size):\n",
    "        print('Symbolic evaluation:')\n",
    "        display_latex(p(sp.symbols([f'x_{i}' for i in range(n)]), [1 for _ in range(fingerprint_size)]))\n",
    "        print('Expanded:')\n",
    "        display_latex(sp.expand(p(sp.symbols([f'x_{i}' for i in range(n)]), [1 for _ in range(fingerprint_size)])))\n",
    "        print('With fingerprint:')\n",
    "        display_latex(sp.expand(p(sp.symbols([f'x_{i}' for i in range(n)]), sp.symbols([f'a_{i}' for i in range(fingerprint_size)]))))\n",
    "\n",
    "        d = int(p([Degree(1) for _ in range(n)], [1 for _ in range(fingerprint_size)]))\n",
    "        print('Max degree:', d)\n",
    "        \n",
    "        self.p = p\n",
    "        self.n = n\n",
    "        self.fingerprint_size = fingerprint_size\n",
    "        self.d = d\n"
   ]
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    "def run(C, k, rand: Random, num_iterations: int=100):\n",
    "    print(f'Running: k={k}')\n",
    "    count = 0\n",
    "    for _ in range(num_iterations):\n",
    "        if solve(C.p, C.n, k, C.d, C.fingerprint_size, rand):\n",
    "            sys.stdout.write('T')\n",
    "            count += 1\n",
    "        else:\n",
    "            sys.stdout.write('F')\n",
    "    print(f'\\n{count}/{num_iterations}')"
   ]
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   "cell_type": "markdown",
   "id": "a461f659-1f84-4b4f-900c-154d0caa78f6",
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   "source": [
    "### Example"
   ]
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     "text": [
      "Symbolic evaluation:\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x_{0}^{4} + x_{2} \\left(x_{0} + x_{1}\\right)$"
      ]
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     "text": [
      "Expanded:\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x_{0}^{4} + x_{0} x_{2} + x_{1} x_{2}$"
      ]
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     "text": [
      "With fingerprint:\n"
     ]
    },
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     "data": {
      "text/latex": [
       "$\\displaystyle a_{0} a_{2} x_{0} x_{2} + a_{1} a_{2} x_{1} x_{2} + a_{3} x_{0}^{4}$"
      ]
     },
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     "text": [
      "Max degree: 4\n",
      "Running: k=4\n",
      "FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF\n",
      "0/100\n",
      "Running: k=3\n",
      "FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF\n",
      "0/100\n",
      "Running: k=2\n",
      "TFTTFTTFFFTTTFTFFTFFTTFTTFFTTFFTTFFTFFTFFTFTFTFTTFTFTFTTFTFTTFFTTFTFTFFTTTTFFTTTTFFTTTTTFTFFTTTTTTFT\n",
      "57/100\n",
      "Running: k=1\n",
      "TFTTTTFFTFFTFFTFFFTFTTFTTFFTTFTTTFTFFFTFTTFTFFTTTTFTTTTTFTTTFFTFTTFFTFTFTTTFFFTTTTFFFTFTFTTTFTTFFTFF\n",
      "55/100\n"
     ]
    }
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   "source": [
    "C = Circuit(lambda X, A: ((X[0] * A[0] + X[1] * A[1]) * X[2]) * A[2] + (X[0] * X[0] * X[0] * X[0]) * A[3], 3, 4)\n",
    "\n",
    "run(C, 4, Random(12345))\n",
    "run(C, 3, Random(12345))\n",
    "run(C, 2, Random(12345))\n",
    "run(C, 1, Random(12345))"
   ]
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