day 51 - rabin-miller.ipynb

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    "from random import randrange"
   ]
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   "source": [
    "## algorithm"
   ]
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    "def rabin_miller(prime, tests):\n",
    "    if prime < 5:\n",
    "        return prime in [2, 3]\n",
    "    \n",
    "    # set: prime = q * 2**r + 1\n",
    "    q, r = prime - 1, 0\n",
    "    while not q & 1:\n",
    "        q >>= 1\n",
    "        r += 1\n",
    "\n",
    "    # test repeatedly\n",
    "    for _ in range(tests):\n",
    "        a = randrange(2, prime - 1)\n",
    "\n",
    "        # pass if: a**q == 1\n",
    "        x = pow(a, q, prime)\n",
    "        if x in [1, prime - 1]:\n",
    "            continue\n",
    "\n",
    "        # pass if: a**(q * 2**s) == -1, s < r\n",
    "        for _ in range(r - 1):\n",
    "            x = pow(x, 2, prime)\n",
    "            if x == prime - 1:\n",
    "                break\n",
    "        else:\n",
    "            return False\n",
    "\n",
    "    return True"
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   "source": [
    "def prime(bits, tests):\n",
    "    while True:\n",
    "        # random number in [2**bits .. 2**(bits+1)-1]\n",
    "        prime = (1 << bits) | randrange(1 << bits) | 1\n",
    "\n",
    "        # primality test\n",
    "        if rabin_miller(prime, tests):\n",
    "            return prime"
   ]
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## primes"
   ]
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       "491"
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   "source": [
    "prime(8, 32)"
   ]
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     "execution_count": 5,
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   "source": [
    "prime(256, 32)"
   ]
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     "execution_count": 6,
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   "source": [
    "prime(1024, 32)"
   ]
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   "execution_count": null,
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