Add Miller-Rabin
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111
TTM4135.ipynb
111
TTM4135.ipynb
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@ -46,7 +46,7 @@
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},
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"cell_type": "code",
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"execution_count": 15,
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"execution_count": 2,
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"metadata": {},
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"outputs": [
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{
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@ -245,14 +245,14 @@
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},
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{
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"cell_type": "code",
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"execution_count": 16,
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"execution_count": 7,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"1105 is composite\n"
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"1105 is probable prime\n"
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]
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}
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],
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@ -283,21 +283,93 @@
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"cell_type": "markdown",
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"metadata": {},
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"outputs": [],
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"source": [
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"# Miller-Rabin\n",
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"n = 137\n",
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"k = 20\n",
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"## Miller-Rabin\n",
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"From https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test \n",
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"This test is not fooled by Charmichael numbers.\n",
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"\n",
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"To test `n`, you choose different `a` values, and test if \n",
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"\n",
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"> $a^{d*2^i} \\equiv 1 \\pmod{n}$ for $i = (0, 1, 2, ... r)$\n",
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"\n",
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"If this happens, look at the congruence on the calculation just before you got `1`. If that value is not -1 (aka n-1), the value is composite aka not prime."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 23,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"168 = 21*2^3 = 168\n",
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"65536 = 1*2^16 = 65536\n",
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"\n",
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"16127 is probably prime.\n",
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"561 is composite.\n",
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"2465 is composite.\n",
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"65537 is probably prime.\n"
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]
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}
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],
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"source": [
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"# Miller-Rabin primality test\n",
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"# This test is not fooled by Charmichael numbers.\n",
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"\n",
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"from math import log2, floor\n",
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"from random import randint\n",
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"def miller_rabin(n):\n",
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" for O in range(k):\n",
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" d = n-1\n",
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" a = randint(2,n-2)\n",
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" x = (a**d) % "
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"\n",
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"def decompose_even_number(n):\n",
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" '''Find the values r and d such that n == d * 2**r.'''\n",
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" r = 0\n",
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" d = n\n",
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" for i in range(1, floor(log2(n)) + 1):\n",
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" if (d/2).is_integer():\n",
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" d = d/2\n",
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" r += 1\n",
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" else:\n",
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" break\n",
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" return (r, int(d))\n",
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"\n",
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"verbose = False # Print during execution\n",
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"\n",
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"def miller_rabin_prime(n):\n",
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" ''':param n: should be odd, and n>3. No primes can be even, for obvious reasons.'''\n",
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" accuracy = 3 # How many different `a` to test\n",
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" r, d = decompose_even_number(n-1) # n is odd, thus n-1 is even\n",
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" verbose and print(f\"Testing {n} a maximum of {accuracy} times...\")\n",
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" \n",
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" for _ in range(accuracy):\n",
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" try:\n",
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" a = randint(2, n-2) # The \"witness\". TODO: don't reuse a witness.\n",
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" verbose and print(f\"\\tTesting {n} with a={a}\")\n",
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"\n",
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" x = (a**d) % n\n",
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" if x == 1 or x == n-1:\n",
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" verbose and print(f\"\\t({a}**{d})%{n} was 1 or -1 already!\")\n",
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" continue # This witness was immediately 1 or -1 (aka n-1), so it can't help us any more.\n",
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" for squaring in range(r-1):\n",
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" x = (x**2) % n\n",
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" verbose and print(f\"\\tx={x}\")\n",
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" if x == n-1:\n",
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" raise Exception(\"Try next witness\") # This witness can't help any more.\n",
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" return \"composite\" # The witness didnt uphold the criteria for a prime, so n is composite!\n",
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" except Exception as tryNext:\n",
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" continue\n",
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" return \"probably prime\"\n",
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" \n",
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"s, d = decompose_even_number(168)\n",
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"print(f\"168 = {d}*2^{s} = {d*(2**s)}\")\n",
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"s, d = decompose_even_number(65536)\n",
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"print(f\"65536 = {d}*2^{s} = {d*(2**s)}\")\n",
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"print()\n",
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"\n",
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"for value in [16127, 561, 2465, 65537]:\n",
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" print(f\"{value} is {miller_rabin_prime(value)}.\")\n"
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]
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},
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{
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@ -371,7 +443,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 14,
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"execution_count": 35,
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"metadata": {},
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"outputs": [
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{
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@ -446,13 +518,6 @@
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"- `P = D(C, K)`: Decryption function"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": []
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},
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{
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"cell_type": "code",
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"execution_count": 10,
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@ -501,5 +566,5 @@
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}
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},
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"nbformat": 4,
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"nbformat_minor": 2
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"nbformat_minor": 4
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}
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