{
 "cells": [
  {
   "cell_type": "markdown",
   "source": [
    "# Permutation Algebra\n",
    "This notebook is a prerequisite to following the DARC tutorial"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "outputs": [],
   "source": [
    "from src.models import OuterKey, InnerKey, SubstitutionKey"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-07-03T17:15:30.365133Z",
     "start_time": "2024-07-03T17:15:30.328857Z"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "## Key Generation Parameters\n",
    "- block size: The number of characters we encode in a block. Block size isn't using in this notebook\n",
    "- key height: The alphabet length. If we want to encode bytes, our alphabet length is 256.\n",
    "    If we want to encode lowercase letters a-z our alphabet length is 26. NKodes 10 key 7 attribute alphabet is 70.\n",
    "- key width: The number of bytes an encrypted charter is in our alphabet.\n"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "outputs": [],
   "source": [
    "height = 4\n",
    "width = 3"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-07-03T17:15:30.365712Z",
     "start_time": "2024-07-03T17:15:30.332512Z"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "## Operand Types\n",
    "### Inner Key\n",
    "An inner key is a list of `height` rows, each a random permutation of an identity arrary of length `width`"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "outputs": [
    {
     "data": {
      "text/plain": "[[1, 2, 0], [0, 1, 2], [2, 1, 0], [2, 0, 1]]"
     },
     "execution_count": 84,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "i0 = InnerKey.init_matrix(width, height)\n",
    "i1 = InnerKey.init_matrix(width, height)\n",
    "i2 = InnerKey.init_matrix(width, height)\n",
    "\n",
    "i0.matrix"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-07-03T17:15:30.366169Z",
     "start_time": "2024-07-03T17:15:30.335990Z"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "### Outer Key\n",
    "An outer key is ..."
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "outputs": [
    {
     "data": {
      "text/plain": "[[1, 2, 3, 0]]"
     },
     "execution_count": 85,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "o0 = OuterKey.init_matrix(height)\n",
    "o1 = OuterKey.init_matrix(height)\n",
    "o2 = OuterKey.init_matrix(height)\n",
    "\n",
    "o0.matrix"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-07-03T17:15:30.405313Z",
     "start_time": "2024-07-03T17:15:30.341422Z"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "### Substitution Key (formerly AlphabetKey or alpha_key)\n",
    "Sub key is ..."
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "execution_count": 106,
   "outputs": [
    {
     "data": {
      "text/plain": "[[237, 92, 1], [165, 114, 152], [38, 71, 114], [52, 116, 61]]"
     },
     "execution_count": 106,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a0 = SubstitutionKey.init_matrix(width, height)\n",
    "a1 = SubstitutionKey.init_matrix(width, height)\n",
    "\n",
    "a0.matrix"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-07-03T17:26:59.359984Z",
     "start_time": "2024-07-03T17:26:59.355497Z"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "## Operators Types\n",
    "### ***<*** Outer Permutation\n",
    "#### Substitution Key < Outer Key"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "execution_count": 107,
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1, 2, 3, 0]]\n",
      "[[237, 92, 1], [165, 114, 152], [38, 71, 114], [52, 116, 61]]\n",
      "[[165, 114, 152], [38, 71, 114], [52, 116, 61], [237, 92, 1]]\n"
     ]
    }
   ],
   "source": [
    "a0_o0 = a0 < o0\n",
    "print(o0.matrix)\n",
    "print(a0.matrix)\n",
    "print(a0_o0.matrix)"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-07-03T17:27:00.733689Z",
     "start_time": "2024-07-03T17:27:00.729585Z"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "#### Inner Key < Outer Key"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "execution_count": 108,
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1, 2, 3, 0]]\n",
      "[[1, 2, 0], [0, 1, 2], [2, 1, 0], [2, 0, 1]]\n",
      "[[0, 1, 2], [2, 1, 0], [2, 0, 1], [1, 2, 0]]\n"
     ]
    }
   ],
   "source": [
    "i0_o0 = i0 < o0\n",
    "print(o0.matrix)\n",
    "print(i0.matrix)\n",
    "print(i0_o0.matrix)"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-07-03T17:27:04.433733Z",
     "start_time": "2024-07-03T17:27:04.429380Z"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "### ***<<*** Inner Permutation\n",
    "#### Outer Key << OuterKey"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "execution_count": 109,
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[2, 1, 3, 0]]\n",
      "[[1, 2, 3, 0]]\n",
      "[[3, 2, 0, 1]]\n"
     ]
    }
   ],
   "source": [
    "o0_o1 = o0 << o1\n",
    "print(o1.matrix)\n",
    "print(o0.matrix)\n",
    "print(o0_o1.matrix)"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-07-03T17:27:08.293098Z",
     "start_time": "2024-07-03T17:27:08.288126Z"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "#### Inner Key << Inner Key"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "execution_count": 110,
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[0, 1, 2], [0, 2, 1], [0, 1, 2], [2, 1, 0]]\n",
      "[[1, 2, 0], [0, 1, 2], [2, 1, 0], [2, 0, 1]]\n",
      "[[1, 2, 0], [0, 2, 1], [2, 1, 0], [1, 0, 2]]\n"
     ]
    }
   ],
   "source": [
    "i0_i1 = i0 << i1\n",
    "print(i1.matrix)\n",
    "print(i0.matrix)\n",
    "print(i0_i1.matrix)"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-07-03T17:27:11.574355Z",
     "start_time": "2024-07-03T17:27:11.568652Z"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "#### Substitution Key << Inner Key"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "execution_count": 111,
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1, 2, 0], [0, 1, 2], [2, 1, 0], [2, 0, 1]]\n",
      "[[237, 92, 1], [165, 114, 152], [38, 71, 114], [52, 116, 61]]\n",
      "[[92, 1, 237], [165, 114, 152], [114, 71, 38], [61, 52, 116]]\n"
     ]
    }
   ],
   "source": [
    "a0_i0 = a0 << i0\n",
    "print(i0.matrix)\n",
    "print(a0.matrix)\n",
    "print(a0_i0.matrix)"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-07-03T17:27:14.277217Z",
     "start_time": "2024-07-03T17:27:14.272132Z"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "### ***^*** Substitution\n",
    "#### Substitution Key ^ Substitution Key"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "execution_count": 114,
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[237, 92, 1], [165, 114, 152], [38, 71, 114], [52, 116, 61]]\n",
      "[[184, 159, 115], [199, 221, 35], [158, 70, 86], [74, 43, 129]]\n",
      "[[85, 195, 114], [98, 175, 187], [184, 1, 36], [126, 95, 188]]\n",
      "85\n"
     ]
    }
   ],
   "source": [
    "a0_a1 = a0 ^ a1\n",
    "print(a0.matrix)\n",
    "print(a1.matrix)\n",
    "print(a0_a1.matrix)\n",
    "print(a0.matrix[0][0] ^ a1.matrix[0][0])"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-07-03T17:29:15.922958Z",
     "start_time": "2024-07-03T17:29:15.915126Z"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "## dddd"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "outputs": [
    {
     "data": {
      "text/plain": "True"
     },
     "execution_count": 88,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a0_i0_o0_v0 = a0 << (i0 < o0)\n",
    "a0_i0_o0_v1 = ((a0 < ~o0) << i0) < o0\n",
    "\n",
    "a0_i0_o0_v0 == a0_i0_o0_v1"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-07-03T17:15:30.408639Z",
     "start_time": "2024-07-03T17:15:30.361902Z"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "outputs": [
    {
     "data": {
      "text/plain": "True"
     },
     "execution_count": 89,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a0_o0_i0_v0 = (a0 < o0) << i0\n",
    "a0_o0_i0_v1 = (a0 << (i0 < ~o0)) < o0\n",
    "a0_i0_o0_v0 == a0_i0_o0_v1"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-07-03T17:15:30.408838Z",
     "start_time": "2024-07-03T17:15:30.365269Z"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "outputs": [
    {
     "data": {
      "text/plain": "True"
     },
     "execution_count": 90,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a0_o0_i0_v0 = (a0 < o0) << i0\n",
    "a0_o0_i0_v1 = (a0 << (i0 < ~o0)) < o0\n",
    "a0_o0_i0_v0 == a0_o0_i0_v1"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-07-03T17:15:30.408892Z",
     "start_time": "2024-07-03T17:15:30.368768Z"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "outputs": [
    {
     "data": {
      "text/plain": "True"
     },
     "execution_count": 91,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(~(i0 << i1)) == (~i1 << ~i0)"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-07-03T17:15:30.408977Z",
     "start_time": "2024-07-03T17:15:30.371377Z"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "outputs": [
    {
     "data": {
      "text/plain": "True"
     },
     "execution_count": 92,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "i0_i1_i2_v0 = i0 << i1 << i2\n",
    "i0_i1_i2_v1 = i0 << (i1 << i2)\n",
    "i0_i1_i2_v2 = (i0 << i1) << i2\n",
    "i0_i1_i2_v0 == i0_i1_i2_v1 == i0_i1_i2_v2"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-07-03T17:15:30.409076Z",
     "start_time": "2024-07-03T17:15:30.375604Z"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "outputs": [
    {
     "data": {
      "text/plain": "True"
     },
     "execution_count": 93,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "i0 == ((i0 << i2 << i1) << ~(i2 << i1)) == ((i0 << i2 << i1) << ~i1 << ~i2)"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-07-03T17:15:30.409139Z",
     "start_time": "2024-07-03T17:15:30.378496Z"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": 94,
   "outputs": [
    {
     "data": {
      "text/plain": "True"
     },
     "execution_count": 94,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "i0_o0_o1_o2_v1 = ((i0 < o0) < o1) < o2\n",
    "i0_o0_o1_o2_v2 = (i0 < (o0 << o1)) < o2\n",
    "i0_o0_o1_o2_v3 = (i0 < o0) < (o1 << o2)\n",
    "i0_o0_o1_o2_v4 = (i0 < (o0 << o1 << o2))\n",
    "\n",
    "i0_o0_o1_o2_v1 == i0_o0_o1_o2_v2 == i0_o0_o1_o2_v3 == i0_o0_o1_o2_v4"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-07-03T17:15:30.409210Z",
     "start_time": "2024-07-03T17:15:30.381727Z"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "### Becareful about your order of operation\n",
    "***Always use parenthesis to control the order of operations***"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "execution_count": 95,
   "outputs": [
    {
     "data": {
      "text/plain": "True"
     },
     "execution_count": 95,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "i0 < (o0 << o1 << o2) != (i0 < (o0 << o1 << o2)) # not equal !!!!!!"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-07-03T17:15:30.409260Z",
     "start_time": "2024-07-03T17:15:30.384147Z"
    }
   }
  }
 ],
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