{
 "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": 35,
   "outputs": [],
   "source": [
    "from src.models import OuterKey, InnerKey, SubstitutionKey"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-07-03T16:38:57.906547Z",
     "start_time": "2024-07-03T16:38:57.867931Z"
    }
   }
  },
  {
   "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": 36,
   "outputs": [],
   "source": [
    "height = 4\n",
    "width = 3"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-07-03T16:38:57.907029Z",
     "start_time": "2024-07-03T16:38:57.872544Z"
    }
   }
  },
  {
   "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": 37,
   "outputs": [
    {
     "data": {
      "text/plain": "[[2, 0, 1], [0, 2, 1], [0, 2, 1], [1, 2, 0]]"
     },
     "execution_count": 37,
     "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": "2024-07-03T16:38:57.907507Z",
     "start_time": "2024-07-03T16:38:57.876023Z"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "### Outer Key\n",
    "An outer key is ..."
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "outputs": [
    {
     "data": {
      "text/plain": "[[3, 2, 1, 0]]"
     },
     "execution_count": 38,
     "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": "2024-07-03T16:38:57.954081Z",
     "start_time": "2024-07-03T16:38:57.880359Z"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "### Substitution Key (formerly AlphabetKey or alpha_key)\n",
    "Sub key is ..."
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "outputs": [
    {
     "data": {
      "text/plain": "[[21, 62, 42], [151, 171, 33], [31, 131, 141], [100, 202, 161]]"
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a0 = SubstitutionKey.init_matrix(width, height)\n",
    "a1 = SubstitutionKey.init_matrix(width, height)\n",
    "a0.matrix"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-07-03T16:38:57.954770Z",
     "start_time": "2024-07-03T16:38:57.884556Z"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "## Operators Types\n",
    "### ***<<*** Inner Permutation"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "outputs": [],
   "source": [],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-07-03T16:38:57.955162Z",
     "start_time": "2024-07-03T16:38:57.887640Z"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "### ***<*** Outer Permutation"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "outputs": [],
   "source": [],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-07-03T16:38:57.955526Z",
     "start_time": "2024-07-03T16:38:57.890158Z"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "### ***^*** Substitution"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "outputs": [],
   "source": [],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-07-03T16:38:57.957721Z",
     "start_time": "2024-07-03T16:38:57.892476Z"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "## dddd"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "outputs": [
    {
     "data": {
      "text/plain": "True"
     },
     "execution_count": 40,
     "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": "2024-07-03T16:38:57.957791Z",
     "start_time": "2024-07-03T16:38:57.896696Z"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "outputs": [
    {
     "data": {
      "text/plain": "True"
     },
     "execution_count": 41,
     "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": "2024-07-03T16:38:57.958249Z",
     "start_time": "2024-07-03T16:38:57.900751Z"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "outputs": [
    {
     "data": {
      "text/plain": "True"
     },
     "execution_count": 42,
     "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": "2024-07-03T16:38:57.958404Z",
     "start_time": "2024-07-03T16:38:57.903708Z"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "outputs": [
    {
     "data": {
      "text/plain": "True"
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(~(i0 << i1)) == (~i1 << ~i0)"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-07-03T16:42:45.756613Z",
     "start_time": "2024-07-03T16:42:45.746766Z"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "outputs": [
    {
     "data": {
      "text/plain": "True"
     },
     "execution_count": 44,
     "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": "2024-07-03T16:38:57.958613Z",
     "start_time": "2024-07-03T16:38:57.910939Z"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "outputs": [
    {
     "data": {
      "text/plain": "True"
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "i0 == ((i0 << i2 << i1) << ~(i2 << i1)) == ((i0 << i2 << i1) << ~i1 << ~i2)"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-07-03T16:41:01.725294Z",
     "start_time": "2024-07-03T16:41:01.718345Z"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "outputs": [
    {
     "data": {
      "text/plain": "True"
     },
     "execution_count": 46,
     "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": "2024-07-03T16:38:57.958859Z",
     "start_time": "2024-07-03T16:38:57.915888Z"
    }
   }
  },
  {
   "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": 47,
   "outputs": [
    {
     "data": {
      "text/plain": "True"
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "i0 < (o0 << o1 << o2) != (i0 < (o0 << o1 << o2)) # not equal !!!!!!"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-07-03T16:38:57.958925Z",
     "start_time": "2024-07-03T16:38:57.920831Z"
    }
   }
  }
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