{
 "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": 69,
   "outputs": [],
   "source": [
    "from src.models import OuterKey, InnerKey, SubstitutionKey"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-07-04T14:12:16.685220Z",
     "start_time": "2024-07-04T14:12:16.446987Z"
    }
   }
  },
  {
   "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": 70,
   "outputs": [],
   "source": [
    "height = 4\n",
    "width = 3"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-07-04T14:12:16.686123Z",
     "start_time": "2024-07-04T14:12:16.453332Z"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "## Operand Types\n",
    "### Inner Permutation Key\n",
    "An inner permutation key (inner key for short) is a list of `height` rows, each a random permutation of an identity array of length `width`"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "outputs": [
    {
     "data": {
      "text/plain": "[[0, 1, 2], [1, 0, 2], [1, 0, 2], [2, 1, 0]]"
     },
     "execution_count": 71,
     "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",
    "i_identity = InnerKey.init_identity_matrix(width, height)\n",
    "\n",
    "i0.matrix"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-07-04T14:12:16.830392Z",
     "start_time": "2024-07-04T14:12:16.460887Z"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "### Outer Permutation Key\n",
    "An outer key is ..."
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "outputs": [
    {
     "data": {
      "text/plain": "[[2, 0, 1, 3]]"
     },
     "execution_count": 72,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "o0 = OuterKey.init_matrix(height)\n",
    "o1 = OuterKey.init_matrix(height)\n",
    "o2 = OuterKey.init_matrix(height)\n",
    "o_identity = OuterKey.init_identity_matrix(height)\n",
    "\n",
    "o0.matrix"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-07-04T14:12:16.831572Z",
     "start_time": "2024-07-04T14:12:16.469310Z"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "### Substitution Key (formerly Alphabet Key or alpha_key)\n",
    "Sub key is ..."
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "outputs": [
    {
     "data": {
      "text/plain": "[[51, 14, 207], [3, 171, 4], [73, 204, 152], [142, 175, 64]]"
     },
     "execution_count": 73,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a0 = SubstitutionKey.init_matrix(width, height)\n",
    "a1 = SubstitutionKey.init_matrix(width, height)\n",
    "a2 = SubstitutionKey.init_matrix(width, height)\n",
    "a_identity = SubstitutionKey.init_identity_matrix(width, height)\n",
    "\n",
    "a0.matrix"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-07-04T14:12:16.832461Z",
     "start_time": "2024-07-04T14:12:16.476713Z"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "## Operators Types\n",
    "### < Outer Permutation\n",
    "#### Substitution Key < Outer Key"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[2, 0, 1, 3]]\n",
      "[[51, 14, 207], [3, 171, 4], [73, 204, 152], [142, 175, 64]]\n",
      "[[73, 204, 152], [51, 14, 207], [3, 171, 4], [142, 175, 64]]\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": "2024-07-04T14:12:16.833320Z",
     "start_time": "2024-07-04T14:12:16.484335Z"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "#### Inner Key < Outer Key"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[2, 0, 1, 3]]\n",
      "[[0, 1, 2], [1, 0, 2], [1, 0, 2], [2, 1, 0]]\n",
      "[[1, 0, 2], [0, 1, 2], [1, 0, 2], [2, 1, 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": "2024-07-04T14:12:16.834157Z",
     "start_time": "2024-07-04T14:12:16.492114Z"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "### << Inner Permutation\n",
    "#### Outer Key << Outer Key"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[0, 1, 3, 2]]\n",
      "[[2, 0, 1, 3]]\n",
      "[[2, 0, 3, 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": "2024-07-04T14:12:16.834973Z",
     "start_time": "2024-07-04T14:12:16.499677Z"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "#### Inner Key << Inner Key"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1, 0, 2], [0, 1, 2], [0, 2, 1], [2, 1, 0]]\n",
      "[[0, 1, 2], [1, 0, 2], [1, 0, 2], [2, 1, 0]]\n",
      "[[1, 0, 2], [1, 0, 2], [1, 2, 0], [0, 1, 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": "2024-07-04T14:12:16.835789Z",
     "start_time": "2024-07-04T14:12:16.507413Z"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "#### Substitution Key << Inner Key"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[0, 1, 2], [1, 0, 2], [1, 0, 2], [2, 1, 0]]\n",
      "[[51, 14, 207], [3, 171, 4], [73, 204, 152], [142, 175, 64]]\n",
      "[[51, 14, 207], [171, 3, 4], [204, 73, 152], [64, 175, 142]]\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": "2024-07-04T14:12:16.854289Z",
     "start_time": "2024-07-04T14:12:16.515159Z"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "### ~Permutation Inversion\n",
    "#### ~Outer Key"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[2, 0, 1, 3]]\n",
      "[[1, 2, 0, 3]]\n"
     ]
    }
   ],
   "source": [
    "print(o0.matrix)\n",
    "print((~o0).matrix)"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-07-04T14:12:16.855384Z",
     "start_time": "2024-07-04T14:12:16.522867Z"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "#### ~Inner Key"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[0, 1, 2], [1, 0, 2], [1, 0, 2], [2, 1, 0]]\n",
      "[[0, 1, 2], [1, 0, 2], [1, 0, 2], [2, 1, 0]]\n"
     ]
    }
   ],
   "source": [
    "print(i0.matrix)\n",
    "print((~i0).matrix)"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-07-04T14:12:16.856225Z",
     "start_time": "2024-07-04T14:12:16.530564Z"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "### ^ Substitution\n",
    "#### Substitution Key ^ Substitution Key"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[51, 14, 207], [3, 171, 4], [73, 204, 152], [142, 175, 64]]\n",
      "[[1, 41, 28], [228, 126, 163], [255, 48, 188], [53, 120, 13]]\n",
      "[[50, 39, 211], [231, 213, 167], [182, 252, 36], [187, 215, 77]]\n",
      "50\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": "2024-07-04T14:12:16.857040Z",
     "start_time": "2024-07-04T14:12:16.538722Z"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "## Substitution Permutation Algebra\n",
    "#### properites:\n",
    "- associative: (a + b) + c = a + (b + c)\n",
    "- commutative: a + b = b + a\n",
    "- identity: a + 0 = a or a * 1 = a\n",
    "- inverse: a + (-a) = 0 or a * 1/a = 1 (for all a ~= 0)\n",
    "- distributive: a(b + c) = ab + ac\n",
    "\n",
    "### Associative\n",
    "#### Outer Key"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "outputs": [
    {
     "data": {
      "text/plain": "True"
     },
     "execution_count": 82,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(o0 << o1 << o2) == ((o0 << o1) << o2) == (o0 << (o1 << o2))"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-07-04T14:12:16.858322Z",
     "start_time": "2024-07-04T14:12:16.546322Z"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "#### Inner Key"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "outputs": [
    {
     "data": {
      "text/plain": "True"
     },
     "execution_count": 83,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(i0 << i1 << i2) == ((i0 << i1) << i2) == (i0 << (i1 << i2))"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-07-04T14:12:16.863346Z",
     "start_time": "2024-07-04T14:12:16.554187Z"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "#### Substitution Key"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "outputs": [
    {
     "data": {
      "text/plain": "True"
     },
     "execution_count": 84,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(a0 ^ a1 ^ a2) == ((a0 ^ a1) ^ a2) == (a0 ^ (a1 ^ a2))"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-07-04T14:12:16.864336Z",
     "start_time": "2024-07-04T14:12:16.562025Z"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "### Commutative Property\n",
    "Substitution is the only key type that is commutative\n",
    "#### Substitution Key"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "outputs": [
    {
     "data": {
      "text/plain": "True"
     },
     "execution_count": 85,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(a0 ^ a1) == (a1 ^ a0)"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-07-04T14:12:16.865178Z",
     "start_time": "2024-07-04T14:12:16.569781Z"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "### Identity Property\n",
    "#### Outer Key"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "outputs": [
    {
     "data": {
      "text/plain": "True"
     },
     "execution_count": 86,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(o0 << o_identity) == o0"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-07-04T14:12:16.866342Z",
     "start_time": "2024-07-04T14:12:16.578046Z"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "#### Inner Key"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "outputs": [
    {
     "data": {
      "text/plain": "True"
     },
     "execution_count": 87,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(i0 << i_identity) == i0"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-07-04T14:12:16.867309Z",
     "start_time": "2024-07-04T14:12:16.589133Z"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "#### Substitution Key"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "outputs": [
    {
     "data": {
      "text/plain": "True"
     },
     "execution_count": 88,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(a0 ^ a_identity) == a0"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-07-04T14:12:16.868143Z",
     "start_time": "2024-07-04T14:12:16.599974Z"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "### Inverse Property\n",
    "#### Outer Key"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "outputs": [
    {
     "data": {
      "text/plain": "True"
     },
     "execution_count": 89,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(o0 << ~o0) == o_identity"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-07-04T14:12:16.868953Z",
     "start_time": "2024-07-04T14:12:16.608436Z"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "#### Inner Key"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "outputs": [
    {
     "data": {
      "text/plain": "True"
     },
     "execution_count": 90,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(i0 << ~i0) == i_identity"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-07-04T14:12:16.869884Z",
     "start_time": "2024-07-04T14:12:16.619030Z"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "#### Substitution Key"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "outputs": [
    {
     "data": {
      "text/plain": "True"
     },
     "execution_count": 91,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(a0 ^ a0) == a_identity"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-07-04T14:12:16.870708Z",
     "start_time": "2024-07-04T14:12:16.629750Z"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "### Distributive\n",
    "#### Inner Key/Outer Key"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "outputs": [
    {
     "data": {
      "text/plain": "True"
     },
     "execution_count": 92,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "((i0 << i1) < o0) == ((i0 < o0) << (i1 < o0))"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-07-04T14:12:16.871520Z",
     "start_time": "2024-07-04T14:12:16.638198Z"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "#### Substitution Key/Outer Key"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "outputs": [
    {
     "data": {
      "text/plain": "True"
     },
     "execution_count": 93,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "((a0 ^ a1) < o0) == ((a0 < o0) ^ (a1 < o0))"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-07-04T14:12:16.872334Z",
     "start_time": "2024-07-04T14:12:16.646323Z"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "#### Substitution Key/Inner Key"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "execution_count": 94,
   "outputs": [
    {
     "data": {
      "text/plain": "True"
     },
     "execution_count": 94,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "((a0 ^ a1) << i0) == ((a0 << i0) ^ (a1 << i0))"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-07-04T14:12:16.873134Z",
     "start_time": "2024-07-04T14:12:16.654286Z"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "#### Substitution Key/Inner Key/Outer Key"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "execution_count": 95,
   "outputs": [
    {
     "data": {
      "text/plain": "True"
     },
     "execution_count": 95,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "((a0 << i0) < o0) == ((a0 < o0) << (i0 < o0))"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-07-04T14:12:16.874410Z",
     "start_time": "2024-07-04T14:12:16.662231Z"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "### Other Examples"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "execution_count": 96,
   "outputs": [
    {
     "data": {
      "text/plain": "True"
     },
     "execution_count": 96,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(a0 << (i0 < o0)) == (((a0 < ~o0) << i0) < o0)"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-07-04T14:12:16.882191Z",
     "start_time": "2024-07-04T14:12:16.670307Z"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": 97,
   "outputs": [
    {
     "data": {
      "text/plain": "True"
     },
     "execution_count": 97,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "((a0 < o0) << i0) == ((a0 << (i0 < ~o0)) < o0)"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-07-04T14:12:16.882308Z",
     "start_time": "2024-07-04T14:12:16.674096Z"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": 98,
   "outputs": [
    {
     "data": {
      "text/plain": "True"
     },
     "execution_count": 98,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(~(i0 << i1)) == (~i1 << ~i0)"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-07-04T14:12:16.882394Z",
     "start_time": "2024-07-04T14:12:16.677061Z"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": 99,
   "outputs": [
    {
     "data": {
      "text/plain": "True"
     },
     "execution_count": 99,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(~(o0 << o1)) == (~o1 << ~o0)"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-07-04T14:12:16.883495Z",
     "start_time": "2024-07-04T14:12:16.683278Z"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": 100,
   "outputs": [
    {
     "data": {
      "text/plain": "True"
     },
     "execution_count": 100,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "i0 == ((i0 << i2 << i1) << ~(i2 << i1)) == ((i0 << i2 << i1) << ~i1 << ~i2)"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-07-04T14:12:16.883574Z",
     "start_time": "2024-07-04T14:12:16.687331Z"
    }
   }
  },
  {
   "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": 101,
   "outputs": [
    {
     "data": {
      "text/plain": "True"
     },
     "execution_count": 101,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "i0 < (o0 << o1 << o2) != (i0 < (o0 << o1 << o2)) # not equal !!!!!!"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2024-07-04T14:12:16.883645Z",
     "start_time": "2024-07-04T14:12:16.690509Z"
    }
   }
  }
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