{
 "cells": [
  {
   "cell_type": "markdown",
   "source": [
    "# Permutation Algebra\n",
    "This notebook is a prerequisite to following the DARC tutorial"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "source": [
    "from src.models import OuterKey, InnerKey, SubstitutionKey"
   ],
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "execution_count": null
  },
  {
   "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",
   "source": [
    "height = 4\n",
    "width = 3"
   ],
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "execution_count": null
  },
  {
   "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",
   "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
   },
   "outputs": [],
   "execution_count": null
  },
  {
   "cell_type": "markdown",
   "source": [
    "### Outer Permutation Key\n",
    "An outer key is ..."
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "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
   },
   "outputs": [],
   "execution_count": null
  },
  {
   "cell_type": "markdown",
   "source": [
    "### Substitution Key\n",
    "Sub key is ..."
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "source": [
    "s0 = SubstitutionKey.init_matrix(width, height)\n",
    "s1 = SubstitutionKey.init_matrix(width, height)\n",
    "s2 = SubstitutionKey.init_matrix(width, height)\n",
    "s_identity = SubstitutionKey.init_identity_matrix(width, height)\n",
    "\n",
    "s0.matrix"
   ],
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "execution_count": null
  },
  {
   "cell_type": "markdown",
   "source": [
    "## Operators Types\n",
    "### < Outer Permutation\n",
    "#### Substitution Key < Outer Key"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "source": [
    "s0_o0 = s0 < o0\n",
    "print(o0.matrix)\n",
    "print(s0.matrix)\n",
    "print(s0_o0.matrix)"
   ],
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "execution_count": null
  },
  {
   "cell_type": "markdown",
   "source": [
    "#### Inner Key < Outer Key"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "source": [
    "i0_o0 = i0 < o0\n",
    "print(o0.matrix)\n",
    "print(i0.matrix)\n",
    "print(i0_o0.matrix)"
   ],
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "execution_count": null
  },
  {
   "cell_type": "markdown",
   "source": [
    "### << Inner Permutation\n",
    "#### Outer Key << Outer Key"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "source": [
    "o0_o1 = o0 << o1\n",
    "print(o1.matrix)\n",
    "print(o0.matrix)\n",
    "print(o0_o1.matrix)"
   ],
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "execution_count": null
  },
  {
   "cell_type": "markdown",
   "source": [
    "#### Inner Key << Inner Key"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "source": [
    "i0_i1 = i0 << i1\n",
    "print(i1.matrix)\n",
    "print(i0.matrix)\n",
    "print(i0_i1.matrix)"
   ],
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "execution_count": null
  },
  {
   "cell_type": "markdown",
   "source": [
    "#### Substitution Key << Inner Key"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "source": [
    "s0_i0 = s0 << i0\n",
    "print(i0.matrix)\n",
    "print(s0.matrix)\n",
    "print(s0_i0.matrix)"
   ],
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "execution_count": null
  },
  {
   "cell_type": "markdown",
   "source": [
    "### ~Permutation Inversion\n",
    "#### ~Outer Key"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "source": [
    "print(o0.matrix)\n",
    "print((~o0).matrix)"
   ],
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "execution_count": null
  },
  {
   "cell_type": "markdown",
   "source": [
    "#### ~Inner Key"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "source": [
    "print(i0.matrix)\n",
    "print((~i0).matrix)"
   ],
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "execution_count": null
  },
  {
   "cell_type": "markdown",
   "source": [
    "### ^ Substitution\n",
    "#### Substitution Key ^ Substitution Key"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "source": [
    "s0_s1 = s0 ^ s1\n",
    "print(s0.matrix)\n",
    "print(s1.matrix)\n",
    "print(s0_s1.matrix)\n",
    "print(s0.matrix[0][0] ^ s1.matrix[0][0])"
   ],
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "execution_count": null
  },
  {
   "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",
   "source": [
    "(o0 << o1 << o2) == ((o0 << o1) << o2) == (o0 << (o1 << o2))"
   ],
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "execution_count": null
  },
  {
   "cell_type": "markdown",
   "source": [
    "#### Inner Key"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "source": [
    "(i0 << i1 << i2) == ((i0 << i1) << i2) == (i0 << (i1 << i2))"
   ],
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "execution_count": null
  },
  {
   "cell_type": "markdown",
   "source": [
    "#### Substitution Key"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "source": "(s0 ^ s1 ^ s2) == ((s0 ^ s1) ^ s2) == (s0 ^ (s1 ^ s2))",
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "execution_count": null
  },
  {
   "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",
   "source": "(s0 ^ s1) == (s1 ^ s0)",
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "execution_count": null
  },
  {
   "cell_type": "markdown",
   "source": [
    "### Identity Property\n",
    "#### Outer Key"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "source": [
    "(o0 << o_identity) == o0"
   ],
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "execution_count": null
  },
  {
   "cell_type": "markdown",
   "source": [
    "#### Inner Key"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "source": [
    "(i0 << i_identity) == i0"
   ],
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "execution_count": null
  },
  {
   "cell_type": "markdown",
   "source": [
    "#### Substitution Key"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "source": "(s0 ^ s_identity) == s0",
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "execution_count": null
  },
  {
   "cell_type": "markdown",
   "source": [
    "### Inverse Property\n",
    "#### Outer Key"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "source": [
    "(o0 << ~o0) == o_identity"
   ],
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "execution_count": null
  },
  {
   "cell_type": "markdown",
   "source": [
    "#### Inner Key"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "source": [
    "(i0 << ~i0) == i_identity"
   ],
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "execution_count": null
  },
  {
   "cell_type": "markdown",
   "source": [
    "#### Substitution Key"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "source": "(s0 ^ s0) == s_identity",
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "execution_count": null
  },
  {
   "cell_type": "markdown",
   "source": [
    "### Distributive\n",
    "#### Inner Key/Outer Key"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "source": [
    "((i0 << i1) < o0) == ((i0 < o0) << (i1 < o0))"
   ],
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "execution_count": null
  },
  {
   "cell_type": "markdown",
   "source": [
    "#### Substitution Key/Outer Key"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "source": "((s0 ^ s1) < o0) == ((s0 < o0) ^ (s1 < o0))",
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "execution_count": null
  },
  {
   "cell_type": "markdown",
   "source": [
    "#### Substitution Key/Inner Key"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "source": "((s0 ^ s1) << i0) == ((s0 << i0) ^ (s1 << i0))",
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "execution_count": null
  },
  {
   "cell_type": "markdown",
   "source": [
    "#### Substitution Key/Inner Key/Outer Key"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "source": "((s0 << i0) < o0) == ((s0 < o0) << (i0 < o0))",
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "execution_count": null
  },
  {
   "cell_type": "markdown",
   "source": [
    "### Other Examples"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "source": "(s0 << (i0 < o0)) == (((s0 < ~o0) << i0) < o0)",
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "execution_count": null
  },
  {
   "cell_type": "code",
   "source": "((s0 < o0) << i0) == ((s0 << (i0 < ~o0)) < o0)",
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "execution_count": null
  },
  {
   "cell_type": "code",
   "source": [
    "(~(i0 << i1)) == (~i1 << ~i0)"
   ],
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "execution_count": null
  },
  {
   "cell_type": "code",
   "source": [
    "(~(o0 << o1)) == (~o1 << ~o0)"
   ],
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "execution_count": null
  },
  {
   "cell_type": "code",
   "source": [
    "i0 == ((i0 << i2 << i1) << ~(i2 << i1)) == ((i0 << i2 << i1) << ~i1 << ~i2)"
   ],
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "execution_count": null
  },
  {
   "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",
   "source": [
    "i0 < (o0 << o1 << o2) != (i0 < (o0 << o1 << o2)) # not equal !!!!!!"
   ],
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "execution_count": null
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.6"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}