2018-03-20-PostScryptum.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Using elliptic curves and isogenies in Sage\n",
    "\n",
    "## Finite fields"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We create finite fields by passing their cardinality"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "Fp = GF(11)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Finite Field of size 11"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Fp"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Finite Field in z2 of size 11^2"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Fq = GF(11^2)\n",
    "Fq"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For extension fields, the generator is obtained with the `.gen()` function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "z2"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z = Fq.gen()\n",
    "z"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z^120"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Same thing in one go"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Finite Field in t of size 340282366920938463463374607431768211507^2"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "K.<t> = GF(next_prime(2^128)^2)\n",
    "K"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Elliptic curves"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Curves over $ℚ$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Elliptic Curve defined by y^2 = x^3 - 10*x + 10 over Rational Field"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E = EllipticCurve([-10,10])\n",
    "E"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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OH5buvVf69FPp7belhx92XREiAcEOAIAg27tXatFCmjtXGjNGuu021xUhUhDsAAAIop07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\nQmCC\n",
      "text/plain": [
       "Graphics object consisting of 2 graphics primitives"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E.plot()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Cuvers over other fields"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Elliptic Curve defined by y^2 = x^3 + x over Finite Field of size 11"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F = EllipticCurve(GF(11), [1, 0])\n",
    "F"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "12"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F.order()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "12"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F.cardinality()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(0 : 0 : 1), (0 : 1 : 0), (5 : 3 : 1), (5 : 8 : 1), (7 : 3 : 1), (7 : 8 : 1), (8 : 5 : 1), (8 : 6 : 1), (9 : 1 : 1), (9 : 10 : 1), (10 : 3 : 1), (10 : 8 : 1)]"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F.points()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(9 : 1 : 1)"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P = F.random_point()\n",
    "P"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P.order()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Group structure"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Additive abelian group isomorphic to Z/12 embedded in Abelian group of points on Elliptic Curve defined by y^2 = x^3 + x over Finite Field of size 11"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F.abelian_group()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(8 : 6 : 1)"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g = F.gens()[0]\n",
    "g"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "12"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g.order()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Construct an isogeny with given kernel"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0 : 0 : 1)"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "origin = 6*g\n",
    "origin"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0 : 0 : 1)"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F.point([0,0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + x over Finite Field of size 11 to Elliptic Curve defined by y^2 = x^3 + 7*x over Finite Field of size 11"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I = F.isogeny(origin)\n",
    "I"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "((x^2 + 1)/x, (x^2*y - y)/x^2)"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I.rational_maps()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "FF = I.codomain()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Elliptic Curve defined by y^2 = x^3 + 7*x over Finite Field of size 11"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "FF"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Additive abelian group isomorphic to Z/6 + Z/2 embedded in Abelian group of points on Elliptic Curve defined by y^2 = x^3 + 7*x over Finite Field of size 11"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "FF.abelian_group()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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DACicoAMAKJygAwAonKADACicoAMAKJygAwAonKADACicoAMAKJygAwAonKADACicoAMA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\n",
      "text/plain": [
       "Graphics object consisting of 1 graphics primitive"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "FF.plot()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The same example, over the rationals"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "E = EllipticCurve([1,0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0 : 0 : 1)"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P = E.lift_x(0)\n",
    "P"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P.order()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Elliptic Curve defined by y^2 = x^3 - 4*x over Rational Field"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "J = E.isogeny(P)\n",
    "EE = J.codomain()\n",
    "EE"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In (very) limited cases, Sage can compute the isogeny given the image curve and the degree"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "JJ = E.isogeny(None, codomain=EE, degree=2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "J == JJ"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Your turn: exercices!\n",
    "\n",
    "### Diffie-Hellman \n",
    "\n",
    "Implement the (naive) Diffie-Helman key exchange scheme with elliptic curves.\n",
    "\n",
    "#### 1. Find a curve of prime order\n",
    "\n",
    "Choose a prime $p$. Take curves at random over $𝔽_p$ until you find one with prime order.\n",
    "\n",
    "**Suggestion:** For a start, take $p$ of ~60 bits. When your code works well enough you can try 160 bits (it may take a few minutes to find a curve)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 2. Implement the scheme\n",
    "\n",
    "Write a function to sample secret keys, and a function to produce public keys. Perform the key exchnage and verify that Alice and Bob obtain the same key."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### ECM Factoring method\n",
    "\n",
    "Implement Lenstra's factoring method. See the [lecture notes](https://arxiv.org/pdf/1711.04062.pdf), Section 5.\n",
    "\n",
    "**Note:** Sage has limited support for curves over $ℤ/Nℤ$, just enough to implement ECM."
   ]
  },
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    "### Generating irreducible polynomials\n",
    "\n",
    "Implement the Couveignes-Lercier method for generating irreducible polynomials. See the [lecture notes](https://arxiv.org/pdf/1711.04062.pdf), Section 8.\n",
    "\n",
    "**Note:** Sage has limited support for curves over $ℤ/Nℤ$, just enough to implement ECM."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
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