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    "# Diffie-Hellman key exchange\n",
    "\n",
    "The following is a simple implementation of the classic [Diffie-Hellman key exchange](https://en.wikipedia.org/wiki/Diffie%E2%80%93Hellman_key_exchange) cryptographic protocol."
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      "Public key: p = 20747 and g = 13428 \n",
      "\n",
      "[[ Alice's secret key: a = 12403 ]]\n",
      "[[ Bob's secret key: b = 17642 ]] \n",
      "\n",
      "Alice sends h1 = 14710 to Bob\n",
      "Bob sends h2 = 10680 to Alice \n",
      "\n",
      "Alice computed 10455 using h2 and her secret a\n",
      "Bob computed 10455 using h1 and his secret b\n"
     ]
    }
   ],
   "source": [
    "# Public information:\n",
    "p = Primes()[10^3 + randint(1,10000)] # random prime\n",
    "g = randint(2, p-1) # random integer\n",
    "\n",
    "print(\"Public key: p =\", p, \"and g =\", g, \"\\n\")\n",
    "\n",
    "a = randint(2, p-1) # Only Alice knows this\n",
    "b = randint(2, p-1) # Only Bob knows this\n",
    "\n",
    "print(\"[[ Alice's secret key: a =\", a, \"]]\")\n",
    "print(\"[[ Bob's secret key: b =\", b, \"]]\", \"\\n\")\n",
    "\n",
    "h1 = (g^a) % p # Alice sends this to Bob\n",
    "h2 = (g^b) % p # Bob sends this to Alice\n",
    "\n",
    "print(\"Alice sends h1 =\", h1, \"to Bob\")\n",
    "print(\"Bob sends h2 =\", h2, \"to Alice\", \"\\n\")\n",
    "\n",
    "secret_a = (h2^a) % p # Alice can compute this because she knows a\n",
    "secret_b = (h1^b) % p # Bob can compute this because he knows b\n",
    "\n",
    "print(\"Alice computed\", secret_a, \"using h2 and her secret a\")\n",
    "print(\"Bob computed\", secret_b, \"using h1 and his secret b\")"
   ]
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   "source": [
    "## General Diffie-Hellman\n",
    "\n",
    "The following code is an implementation of a generic Diffie-Hellman key exchange protocol that uses a group $G$ instead of $(\\mathbb Z/p \\mathbb Z)^\\times$."
   ]
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     "text": [
      "Public key:\n",
      "G = Additive abelian group isomorphic to Z/171 embedded in Abelian group of points on Elliptic Curve defined by y^2 = x^3 + x + 156 over Finite Field of size 157 \n",
      "g = (155 : 60 : 1) \n",
      "\n",
      "[[ Alice's secret key: a = 141 ]]\n",
      "[[ Bob's secret key: b = 158 ]] \n",
      "\n",
      "Alice sends h1 = (29 : 125 : 1) to Bob\n",
      "Bob sends h2 = (60 : 59 : 1) to Alice \n",
      "\n",
      "Alice computed (109 : 94 : 1) using h2 and her secret a\n",
      "Bob computed (109 : 94 : 1) using h1 and his secret b\n"
     ]
    }
   ],
   "source": [
    "def genericDH(G):\n",
    "    if G.cardinality() == 1:\n",
    "        print(\"Group is trivial, can't do anything\")\n",
    "        return\n",
    "    g = G.random_element()\n",
    "    while g == G.identity(): # Make sure g is not trivial\n",
    "        g = G.random_element()\n",
    "        \n",
    "    print(\"Public key:\\nG =\", G, \"\\ng =\", g, \"\\n\")\n",
    "    \n",
    "    a = randint(2, G.exponent()-1) # Only Alice knows this\n",
    "    b = randint(2, G.exponent()-1) # Only Bob knows this\n",
    "\n",
    "    print(\"[[ Alice's secret key: a =\", a, \"]]\")\n",
    "    print(\"[[ Bob's secret key: b =\", b, \"]]\", \"\\n\")\n",
    "    \n",
    "    # \"Ternary operator\", I did not explain this\n",
    "    # https://docs.python.org/3/reference/expressions.html#conditional-expressions\n",
    "    h1 = g^a if G.is_multiplicative() else a*g # Alice sends this to Bob\n",
    "    h2 = g^b if G.is_multiplicative() else b*g # Bob sends this to Alice\n",
    "\n",
    "    print(\"Alice sends h1 =\", h1, \"to Bob\")\n",
    "    print(\"Bob sends h2 =\", h2, \"to Alice\", \"\\n\")\n",
    "    \n",
    "    secret_a = h2^a if G.is_multiplicative() else a*h2 # Alice can compute this because she knows a\n",
    "    secret_b = h1^b if G.is_multiplicative() else b*h1 # Bob can compute this because he knows b\n",
    "\n",
    "    print(\"Alice computed\", secret_a, \"using h2 and her secret a\")\n",
    "    print(\"Bob computed\", secret_b, \"using h1 and his secret b\")\n",
    "    \n",
    "E = EllipticCurve(GF(157), [1,-1])\n",
    "G = E.abelian_group()\n",
    "genericDH(G)"
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   "source": [
    "# Numerical methods for PDEs"
   ]
  },
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   "execution_count": null,
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   "outputs": [],
   "source": []
  }
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