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+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# 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."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 9,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "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\")"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "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 $(\\mathbb Z/p \\mathbb Z)^\\times$."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 24,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "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)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Numerical methods for PDEs"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "SageMath 9.0",
+   "language": "sage",
+   "name": "sagemath"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.8.5"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 4
+}