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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
}
|
