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WitherOrNot/pidgenx.ipynb

Last active February 16, 2025 03:27
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PIDGENX validation implementation in SageMath (works on SageMath 9.0)
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pidgenx.ipynb
{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The math in this notebook is described in [this patent](https://patentimages.storage.googleapis.com/a3/27/c1/3c0948a078cb28/US7587605.pdf). Be warned, the math is very complicated."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"# Windows 7 Retail/GVLK, public key ID 170\n",
"\n",
"pubkey_data = {\n",
" \"field\": {\n",
" \"modulus\": 886368969471450739924935101400677,\n",
" \"ec_base_order\": 886368969471450710152985728350703,\n",
" \"k3_minpoly\": [\n",
" 4,\n",
" 1,\n",
" 0,\n",
" 1\n",
" ],\n",
" \"k6_minpoly\": [\n",
" 2,\n",
" 0,\n",
" 1\n",
" ]\n",
" },\n",
" \"h1_bases\": [\n",
" 1,\n",
" 3,\n",
" 8,\n",
" 15,\n",
" 15,\n",
" 15,\n",
" 31,\n",
" 3,\n",
" 3,\n",
" 3,\n",
" 3,\n",
" 3,\n",
" 7,\n",
" 63\n",
" ],\n",
" \"curve\": {\n",
" \"a\": 26392827536965106777121445123290,\n",
" \"b\": 372325368096095544195525883520589\n",
" },\n",
" \"points\": [\n",
" {\n",
" \"x\": [\n",
" 365236101742748463929673543888206,\n",
" 858097895593939865996182272259769,\n",
" 148438159087534462792506738986740\n",
" ],\n",
" \"y\": [\n",
" 776418047571862972603801173382237,\n",
" 873677028107508092012208744232957,\n",
" 622138327043805563266794621920098\n",
" ]\n",
" },\n",
" {\n",
" \"x\": [\n",
" 940136574680879136511599445781,\n",
" 566978253317108608042529258054523,\n",
" 176284220413545220121710961573292\n",
" ],\n",
" \"y\": [\n",
" 828856809691743749590800150937649,\n",
" 225146018128364550960496522448712,\n",
" 348601659612301002638949468744847\n",
" ]\n",
" },\n",
" {\n",
" \"x\": [\n",
" 733747358623171948496764545320051,\n",
" 728506535527490173098593825125337,\n",
" 82462451162574422717677160727098\n",
" ],\n",
" \"y\": [\n",
" 416331132638004657079841565104549,\n",
" 366794872410090667339979925100938,\n",
" 154519017608105570119249112044121\n",
" ]\n",
" },\n",
" {\n",
" \"x\": [\n",
" 849718119860311390685018324089317,\n",
" 69736499980142833460080381132368,\n",
" 72139323263966224829624934948858\n",
" ],\n",
" \"y\": [\n",
" 122550620604034160835298121626961,\n",
" 232865179257577260620478614346661,\n",
" 96495922331236902442197840422963\n",
" ]\n",
" },\n",
" {\n",
" \"x\": [\n",
" 737949449696062407373114808812927,\n",
" 526576673882551145431025311648593,\n",
" 577710732700754839750249914833193\n",
" ],\n",
" \"y\": [\n",
" 245977198113437420529250724111432,\n",
" 316396368275232555978824338443046,\n",
" 755792900000892204654488821885538\n",
" ]\n",
" },\n",
" {\n",
" \"x\": [\n",
" 712586405875738967442641545880322,\n",
" 615445286425878710157557053762371,\n",
" 734183236086095230968388017605820\n",
" ],\n",
" \"y\": [\n",
" 851284376759840359812981263306021,\n",
" 769237654873203944088649987250083,\n",
" 359324880331507581802773028306633\n",
" ]\n",
" },\n",
" {\n",
" \"x\": [\n",
" 579665839598807564349118802507078,\n",
" 793103874095793223248478622956780,\n",
" 502860226530799804560661048077280\n",
" ],\n",
" \"y\": [\n",
" 526775274489316486107329470634542,\n",
" 828721161962151275145535457964404,\n",
" 204415317809040518371881977645416\n",
" ]\n",
" },\n",
" {\n",
" \"x\": [\n",
" 804790447447351785544412956578788,\n",
" 119046642031064430140912082580578,\n",
" 475159529884254928674792290619954\n",
" ],\n",
" \"y\": [\n",
" 458245266057063984580129835988070,\n",
" 338411981227059768831710308435687,\n",
" 577923375329917551735757167190702\n",
" ]\n",
" },\n",
" {\n",
" \"x\": [\n",
" 448295070796654878810211055051604,\n",
" 482910785083759911781193909334072,\n",
" 795628820954832750108065551162801\n",
" ],\n",
" \"y\": [\n",
" 417757375223493128894380427308216,\n",
" 755520039102173573177271365439537,\n",
" 863842006193777913816171128026446\n",
" ]\n",
" },\n",
" {\n",
" \"x\": [\n",
" 663389221842281261857262032548436,\n",
" 846447543951704162020988219326272,\n",
" 686142287698732386980948449542167\n",
" ],\n",
" \"y\": [\n",
" 769015970121598916167134609518482,\n",
" 738460771147019950148429256265493,\n",
" 613009789239563486872501072748270\n",
" ]\n",
" },\n",
" {\n",
" \"x\": [\n",
" 23530113060362511985797534739195,\n",
" 718131004725002854064127778364823,\n",
" 140870968646848990835780066321375\n",
" ],\n",
" \"y\": [\n",
" 641031697928634900295866764583620,\n",
" 295544383156746469642549388283327,\n",
" 133766761871461067699690599056442\n",
" ]\n",
" },\n",
" {\n",
" \"x\": [\n",
" 7518354584460889742005963384331,\n",
" 340825540582760123772991939806390,\n",
" 525549834323799848592419044187971\n",
" ],\n",
" \"y\": [\n",
" 585295007893871934790357000030208,\n",
" 117490751031779271453224407217079,\n",
" 838852298106199238437827740364400\n",
" ]\n",
" },\n",
" {\n",
" \"x\": [\n",
" 806036388470182281562651653929939,\n",
" 266085928879449679004785507000719,\n",
" 201474020142460453395308745398496\n",
" ],\n",
" \"y\": [\n",
" 573468377549807523205344415925956,\n",
" 667459718759242575444856430313959,\n",
" 226975716159080217447594275999935\n",
" ]\n",
" },\n",
" {\n",
" \"x\": [\n",
" 794167987155642331621801361756614,\n",
" 809201520617560616339201020039820,\n",
" 198696155869194654384403079624544\n",
" ],\n",
" \"y\": [\n",
" 725959545288387914551997303844726,\n",
" 49262476800238214847233993847181,\n",
" 537326577113493149345527624223733\n",
" ]\n",
" }\n",
" ],\n",
" \"pairing_val\": [\n",
" [\n",
" 242940802691096077821709859741616,\n",
" 178851543248946074944443141484182,\n",
" 802059826004050667481466713086225\n",
" ],\n",
" [\n",
" 701042518368651902930590425782509,\n",
" 265571225406900742458432149860962,\n",
" 699432283102586243018242179516873\n",
" ]\n",
" ]\n",
"}"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"p = pubkey_data[\"field\"][\"modulus\"]\n",
"a = pubkey_data[\"curve\"][\"a\"]\n",
"b = pubkey_data[\"curve\"][\"b\"]\n",
"order = pubkey_data[\"field\"][\"ec_base_order\"]\n",
"h1_bases = list(map(lambda x: x+1, pubkey_data[\"h1_bases\"]))\n",
"KCHARS = \"BCDFGHJKMPQRTVWXY2346789\"\n",
"\n",
"def decode_pkey(k):\n",
" k = k.replace(\"-\", \"\")\n",
" out = 0\n",
" \n",
" for c in k:\n",
" out *= 24\n",
" out += KCHARS.index(c)\n",
" \n",
" return out\n",
"\n",
"K = GF(p)\n",
"Kx.<x> = K[]\n",
"K3.<u> = K.extension(Kx(pubkey_data[\"field\"][\"k3_minpoly\"]))\n",
"K3y.<y> = K3[]\n",
"K6.<t> = K3.extension(K3y(pubkey_data[\"field\"][\"k6_minpoly\"]))\n",
"\n",
"E = EllipticCurve(K, [a, b])\n",
"E6 = EllipticCurve(K6, [a, b])\n",
"\n",
"Qi = [E6(K3(point[\"x\"]) * t^-2, K3(point[\"y\"]) * t^-3) for point in pubkey_data[\"points\"]]\n",
"\n",
"# pairing_val = e_m(P, S)\n",
"pairing_val = K6([K3(pubkey_data[\"pairing_val\"][0]), K3(pubkey_data[\"pairing_val\"][1])])\n",
"\n",
"assert is_prime(p)\n",
"assert len(h1_bases) == len(Qi)\n",
"assert h1_bases[0] == 2\n",
"assert pairing_val^order == 1"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"pkey_chars = \"33PXH-7Y6KF-2VJC9-XBBR8-HVTHH\"\n",
"\n",
"# pkey = HASH(M)\n",
"# HASH is a currently unknown hash function\n",
"pkey = decode_pkey(pkey_chars)\n",
"\n",
"# h1_coeffs = H1(M)\n",
"# During validation, coeffs must be found by a search that i havent implemented\n",
"# h1_coeffs = [1, 0, 7, 1, 4, 15, 9, 1, 1, 0, 2, 1, 0, 19]\n",
"\n",
"# 10 bits unknown, 30 bits product ID, 1 bit unknown (upgrade?)\n",
"key_data = (342 << 31 | 918500000 << 1 | 0)\n",
"h1_coeffs = [1]\n",
"\n",
"for i in range(len(h1_bases) - 1):\n",
" h1_coeffs.append(key_data % h1_bases[i + 1])\n",
" key_data //= h1_bases[i + 1]\n",
"\n",
"print(h1_coeffs)\n",
"\n",
"# H2(M) = E.lift_x(HASH(M) % p)\n",
"T = E6(E.lift_x(pkey % p))\n",
"Q = sum(map(lambda x: x[0] * x[1], zip(h1_coeffs, Qi))) \n",
"\n",
"test_pairing = T.tate_pairing(Q, order, 6, q=p)\n",
"\n",
"print(test_pairing == pairing_val or test_pairing == 1/pairing_val)\n",
"\n",
"key_data = 0\n",
"\n",
"for i in range(len(h1_bases) - 1, 0, -1):\n",
" key_data *= h1_bases[i]\n",
" key_data += h1_coeffs[i]\n",
" print(h1_bases[i], h1_coeffs[i], key_data)\n",
"\n",
"pid = (key_data & ((1 << 31) - 1)) >> 1"
]
}
],
"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": "3.8.10"
}
},
"nbformat": 4,
"nbformat_minor": 4
}
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