{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "bafc2f4e-05a3-4120-bcd6-5d1f5fb91cd9",
   "metadata": {},
   "source": [
    "# Distinguishing countermeasures by output"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "33ee6084-2ac3-4f95-9610-0fbc06026538",
   "metadata": {},
   "outputs": [],
   "source": [
    "import io\n",
    "import random\n",
    "import itertools\n",
    "import cypari2\n",
    "\n",
    "from matplotlib import pyplot as plt\n",
    "from collections import Counter\n",
    "from tqdm.auto import tqdm, trange\n",
    "\n",
    "from pyecsca.misc.utils import TaskExecutor\n",
    "from pyecsca.ec.mod import mod, RandomModAction\n",
    "from pyecsca.ec.point import Point\n",
    "from pyecsca.ec.model import ShortWeierstrassModel\n",
    "from pyecsca.ec.params import load_params_ectester\n",
    "from pyecsca.ec.mult import LTRMultiplier\n",
    "from pyecsca.ec.context import local, DefaultContext\n",
    "from pyecsca.ec.countermeasures import GroupScalarRandomization, AdditiveSplitting, MultiplicativeSplitting, EuclideanSplitting, BrumleyTuveri\n",
    "\n",
    "%matplotlib ipympl"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "b1b9596c-1eba-4ace-af84-8cb279d84cc2",
   "metadata": {},
   "outputs": [],
   "source": [
    "model = ShortWeierstrassModel()\n",
    "coords = model.coordinates[\"projective\"]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "b0afb195-8390-44c5-931e-75a70ccd4e9e",
   "metadata": {},
   "outputs": [],
   "source": [
    "add = coords.formulas[\"add-2015-rcb\"]\n",
    "dbl = coords.formulas[\"dbl-2015-rcb\"]\n",
    "mult = LTRMultiplier(add, dbl, complete=False)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "52c877e1-5021-4ec2-9daa-dd20bec6bcb2",
   "metadata": {},
   "outputs": [],
   "source": [
    "gsr = GroupScalarRandomization(mult)\n",
    "asplit = AdditiveSplitting(mult)\n",
    "msplit = MultiplicativeSplitting(mult)\n",
    "esplit = EuclideanSplitting(mult)\n",
    "bt = BrumleyTuveri(mult)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "27626337-dcbc-497c-a54e-02d50e2b8f34",
   "metadata": {},
   "source": [
    "## 3n test"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 208,
   "id": "c3088419-161b-4193-a1b6-6f623f217fcd",
   "metadata": {},
   "outputs": [],
   "source": [
    "key3n = 0x20959f2b437de1e522baf6d814911938157390d3ea5118660b852ab0d5387006\n",
    "params3n = load_params_ectester(io.BytesIO(b\"0xc381bb0394f34b5ed061c9107b66974f4d0a8ec89b9fe73b98b6d1368c7d974d,0x5ca6c5ee0a10097af291a8f125303fb1a3e35e8100411902245d691e0e5cb497,0x385a5a8bb8af94721f6fd10b562606d9b9df931f7fd966e96859bb9bd7c05836,0x4616af1898b92cac0f902a9daee24bbae63571cead270467c6a7886ced421f5e,0x34e896bdb1337e0ae5960fa3389fb59c2c8d6c7dbfd9aac33a844f8f98e433ef,0x412b3e5686fbc3ca4575edb0292232702ae721a7d4a230cc170a5561aa70e00f,0x01\"), \"projective\")\n",
    "bits3n = params3n.full_order.bit_length()\n",
    "point3n = Point(X=mod(0x4a48addb2e471767b7cd0f6f1d4c27fe46f4a828fc20f950bd1f72c939b36a84, params3n.curve.prime),\n",
    "                Y=mod(0x13384d38c353f862832c0f067e46a3e510bb6803c20745dfb31929f4a18d890d, params3n.curve.prime),\n",
    "                Z=mod(1, params3n.curve.prime), model=coords)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 209,
   "id": "a8dde7e6-cd48-4f99-9677-23a19e4c2e5b",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "prime:\t0xc381bb0394f34b5ed061c9107b66974f4d0a8ec89b9fe73b98b6d1368c7d974d\n",
      "a:\t0x5ca6c5ee0a10097af291a8f125303fb1a3e35e8100411902245d691e0e5cb497\n",
      "b:\t0x385a5a8bb8af94721f6fd10b562606d9b9df931f7fd966e96859bb9bd7c05836\n",
      "G:\t[0x4616af1898b92cac0f902a9daee24bbae63571cead270467c6a7886ced421f5e,\n",
      "\t 0x34e896bdb1337e0ae5960fa3389fb59c2c8d6c7dbfd9aac33a844f8f98e433ef]\n",
      "n:\t0x412b3e5686fbc3ca4575edb0292232702ae721a7d4a230cc170a5561aa70e00f\n",
      "3n:\t0xc381bb0394f34b5ed061c9107b66975080b564f77de69264451f0024ff52a02d\n",
      "\n",
      "P:\t[0x4a48addb2e471767b7cd0f6f1d4c27fe46f4a828fc20f950bd1f72c939b36a84,\n",
      "\t 0x13384d38c353f862832c0f067e46a3e510bb6803c20745dfb31929f4a18d890d]\n"
     ]
    }
   ],
   "source": [
    "print(f\"prime:\\t0x{params3n.curve.prime:x}\")\n",
    "print(f\"a:\\t0x{params3n.curve.parameters['a']:x}\")\n",
    "print(f\"b:\\t0x{params3n.curve.parameters['b']:x}\")\n",
    "print(f\"G:\\t[0x{params3n.generator.X:x},\\n\\t 0x{params3n.generator.Y:x}]\")\n",
    "print(f\"n:\\t0x{params3n.order:x}\")\n",
    "print(f\"3n:\\t0x{3 * params3n.order:x}\")\n",
    "print(f\"\\nP:\\t[0x{point3n.X:x},\\n\\t 0x{point3n.Y:x}]\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 210,
   "id": "cd6f8500-7509-45b0-8b23-471ee5014f42",
   "metadata": {},
   "outputs": [],
   "source": [
    "def generate_scalars_mod3(rem, samples):\n",
    "    scalars = []\n",
    "    while True:\n",
    "        scalar = random.randint(0, params3n.full_order)\n",
    "        if scalar % 3 == rem:\n",
    "            scalars.append(scalar)\n",
    "        if len(scalars) == samples:\n",
    "            break\n",
    "    return scalars\n",
    "\n",
    "def test_3n(countermeasure, scalars):\n",
    "    ctr = Counter()\n",
    "    for k in tqdm(scalars, leave=False):\n",
    "        mult.init(params3n, point3n)\n",
    "        kP = mult.multiply(k).to_affine()\n",
    "        mult.init(params3n, point3n)\n",
    "        knP = mult.multiply(k + params3n.full_order).to_affine()\n",
    "        mult.init(params3n, point3n)\n",
    "        k2nP = mult.multiply(k + 2 * params3n.full_order).to_affine()\n",
    "\n",
    "        countermeasure.init(params3n, point3n)\n",
    "        res = countermeasure.multiply(k)\n",
    "        aff = res.to_affine()\n",
    "        if aff.equals(kP):\n",
    "            ctr[\"k\"] += 1\n",
    "        elif aff.equals(knP):\n",
    "            ctr[\"k + 1n\"] += 1\n",
    "        elif aff.equals(k2nP):\n",
    "            ctr[\"k + 2n\"] += 1\n",
    "        else:\n",
    "            ctr[aff] += 1\n",
    "    for name, count in sorted(ctr.items()):\n",
    "        print(f\"{name}:\\t{count}\")\n",
    "\n",
    "def test_3n_fixed_scalar(countermeasure, samples):\n",
    "    test_3n(countermeasure, [key3n for _ in range(samples)])\n",
    "\n",
    "def test_3n_random_scalar(countermeasure, samples):\n",
    "    test_3n(countermeasure, [random.randint(0, params3n.full_order) for _ in range(samples)])\n",
    "\n",
    "def test_3n_random_scalar_projected(countermeasure, samples):\n",
    "    print(\"k = 0 mod 3\")\n",
    "    test_3n(countermeasure, generate_scalars_mod3(0, samples))\n",
    "    print()\n",
    "    print(\"k = 1 mod 3\")\n",
    "    test_3n(countermeasure, generate_scalars_mod3(1, samples))\n",
    "    print()\n",
    "    print(\"k = 2 mod 3\")\n",
    "    test_3n(countermeasure, generate_scalars_mod3(2, samples))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "46b8f74a-433d-48c9-b5b9-6bb7d2731246",
   "metadata": {},
   "source": [
    "### Fixed scalar experiments"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fc82d4b9-91cd-423c-83aa-89721efa1ae9",
   "metadata": {},
   "source": [
    "#### Group scalar randomization"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 211,
   "id": "86532d50-2db7-4370-b449-c545b330a852",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "b0a80fc9b8f14b2992f10c7bb66a9a83",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
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      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "k:\t326\n",
      "k + 1n:\t341\n",
      "k + 2n:\t333\n"
     ]
    }
   ],
   "source": [
    "test_3n_fixed_scalar(gsr, 1000)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "aba3e713-6246-435d-93af-2e8b42ee9582",
   "metadata": {},
   "source": [
    "#### Additive splitting"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 212,
   "id": "ad421630-606f-4666-9bbf-1a446eec1b59",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "0d0f2e87f3044516924fbd74ba68f983",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "  0%|          | 0/1000 [00:00<?, ?it/s]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "k:\t500\n",
      "k + 1n:\t500\n"
     ]
    }
   ],
   "source": [
    "test_3n_fixed_scalar(asplit, 1000)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a1284a11-4ace-437d-9826-2035cce36756",
   "metadata": {},
   "source": [
    "#### Multiplicative splitting"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 213,
   "id": "3ed5d7f3-0ba1-4b62-9635-aeb492499175",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "9eb1217838fa4931bb0f0f9cd86aba09",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
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      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "k:\t221\n",
      "k + 1n:\t544\n",
      "k + 2n:\t235\n"
     ]
    }
   ],
   "source": [
    "test_3n_fixed_scalar(msplit, 1000)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5e40f8e9-d26f-41e7-adbf-a0fbdb680677",
   "metadata": {},
   "source": [
    "#### Euclidean splitting"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 214,
   "id": "314447c6-a1fb-4d3a-8988-b34c8912dd5e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "1c6788e6f3614ea1a7a6c8c6cd761bb5",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
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      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "k:\t1000\n"
     ]
    }
   ],
   "source": [
    "test_3n_fixed_scalar(esplit, 1000)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ff7185a2-3cd9-44d7-b1a9-982eaed561dc",
   "metadata": {},
   "source": [
    "#### Brumley and Tuveri bit-length fixing"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 215,
   "id": "f41dfc1d-1017-4aa0-bcd4-6569c53bf81e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "9290687728e144329fd7dca7dcebabd8",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "  0%|          | 0/1000 [00:00<?, ?it/s]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "k + 2n:\t1000\n"
     ]
    }
   ],
   "source": [
    "test_3n_fixed_scalar(bt, 1000)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "28915553-b5e6-4108-bc23-e5844b6a63b8",
   "metadata": {},
   "source": [
    "### Random scalar experiments"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "566ddd10-2d0e-4b32-9b27-60770ab68155",
   "metadata": {},
   "source": [
    "#### Group scalar randomization"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 216,
   "id": "7255321a-6ad6-4938-8ec9-dd8d977686db",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "63db299beb5b47679019f469b446d2c9",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "  0%|          | 0/1000 [00:00<?, ?it/s]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "k:\t324\n",
      "k + 1n:\t337\n",
      "k + 2n:\t339\n"
     ]
    }
   ],
   "source": [
    "test_3n_random_scalar(gsr, 1000)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "376c0da8-b92d-4151-ac30-839ab5c0ceae",
   "metadata": {},
   "source": [
    "#### Additive splitting"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 217,
   "id": "b0146a9a-0803-43c4-ab29-8ba6e15934b5",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "639cadec2edd4719b7bcbd5f6d9b80ae",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "  0%|          | 0/1000 [00:00<?, ?it/s]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "k:\t491\n",
      "k + 1n:\t509\n"
     ]
    }
   ],
   "source": [
    "test_3n_random_scalar(asplit, 1000)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "96823352-cd63-4cf8-8ab7-cf6e025837b9",
   "metadata": {},
   "source": [
    "#### Multiplicative splitting"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 218,
   "id": "5645ae6f-f5f4-419d-ba47-248532dc2114",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "f3c8933bed464791b7d36ef8dbf8bf2b",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "  0%|          | 0/1000 [00:00<?, ?it/s]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "k:\t341\n",
      "k + 1n:\t308\n",
      "k + 2n:\t351\n"
     ]
    }
   ],
   "source": [
    "test_3n_random_scalar(msplit, 1000)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1784d763-932a-4f66-a1da-d7ef51cbc88a",
   "metadata": {},
   "source": [
    "#### Euclidean splitting"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 219,
   "id": "c9fc4f35-1c25-4cac-bb63-8bd70263db47",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "85635506a91c4665b65d5377c06549b1",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "  0%|          | 0/1000 [00:00<?, ?it/s]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "k:\t1000\n"
     ]
    }
   ],
   "source": [
    "test_3n_random_scalar(esplit, 1000)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8917dfd5-1548-414b-846a-685857bfb427",
   "metadata": {},
   "source": [
    "#### Brumley and Tuveri bit-length fixing"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 220,
   "id": "4fd6b288-08a9-4dbe-9145-e96401805315",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "c46f2d9115ca4f17b229c5b0da6baf7b",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "  0%|          | 0/1000 [00:00<?, ?it/s]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "k + 1n:\t36\n",
      "k + 2n:\t964\n"
     ]
    }
   ],
   "source": [
    "test_3n_random_scalar(bt, 1000)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ee9e23f6-edd9-4fc2-9e7c-b5c176991071",
   "metadata": {},
   "source": [
    "### Random scalar experiments projected to scalar divisor classes mod 3"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e17fcb12-1e02-4bcf-a2b7-785c16c03028",
   "metadata": {},
   "source": [
    "#### Group scalar randomization"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 221,
   "id": "6c46fdbb-2ffb-4169-8e00-6d93b8407ee5",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "k = 0 mod 3\n"
     ]
    },
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "7561669af4664bfebcbcf2d5e5629d2e",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
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      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "k:\t340\n",
      "k + 1n:\t308\n",
      "k + 2n:\t352\n",
      "\n",
      "k = 1 mod 3\n"
     ]
    },
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "25e114540bef47d89efe10138dbe13cf",
       "version_major": 2,
       "version_minor": 0
      },
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      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "k:\t325\n",
      "k + 1n:\t337\n",
      "k + 2n:\t338\n",
      "\n",
      "k = 2 mod 3\n"
     ]
    },
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "a8a2dcd9e5954ecd8a0d2f6b54f4d436",
       "version_major": 2,
       "version_minor": 0
      },
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      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "k:\t312\n",
      "k + 1n:\t356\n",
      "k + 2n:\t332\n"
     ]
    }
   ],
   "source": [
    "test_3n_random_scalar_projected(gsr, 1000)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b9d635c8-f788-4700-876b-ac48e89557a7",
   "metadata": {},
   "source": [
    "#### Additive splitting"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 222,
   "id": "344a4f90-3470-40e9-a75f-b925a88c2480",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "k = 0 mod 3\n"
     ]
    },
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "bc8d0b7c8f414943af7d3f133e43bd78",
       "version_major": 2,
       "version_minor": 0
      },
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      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "k:\t482\n",
      "k + 1n:\t518\n",
      "\n",
      "k = 1 mod 3\n"
     ]
    },
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "df03bde64059413fbe6a5208c458f5f9",
       "version_major": 2,
       "version_minor": 0
      },
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      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "k:\t513\n",
      "k + 1n:\t487\n",
      "\n",
      "k = 2 mod 3\n"
     ]
    },
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "d953bea042794219bf4aa32c1d16ad92",
       "version_major": 2,
       "version_minor": 0
      },
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      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "k:\t497\n",
      "k + 1n:\t503\n"
     ]
    }
   ],
   "source": [
    "test_3n_random_scalar_projected(asplit, 1000)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "78aaf83f-a689-40ad-ae88-58ab20e5e6f9",
   "metadata": {},
   "source": [
    "#### Multiplicative splitting"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 223,
   "id": "616a7726-01e6-4e9c-b7f2-fe8f14b60071",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "k = 0 mod 3\n"
     ]
    },
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "4f329714588d4127bd4273dda2695222",
       "version_major": 2,
       "version_minor": 0
      },
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      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "k:\t563\n",
      "k + 1n:\t225\n",
      "k + 2n:\t212\n",
      "\n",
      "k = 1 mod 3\n"
     ]
    },
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "5f8e4efe89454eb5817d889b2fe77210",
       "version_major": 2,
       "version_minor": 0
      },
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      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "k:\t234\n",
      "k + 1n:\t221\n",
      "k + 2n:\t545\n",
      "\n",
      "k = 2 mod 3\n"
     ]
    },
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "0712286b1d8242c7ae0ea2255eda2f37",
       "version_major": 2,
       "version_minor": 0
      },
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      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "k:\t210\n",
      "k + 1n:\t578\n",
      "k + 2n:\t212\n"
     ]
    }
   ],
   "source": [
    "test_3n_random_scalar_projected(msplit, 1000)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "018c8f31-d7c3-497b-bdaf-580c6465753f",
   "metadata": {},
   "source": [
    "#### Euclidean splitting"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 224,
   "id": "adced4e4-37a7-43ed-97b5-01cb5d274d6b",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "k = 0 mod 3\n"
     ]
    },
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "9588400d40a041e088d86352e7512a06",
       "version_major": 2,
       "version_minor": 0
      },
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      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "k:\t1000\n",
      "\n",
      "k = 1 mod 3\n"
     ]
    },
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "e3014e90730b44209d9bf87cd89598a3",
       "version_major": 2,
       "version_minor": 0
      },
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      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "k:\t1000\n",
      "\n",
      "k = 2 mod 3\n"
     ]
    },
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "1fe4739a9dd3482a9e0f1e7d379c57b3",
       "version_major": 2,
       "version_minor": 0
      },
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      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "k:\t1000\n"
     ]
    }
   ],
   "source": [
    "test_3n_random_scalar_projected(esplit, 1000)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8d7b0a42-7be7-464d-932d-8386ad912034",
   "metadata": {},
   "source": [
    "#### Brumley and Tuveri bit-length fixing"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 225,
   "id": "fe8d8295-3e69-4b60-b8c3-5710deaeb0b3",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "k = 0 mod 3\n"
     ]
    },
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "2101fd299ebd49358729bf7dc7aa204a",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
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      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "k + 1n:\t39\n",
      "k + 2n:\t961\n",
      "\n",
      "k = 1 mod 3\n"
     ]
    },
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "ae5b94fb0d5b4395b98a6f905ea29a80",
       "version_major": 2,
       "version_minor": 0
      },
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      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "k + 1n:\t29\n",
      "k + 2n:\t971\n",
      "\n",
      "k = 2 mod 3\n"
     ]
    },
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "b3155abae413412bae324c579e5c7de5",
       "version_major": 2,
       "version_minor": 0
      },
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      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "k + 1n:\t33\n",
      "k + 2n:\t967\n"
     ]
    }
   ],
   "source": [
    "test_3n_random_scalar_projected(bt, 1000)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "43b309af-5683-4384-9623-d7633723177c",
   "metadata": {},
   "source": [
    "## Mask recovery\n",
    "Using a composite order curve we can recover the size and the actual mask values (in a known key scenario) in both GSR and multiplicative splitting. However, real-world targets do not like composite order curves and may either check the order or otherwise fail to compute on such curves. Thus, we lie to them and set the order to the next-prime of the true order, in this case $n + 92$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "20a26f27-620d-4d7f-92bd-b949482b5c9a",
   "metadata": {},
   "outputs": [],
   "source": [
    "pari = cypari2.Pari(256_000_000, 2_000_000_000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "144340bd-5372-4beb-a46e-fd60c596b254",
   "metadata": {},
   "outputs": [],
   "source": [
    "real_n = 0xa9fa3419aca88bade2cba14e317816c6828910c6ce04fcd2a2e857d25df50775\n",
    "#      = 2898786277 * 2916913393 * 3067509271 * 3248233993 * 3894099889 * 4099407227 * 4101666977 * 13936975277\n",
    "real_n_facts = pari.factor(real_n)\n",
    "params92pn = load_params_ectester(io.BytesIO(b\"0xa9fa3419aca88bade2cba14e317816c79d52481d463dc9bcb12c37f45aa3b4e1,0x2ea3bfe6659f8e035735349b91fbfa2baf0cf8e640315f0fe03c1136813dec99,0x2b07c518e04b02158651e3dbbef7720015dd496bf15af02f8439f8e1503b8370,0x90fb04b1af19e8e20396ac052f260a9fb5f736b97e3cd4af08fe81a1e75dac6d,0x2302bcf700d3d5899f04d0c7441f5017c9758bfafd6ce15dbe36fb4eea76baec,0xa9fa3419aca88bade2cba14e317816c6828910c6ce04fcd2a2e857d25df507d1,0x01\"), \"projective\")\n",
    "e = pari.ellinit([int(params92pn.curve.parameters[\"a\"]), int(params92pn.curve.parameters[\"b\"])], int(params92pn.curve.prime))\n",
    "e[15][0] = real_n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "f103129c-17d3-4217-999b-94ecb4ec523d",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "prime:\t0xa9fa3419aca88bade2cba14e317816c79d52481d463dc9bcb12c37f45aa3b4e1\n",
      "a:\t0x2ea3bfe6659f8e035735349b91fbfa2baf0cf8e640315f0fe03c1136813dec99\n",
      "b:\t0x2b07c518e04b02158651e3dbbef7720015dd496bf15af02f8439f8e1503b8370\n",
      "G:\t[0x90fb04b1af19e8e20396ac052f260a9fb5f736b97e3cd4af08fe81a1e75dac6d,\n",
      "\t 0x2302bcf700d3d5899f04d0c7441f5017c9758bfafd6ce15dbe36fb4eea76baec]\n",
      "n+92:\t0xa9fa3419aca88bade2cba14e317816c6828910c6ce04fcd2a2e857d25df507d1 (fake order, given to the target, prime)\n",
      "n:\t0xa9fa3419aca88bade2cba14e317816c6828910c6ce04fcd2a2e857d25df50775 (real order, composite)\n"
     ]
    }
   ],
   "source": [
    "print(f\"prime:\\t0x{params92pn.curve.prime:x}\")\n",
    "print(f\"a:\\t0x{params92pn.curve.parameters['a']:x}\")\n",
    "print(f\"b:\\t0x{params92pn.curve.parameters['b']:x}\")\n",
    "print(f\"G:\\t[0x{params92pn.generator.X:x},\\n\\t 0x{params92pn.generator.Y:x}]\")\n",
    "print(f\"n+92:\\t0x{params92pn.order:x} (fake order, given to the target, prime)\")\n",
    "print(f\"n:\\t0x{real_n:x} (real order, composite)\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "322d2e68-5259-4ea6-9748-2b0aa21b557f",
   "metadata": {},
   "source": [
    "### Group scalar randomization\n",
    "In GSR getting the mask out this way is quite simple. The target believes it is operating on a curve of order $n+92$ so it will use that value multiplied with the mask to randomize the scalar. Thus as a result we get:\n",
    "$$ P = [k + r(n + 92)]G $$\n",
    "\n",
    "However, the curve is truly of order $n$, thus arithmetic on its group will make this equal to: \n",
    "$$ P = [k + r 92]G $$\n",
    "\n",
    "Since this is a composite order curve, we can solve the dlog and obtain $k + r 92$ and since we assume we know $k$ we can easily compute both the mask size and mask value $r$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "08d99bd5-2b87-4a04-995d-7a87f9b67102",
   "metadata": {},
   "outputs": [],
   "source": [
    "key = 0x20959f2b437de1e522baf6d814911938157390d3ea5118660b852ab0d5387006  # any key works ofc\n",
    "gsr.init(params92pn, params92pn.generator)\n",
    "res = gsr.multiply(key)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "2a869bed-8e21-46af-8f70-065f4afd6a82",
   "metadata": {},
   "outputs": [],
   "source": [
    "affine_gen = params92pn.generator.to_affine()\n",
    "affine_res = res.to_affine()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "e440399a-bc01-488b-8822-08cc0bf1672d",
   "metadata": {},
   "outputs": [],
   "source": [
    "dlog = pari.elllog(e,\n",
    "                   [int(affine_res.x), int(affine_res.y)],\n",
    "                   [int(affine_gen.x), int(affine_gen.y)],\n",
    "                   real_n)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "7ea6d6ae-a6f5-4b53-8c40-787d79970cb6",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3188254931\n",
      "32\n"
     ]
    }
   ],
   "source": [
    "mask = int((dlog - key) / 92)\n",
    "mask_len = mask.bit_length()\n",
    "print(mask)\n",
    "print(mask_len)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d40ec035-0656-4eda-8ef4-c14f9d53f49f",
   "metadata": {},
   "source": [
    "### Multiplicative splitting\n",
    "In multiplicative splitting the situation is a bit more complicated. Doing the same computation, where the target thinks the curve order is $n+92$ leads to:\n",
    "$$ P = [k r^{-1}\\pmod{n+92}][r \\mod n]G $$\n",
    "\n",
    "Since the curve is composite order we can easily compute the dlog $d$ of P to G, we get:\n",
    "$$ d = (k r^{-1})\\pmod{n+92}\\: r  = k + t (n + 92) $$\n",
    "\n",
    "However, the dlog is computed $\\mod n$ so we really get: $ d = k + t 92$. We extract the $t$ out of this.\n",
    "Note that $t$ will have roughly the same size as the mask $r$, since at the left side we have $(k r^{-1})_{\\mod (n+92)}$\n",
    "of size $n$ and $r$ of size of the mask and on the right size we have $t n$ that dominates.\n",
    "Thus at this point we have recovered the mask size.\n",
    "However, $t$ is always smaller than $r$, sometimes also in bitsize.\n",
    "\n",
    "Now that we have $s$ we can go back to the original equation and get:\n",
    "$$ (k r^{-1})\\pmod{n+92}\\: r = k + t (n + 92) $$\n",
    "\n",
    "We can then factor this value, lets call it $full$, and look for divisors that are larger than $t$ but smaller than the mask length\n",
    "that we recovered before. There may be multiple candidates here and we don't know how to distinguish between\n",
    "them. It holds for all of the candidates $c$ that the rest of the value $full$ is equal to the inverse of $c \\mod (n+92)$.\n",
    "However, sometimes there is only one candidate, which is equal to the true mask value $r$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "b5f398fc-90d7-455e-97bd-62b682d55961",
   "metadata": {},
   "outputs": [],
   "source": [
    "def divisors(primes, powers):\n",
    "    for comb in itertools.product(*[range(power+1) for power in powers]):\n",
    "        value = 1\n",
    "        for prime, power in zip(primes, comb):\n",
    "            value *= prime**power\n",
    "        yield value\n",
    "\n",
    "def pari_factor(number):\n",
    "    pari = cypari2.Pari(256_000_000, 2_000_000_000)\n",
    "    factors = pari.factor(number)\n",
    "    primes = list(map(int, factors[0]))\n",
    "    powers = list(map(int, factors[1]))\n",
    "    return primes, powers\n",
    "\n",
    "def pari_dlog(params, P, G, real_n, facts_str):\n",
    "    pari = cypari2.Pari(256_000_000, 2_000_000_000)\n",
    "    e = pari.ellinit([int(params.curve.parameters[\"a\"]), int(params.curve.parameters[\"b\"])], int(params.curve.prime))\n",
    "    e[15][0] = real_n\n",
    "    facts = pari(facts_str)\n",
    "    dlog = pari.elllog(e, P, G, facts)\n",
    "    return int(dlog)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "5f03e586-33df-4525-a722-f5f63d6ca28d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "f6984b3117d940a98a6a79a427492f2d",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "Collecting scalarmults:   0%|          | 0/100 [00:00<?, ?it/s]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "d646ad0b77344351a0cf9acbbbaac484",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "Computing dlogs:   0%|          | 0/100 [00:00<?, ?it/s]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "32\n"
     ]
    }
   ],
   "source": [
    "key = 0x20959f2b437de1e522baf6d814911938157390d3ea5118660b852ab0d5387006  # any key works\n",
    "msplit = MultiplicativeSplitting(mult, rand_bits=32)  # change the mask size here to your liking\n",
    "tries = 100\n",
    "num_workers = 20\n",
    "\n",
    "blens = [None for _ in range(tries)]\n",
    "ts = [None for _ in range(tries)]\n",
    "\n",
    "results = []\n",
    "rs = []\n",
    "\n",
    "with TaskExecutor(max_workers=num_workers) as pool:\n",
    "    for i in trange(tries, desc=\"Collecting scalarmults\"):\n",
    "        msplit.init(params92pn, params92pn.generator)\n",
    "        with local(DefaultContext()) as ctx:\n",
    "            res = msplit.multiply(key)\n",
    "    \n",
    "        affine_res = res.to_affine()\n",
    "        affine_gen = params92pn.generator.to_affine()\n",
    "        results.append(affine_res)\n",
    "        ctx.actions[0].walk(lambda action: rs.append(int(action.result)) if isinstance(action, RandomModAction) else None)\n",
    "        \n",
    "        pool.submit_task(i,\n",
    "                         pari_dlog,\n",
    "                         params92pn,\n",
    "                         [int(affine_res.x), int(affine_res.y)],\n",
    "                         [int(affine_gen.x), int(affine_gen.y)],\n",
    "                         real_n,\n",
    "                         repr(real_n_facts))\n",
    "    \n",
    "    for i, future in tqdm(pool.as_completed(), desc=\"Computing dlogs\", total=len(pool.tasks)):\n",
    "        dlog = future.result()\n",
    "        t = int((dlog - key) / 92)\n",
    "        ts[i] = t\n",
    "        blens[i] = t.bit_length()\n",
    "\n",
    "mask_len = max(blens)\n",
    "print(mask_len)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "5fbf8a38-983d-49a6-9cac-5350f960dc3e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "4a1915bb130e4c68bd8c450c37f9fd25",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "Factoring:   0%|          | 0/100 [00:00<?, ?it/s]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "with TaskExecutor(max_workers=num_workers) as pool:\n",
    "    fulls = []\n",
    "    for t in ts:\n",
    "        full = t * (real_n + 92) + key\n",
    "        fulls.append(full)\n",
    "        pool.submit_task(t,\n",
    "                         pari_factor,\n",
    "                         full)\n",
    "    facts = [None for _ in ts]\n",
    "    for t, future in tqdm(pool.as_completed(), desc=\"Factoring\", total=len(ts)):\n",
    "        result = future.result()\n",
    "        facts[ts.index(t)] = result"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "id": "0973fe4b-cdf5-4e91-850b-25375eeabb7e",
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Only one candidate, we got the mask: 2921741201 True\n",
      "Only one candidate, we got the mask: 2072087159 True\n",
      "Several candidates for r\n",
      "true r = 4039527236\n",
      "t = 713461330\n",
      "full = k + t (n + 92) = 54853006610429443711102562796765991385990914712382307462633349524199986834658739412728\n",
      "-----\n",
      "candidate = 3539175297\n",
      "candidate^-1 = 26490707025138842871232299283028891918970647143044648555061130998090929713163\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 54853006610429443711102562796765991385990914712382307462633349524199986834658739412728\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 1010980488\n",
      "candidate^-1 = 65382395850142385046254203937937214383274413673251102786718616383197090673492\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 54853006610429443711102562796765991385990914712382307462633349524199986834658739412728\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 1131335308\n",
      "candidate^-1 = 68241665580096390720464305945366132997268084763730630597092100846220368619186\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 54853006610429443711102562796765991385990914712382307462633349524199986834658739412728\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 1468796812\n",
      "candidate^-1 = 42974788306596910581678149765329273626138664755529162589164596291869762059227\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 54853006610429443711102562796765991385990914712382307462633349524199986834658739412728\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 2262670616\n",
      "candidate^-1 = 34120832790048195360232152972683066498634042381865315298546050423110184309593\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 54853006610429443711102562796765991385990914712382307462633349524199986834658739412728\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 2937593624\n",
      "candidate^-1 = 59928865533177637950767892719235074044242216512664024713011202452441507430870\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 54853006610429443711102562796765991385990914712382307462633349524199986834658739412728\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 1123577112\n",
      "candidate^-1 = 28797267645146503647782014101216400393084893568537834892905377895952751088363\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 54853006610429443711102562796765991385990914712382307462633349524199986834658739412728\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 2019763618\n",
      "candidate^-1 = 14492196383530008016252425678693741785643004999654181923797869651794644589503\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 54853006610429443711102562796765991385990914712382307462633349524199986834658739412728\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 3394005924\n",
      "candidate^-1 = 22747221860032130240154768648455377665756028254576876865697366948740122873062\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 54853006610429443711102562796765991385990914712382307462633349524199986834658739412728\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 1101597609\n",
      "candidate^-1 = 57299717742129214108904199687105698168184886340705550118886128389159682745636\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 54853006610429443711102562796765991385990914712382307462633349524199986834658739412728\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 1179725099\n",
      "candidate^-1 = 2589178315658163293839262175945801294566173159335058828325584381259536336976\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 54853006610429443711102562796765991385990914712382307462633349524199986834658739412728\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 3634715564\n",
      "candidate^-1 = 63500943916056123744437126756972853187061581127739385731937263938095743759671\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 54853006610429443711102562796765991385990914712382307462633349524199986834658739412728\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 1804895148\n",
      "candidate^-1 = 32692374163422277729585303046592231763960313081236584094052860741701608831649\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 54853006610429443711102562796765991385990914712382307462633349524199986834658739412728\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 4044771768\n",
      "candidate^-1 = 52010472181905874507274875250078108352478979351338650049500167853209533299960\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 54853006610429443711102562796765991385990914712382307462633349524199986834658739412728\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 2726036673\n",
      "candidate^-1 = 33412630048235921446011078560536554607851595990453256418011146175452156477886\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 54853006610429443711102562796765991385990914712382307462633349524199986834658739412728\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 4039527236\n",
      "candidate^-1 = 45687569571644186668055030675917308123994386634726534380327839132403948696008\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 54853006610429443711102562796765991385990914712382307462633349524199986834658739412728\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 734398406\n",
      "candidate^-1 = 9066633853435455843498663857517672789931561241259438341471383970726271315941\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 54853006610429443711102562796765991385990914712382307462633349524199986834658739412728\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 1203263432\n",
      "candidate^-1 = 10597089865254233934449136733317910414767585486955432722650386806045786846217\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 54853006610429443711102562796765991385990914712382307462633349524199986834658739412728\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 1009881809\n",
      "candidate^-1 = 28984392767060016032504851357387483571286009999308363847595739303589289179006\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 54853006610429443711102562796765991385990914712382307462633349524199986834658739412728\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 1697002962\n",
      "candidate^-1 = 45494443720064260480309537296910755331512056509153753731394733897480245746124\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 54853006610429443711102562796765991385990914712382307462633349524199986834658739412728\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 2203195218\n",
      "candidate^-1 = 28649858871064607054452099843552849084092443170352775059443064194579841372818\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 54853006610429443711102562796765991385990914712382307462633349524199986834658739412728\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 1817357782\n",
      "candidate^-1 = 50118945072353882169016617840804831911777393985679884627016719263178234716829\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 54853006610429443711102562796765991385990914712382307462633349524199986834658739412728\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 2359450198\n",
      "candidate^-1 = 1294589157829081646919631087972900647283086579667529414162792190629768168488\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 54853006610429443711102562796765991385990914712382307462633349524199986834658739412728\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 902447574\n",
      "candidate^-1 = 65384748326844555459170606093184463527920626162473168188105721483403217663298\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 54853006610429443711102562796765991385990914712382307462633349524199986834658739412728\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 3609790296\n",
      "candidate^-1 = 54787658461590321524721469359866553113153040675517735465455334677357430817081\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 54853006610429443711102562796765991385990914712382307462633349524199986834658739412728\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 2022385884\n",
      "candidate^-1 = 27138001604053383694692114827015342242612190432878413262142527093405813797407\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 54853006610429443711102562796765991385990914712382307462633349524199986834658739412728\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 1348257256\n",
      "candidate^-1 = 2265531026200892882109354403952576132745401514418176474784886333602094294854\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 54853006610429443711102562796765991385990914712382307462633349524199986834658739412728\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 848501481\n",
      "candidate^-1 = 14105944680370155640761438920680636200678344748508620625931659181947238689735\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 54853006610429443711102562796765991385990914712382307462633349524199986834658739412728\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 3959673578\n",
      "candidate^-1 = 8514341200062059445867282316798770218884343036808910622475199011346783490837\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 54853006610429443711102562796765991385990914712382307462633349524199986834658739412728\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 908678891\n",
      "candidate^-1 = 23354947384949399018175600008468789361209019701560882417175629913343216631145\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 54853006610429443711102562796765991385990914712382307462633349524199986834658739412728\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 1011192942\n",
      "candidate^-1 = 54276003208106767389384229654030684485224380865756826524285054186811627594814\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 54853006610429443711102562796765991385990914712382307462633349524199986834658739412728\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 3029645427\n",
      "candidate^-1 = 9661464255686672010834950452462494523762003333102787949198579767863096393002\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 54853006610429443711102562796765991385990914712382307462633349524199986834658739412728\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 1979836789\n",
      "candidate^-1 = 17028682400124118891734564633597540437768686073617821244950398022693566981674\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 54853006610429443711102562796765991385990914712382307462633349524199986834658739412728\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 1258968696\n",
      "candidate^-1 = 11695781644483879685362356889647048665988891413217912935596458034008096001175\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 54853006610429443711102562796765991385990914712382307462633349524199986834658739412728\n",
      "above == full? True\n",
      "-----\n",
      "---------------------\n",
      "Several candidates for r\n",
      "true r = 2546266567\n",
      "t = 1548864186\n",
      "full = k + t (n + 92) = 119081236569616103969202985151515360402172033984486476071765168905412656991728103589856\n",
      "-----\n",
      "candidate = 3968039777\n",
      "candidate^-1 = 47199999723628001336554368590553381329909688423987679279501585692741942222130\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 119081236569616103969202985151515360402172033984486476071765168905412656991728103589856\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 2546266567\n",
      "candidate^-1 = 53261354779930768597409107118256662925077636528602413316194826191853234093722\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 119081236569616103969202985151515360402172033984486476071765168905412656991728103589856\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 2441870632\n",
      "candidate^-1 = 57479263860955910841936440041364026045516801621530257119975624597452342910333\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 119081236569616103969202985151515360402172033984486476071765168905412656991728103589856\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 2857178584\n",
      "candidate^-1 = 75748083438556289408418666080557136798495350712469032724680328085740307176009\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 119081236569616103969202985151515360402172033984486476071765168905412656991728103589856\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 1757828696\n",
      "candidate^-1 = 17227168816999813895475622948114925759056198231882304548407742991291781760271\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 119081236569616103969202985151515360402172033984486476071765168905412656991728103589856\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 3736310456\n",
      "candidate^-1 = 48879952893042060686454552217703589967985177807205861867478155758152793393123\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 119081236569616103969202985151515360402172033984486476071765168905412656991728103589856\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 3133866544\n",
      "candidate^-1 = 33664482913723953820412695074440929318832358645764599964801070204254096100835\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 119081236569616103969202985151515360402172033984486476071765168905412656991728103589856\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 3515657392\n",
      "candidate^-1 = 47055055788379089607666629310627900110700983250840595692632775802152517281392\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 119081236569616103969202985151515360402172033984486476071765168905412656991728103589856\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 3329733203\n",
      "candidate^-1 = 63341901525758342256800386523708293549278712718872106311460104915244606307703\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 119081236569616103969202985151515360402172033984486476071765168905412656991728103589856\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 1566933272\n",
      "candidate^-1 = 67328965827447907640825390148881858637664717291529199929602140408508192201670\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 119081236569616103969202985151515360402172033984486476071765168905412656991728103589856\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 1868155228\n",
      "candidate^-1 = 20876963026325756053051468762266305473624587344612836898098502903292333983733\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 119081236569616103969202985151515360402172033984486476071765168905412656991728103589856\n",
      "above == full? True\n",
      "-----\n",
      "---------------------\n",
      "Several candidates for r\n",
      "true r = 586356740\n",
      "t = 404263194\n",
      "full = k + t (n + 92) = 31080944018917461343795538061263727736823600975872546557801979004011543724824943696960\n",
      "-----\n",
      "candidate = 938170784\n",
      "candidate^-1 = 52975562029403857607630522378170505964905977660731390116544502276802695405629\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 31080944018917461343795538061263727736823600975872546557801979004011543724824943696960\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 1876341568\n",
      "candidate^-1 = 64929252394581111463744079025655690213625872965265138476701155444907974104071\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 31080944018917461343795538061263727736823600975872546557801979004011543724824943696960\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 437122240\n",
      "candidate^-1 = 34960422445902541801696735185321223222583924317096364308996315884209127750875\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 31080944018917461343795538061263727736823600975872546557801979004011543724824943696960\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 1222440419\n",
      "candidate^-1 = 68600022163021642948955277628963692751947775934147947283493669830758786832741\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 31080944018917461343795538061263727736823600975872546557801979004011543724824943696960\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 586356740\n",
      "candidate^-1 = 23254545039239479916322727266560109973972949641331114716984956752473710406996\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 31080944018917461343795538061263727736823600975872546557801979004011543724824943696960\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 2444880838\n",
      "candidate^-1 = 72741482461390004134406456651052283607146772101973417060175739221886019817627\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 31080944018917461343795538061263727736823600975872546557801979004011543724824943696960\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 1172713480\n",
      "candidate^-1 = 11627272519619739958161363633280054986986474820665557358492478376236855203498\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 31080944018917461343795538061263727736823600975872546557801979004011543724824943696960\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 469085392\n",
      "candidate^-1 = 29068181299049349895403409083200137467466187051663893396231195940592138008745\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 31080944018917461343795538061263727736823600975872546557801979004011543724824943696960\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 2345426960\n",
      "candidate^-1 = 5813636259809869979080681816640027493493237410332778679246239188118427601749\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 31080944018917461343795538061263727736823600975872546557801979004011543724824943696960\n",
      "above == full? True\n",
      "-----\n",
      "---------------------\n",
      "Several candidates for r\n",
      "true r = 4099981182\n",
      "t = 2642778076\n",
      "full = k + t (n + 92) = 203184555558590712836289545636923662903278351877540745889142089702149488986478501195426\n",
      "-----\n",
      "candidate = 3228686923\n",
      "candidate^-1 = 67416396472256627079998760927485239166988608007947808690086396876984288458738\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 203184555558590712836289545636923662903278351877540745889142089702149488986478501195426\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 3923474478\n",
      "candidate^-1 = 53037483928698273854873915437268008099102139343418152750051187589693742832853\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 203184555558590712836289545636923662903278351877540745889142089702149488986478501195426\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 2767445934\n",
      "candidate^-1 = 27397167377793821380093463966638862719922863829406518913862257281139501333519\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 203184555558590712836289545636923662903278351877540745889142089702149488986478501195426\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 3082729947\n",
      "candidate^-1 = 67502252272888712178930437829250192126129995527986739863701511477792036332722\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 203184555558590712836289545636923662903278351877540745889142089702149488986478501195426\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 4099981182\n",
      "candidate^-1 = 53728758251263460792603093081868749337569947904541549081333778030796416192083\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 203184555558590712836289545636923662903278351877540745889142089702149488986478501195426\n",
      "above == full? True\n",
      "-----\n",
      "---------------------\n",
      "Only one candidate, we got the mask: 4240436952 True\n",
      "Several candidates for r\n",
      "true r = 1054759622\n",
      "t = 694271292\n",
      "full = k + t (n + 92) = 53377620017317855809796612613608262425705269850145672571876348376319535218874908447234\n",
      "-----\n",
      "candidate = 2082978083\n",
      "candidate^-1 = 70926800923833726814245799453581520027922406161928168992184434104522114548589\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 53377620017317855809796612613608262425705269850145672571876348376319535218874908447234\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 1465316453\n",
      "candidate^-1 = 60239579291535863658082545576886406273278056948782152971212929977157790371263\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 53377620017317855809796612613608262425705269850145672571876348376319535218874908447234\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 4165956166\n",
      "candidate^-1 = 73904871841796046067051717563361197245134087215863527914521121358767683675551\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 53377620017317855809796612613608262425705269850145672571876348376319535218874908447234\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 1054759622\n",
      "candidate^-1 = 32916166462770436865868266168928274812999826737500918380668592970627433533542\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 53377620017317855809796612613608262425705269850145672571876348376319535218874908447234\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 1150690474\n",
      "candidate^-1 = 71499038532384546579941614468655340824282611408865892492623092957903951282093\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 53377620017317855809796612613608262425705269850145672571876348376319535218874908447234\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 2930632906\n",
      "candidate^-1 = 68561261025647114488970090625013640367811912609290519904035369295085521586888\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 53377620017317855809796612613608262425705269850145672571876348376319535218874908447234\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 891931754\n",
      "candidate^-1 = 71506829279038074109757883564477355140976176319499648868114881886397351037606\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 53377620017317855809796612613608262425705269850145672571876348376319535218874908447234\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 1582042478\n",
      "candidate^-1 = 3588093433295855163732315625904528984462656656446659629430225432413461610135\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 53377620017317855809796612613608262425705269850145672571876348376319535218874908447234\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 2863810801\n",
      "candidate^-1 = 68817051866482904589958336845935410415616665963668120410224486154991202020843\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 53377620017317855809796612613608262425705269850145672571876348376319535218874908447234\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 4027416659\n",
      "candidate^-1 = 9445019186430103977146513323452829598031350649704699735484053900399235680239\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 53377620017317855809796612613608262425705269850145672571876348376319535218874908447234\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 791021239\n",
      "candidate^-1 = 7176186866591710327464631251809057968925313312893319258860450864826923220270\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 53377620017317855809796612613608262425705269850145672571876348376319535218874908447234\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 1328364218\n",
      "candidate^-1 = 61241335234879901001880976060972940299803372432205022907074540241147849493070\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 53377620017317855809796612613608262425705269850145672571876348376319535218874908447234\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 2538649658\n",
      "candidate^-1 = 40460798138314952610978306328523747705796793263233053748780937033651947724626\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 53377620017317855809796612613608262425705269850145672571876348376319535218874908447234\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 1269324829\n",
      "candidate^-1 = 4038653516871539902098976983906620949247818256667220660704065454290642646739\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 53377620017317855809796612613608262425705269850145672571876348376319535218874908447234\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 2131580798\n",
      "candidate^-1 = 13269399537091224226648020073813398658821185281376945290634379651910514289428\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 53377620017317855809796612613608262425705269850145672571876348376319535218874908447234\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 1065790399\n",
      "candidate^-1 = 26538799074182448453296040147626797317642370562753890581268759303821028578856\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 53377620017317855809796612613608262425705269850145672571876348376319535218874908447234\n",
      "above == full? True\n",
      "-----\n",
      "---------------------\n",
      "Several candidates for r\n",
      "true r = 3634113672\n",
      "t = 2680190962\n",
      "full = k + t (n + 92) = 206060968331406077945547732896592415701478405398401148885296351493997971930068100577944\n",
      "-----\n",
      "candidate = 3634113672\n",
      "candidate^-1 = 37634033022565823810085029464398851788513344511265489647294862481459002347491\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 206060968331406077945547732896592415701478405398401148885296351493997971930068100577944\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 4048089192\n",
      "candidate^-1 = 71468722554076888672183327062398623529815128497721212450005095375489413086598\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 206060968331406077945547732896592415701478405398401148885296351493997971930068100577944\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 3735061274\n",
      "candidate^-1 = 44928566482470354552499773065160058438807120959047383098649899561745326911344\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 206060968331406077945547732896592415701478405398401148885296351493997971930068100577944\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 3201481092\n",
      "candidate^-1 = 1161365723043170098011311460592818537044462272356022390519677080027379528226\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 206060968331406077945547732896592415701478405398401148885296351493997971930068100577944\n",
      "above == full? True\n",
      "-----\n",
      "---------------------\n",
      "Several candidates for r\n",
      "true r = 2376091517\n",
      "t = 1953095034\n",
      "full = k + t (n + 92) = 150159693718128688257542828458944146909547607961398772887284238536472708007326310110880\n",
      "-----\n",
      "candidate = 3246914120\n",
      "candidate^-1 = 44590844092197648231531019613425704807242597656472481439265521332832917097606\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 150159693718128688257542828458944146909547607961398772887284238536472708007326310110880\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 2376091517\n",
      "candidate^-1 = 19772973002144164969938083673165593098966704920762394800382781683851452496636\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 150159693718128688257542828458944146909547607961398772887284238536472708007326310110880\n",
      "above == full? True\n",
      "-----\n",
      "candidate = 2597531296\n",
      "candidate^-1 = 17297083735367877629484956680211693777880362935691158380652997359534519970751\n",
      "(key * candidate^-1)_mod(n+92) * candidate = 150159693718128688257542828458944146909547607961398772887284238536472708007326310110880\n",
      "above == full? True\n",
      "-----\n",
      "---------------------\n",
      "Only one candidate, we got the mask: 3341139537 True\n",
      "Only one candidate, we got the mask: 3722808561 True\n",
      "Only one candidate, we got the mask: 2045751181 True\n",
      "Only one candidate, we got the mask: 2166075112 True\n",
      "Only one candidate, we got the mask: 1985265929 True\n",
      "Only one candidate, we got the mask: 4284624218 True\n",
      "Only one candidate, we got the mask: 3985300601 True\n",
      "Only one candidate, we got the mask: 3864453118 True\n",
      "Only one candidate, we got the mask: 2479654721 True\n",
      "Total recovered masks: 12 out of 100\n"
     ]
    }
   ],
   "source": [
    "candidate_amounts = []\n",
    "for t, blen, r, (primes, powers), full, result in zip(ts, blens, rs, facts, fulls, results):\n",
    "    #print(primes, powers)\n",
    "    #print(s, blen, r, r.bit_length())\n",
    "    candidates = set()\n",
    "    for divisor in divisors(primes, powers):\n",
    "        if blen <= divisor.bit_length()  <= mask_len and divisor > t:\n",
    "            candidates.add(divisor)\n",
    "    #print(f\"Candidates: {len(candidates)}, {r in candidates}\")\n",
    "    candidate_amounts.append(len(candidates))\n",
    "    if len(candidates) == 1:\n",
    "        candidate = candidates.pop()\n",
    "        print(\"Only one candidate, we got the mask:\", candidate, candidate == r)\n",
    "    else:\n",
    "        if len(candidate_amounts) > 10:\n",
    "            # Do not print everything\n",
    "            continue\n",
    "        print(\"Several candidates for r\")\n",
    "        print(f\"true r = {r}\")\n",
    "        print(f\"t = {t}\")\n",
    "        print(f\"full = k + t (n + 92) = {full}\")\n",
    "        print(\"-----\")\n",
    "        for candidate in candidates:\n",
    "            print(f\"candidate = {candidate}\")            \n",
    "            candidate_inverse = mod(candidate, real_n + 92).inverse() \n",
    "            print(f\"candidate^-1 = {candidate_inverse}\")\n",
    "            multiplied = candidate * int(candidate_inverse * key)\n",
    "            print(f\"(key * candidate^-1)_mod(n+92) * candidate = {multiplied}\")\n",
    "            print(f\"above == full? {multiplied == full}\")\n",
    "            print(\"-----\")\n",
    "        print(\"---------------------\")\n",
    "    #print(\"--\")\n",
    "print(f\"Total recovered masks: {len(list(filter(lambda a: a == 1, candidate_amounts)))} out of {tries}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "id": "6274ff91-325f-4c6b-a4d7-d66b994d730f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "aba07231a8d44a2f92c75ef741f73f50",
       "version_major": 2,
       "version_minor": 0
      },
      "image/png": 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",
      "text/html": [
       "\n",
       "            <div style=\"display: inline-block;\">\n",
       "                <div class=\"jupyter-widgets widget-label\" style=\"text-align: center;\">\n",
       "                    Figure\n",
       "                </div>\n",
       "                <img 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' width=640.0/>\n",
       "            </div>\n",
       "        "
      ],
      "text/plain": [
       "Canvas(toolbar=Toolbar(toolitems=[('Home', 'Reset original view', 'home', 'home'), ('Back', 'Back to previous …"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "max_amount = max(candidate_amounts)\n",
    "fig = plt.subplots()\n",
    "plt.hist(candidate_amounts, range=(1, max_amount), align=\"left\", density=True, bins=range(1, max_amount))#, bins=list(range(20)) + list(range(20, 100, 5)) + list(range(100, max(candidate_amounts), 10)))\n",
    "plt.xlabel(\"candidate masks\")\n",
    "plt.ylabel(\"proportion\")\n",
    "plt.xticks(range(max_amount))\n",
    "plt.xlim(0, 20);\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "id": "9f22ca9d-bdc2-4ea5-b2bc-249a256bb8ad",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "7aebc586f3684a8dbc8977aa2bcb8c38",
       "version_major": 2,
       "version_minor": 0
      },
      "image/png": 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",
      "text/html": [
       "\n",
       "            <div style=\"display: inline-block;\">\n",
       "                <div class=\"jupyter-widgets widget-label\" style=\"text-align: center;\">\n",
       "                    Figure\n",
       "                </div>\n",
       "                <img 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' width=640.0/>\n",
       "            </div>\n",
       "        "
      ],
      "text/plain": [
       "Canvas(toolbar=Toolbar(toolitems=[('Home', 'Reset original view', 'home', 'home'), ('Back', 'Back to previous …"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "max_amount = max(candidate_amounts)\n",
    "fig = plt.subplots()\n",
    "plt.hist(candidate_amounts, range=(1, max_amount), align=\"left\", density=True, bins=range(1, max_amount))#, bins=list(range(20)) + list(range(20, 100, 5)) + list(range(100, max(candidate_amounts), 10)))\n",
    "plt.xlabel(\"candidate masks\")\n",
    "plt.ylabel(\"proportion\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "1009372d-08af-4dbd-9382-be1f02b4ad42",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.12.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}