{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "bafc2f4e-05a3-4120-bcd6-5d1f5fb91cd9",
   "metadata": {},
   "source": [
    "# Distinguishing countermeasures by output"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 204,
   "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": 205,
   "id": "b1b9596c-1eba-4ace-af84-8cb279d84cc2",
   "metadata": {},
   "outputs": [],
   "source": [
    "model = ShortWeierstrassModel()\n",
    "coords = model.coordinates[\"projective\"]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 206,
   "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": 207,
   "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": [
       "  0%|          | 0/1000 [00:00<?, ?it/s]"
      ]
     },
     "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": [
       "  0%|          | 0/1000 [00:00<?, ?it/s]"
      ]
     },
     "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": 226,
   "id": "20a26f27-620d-4d7f-92bd-b949482b5c9a",
   "metadata": {},
   "outputs": [],
   "source": [
    "pari = cypari2.Pari(256_000_000, 2_000_000_000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 227,
   "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": 228,
   "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": 229,
   "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": 230,
   "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": 231,
   "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": 232,
   "id": "7ea6d6ae-a6f5-4b53-8c40-787d79970cb6",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3752128619\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": 233,
   "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": 234,
   "id": "5f03e586-33df-4525-a722-f5f63d6ca28d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "e4879058a89a44af9abf2e7a58b1022f",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "Collecting scalarmults:   0%|          | 0/1000 [00:00<?, ?it/s]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "69ff8872b4454bacb5f1c4a7503e6d3b",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "Computing dlogs:   0%|          | 0/1000 [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 = 1000\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] = s.bit_length()\n",
    "\n",
    "mask_len = max(blens)\n",
    "print(mask_len)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 235,
   "id": "5fbf8a38-983d-49a6-9cac-5350f960dc3e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "89d1369c99644e88bd73b6230963716a",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "Factoring:   0%|          | 0/1000 [00:00<?, ?it/s]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "num_workers = 25\n",
    "\n",
    "with TaskExecutor(max_workers=num_workers) as pool:\n",
    "    for t in ts:\n",
    "        full = t * (real_n + 92) + key\n",
    "        pool.submit_task(t,\n",
    "                         pari_factor,\n",
    "                         full)\n",
    "    facts = [None for _ in ss]\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": 236,
   "id": "0973fe4b-cdf5-4e91-850b-25375eeabb7e",
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Only one candidate, we got the mask: 3223834487 True\n",
      "Only one candidate, we got the mask: 2377873873 True\n",
      "Only one candidate, we got the mask: 2549271668 True\n",
      "Only one candidate, we got the mask: 2536406777 True\n",
      "Only one candidate, we got the mask: 3474268328 True\n",
      "Only one candidate, we got the mask: 3550639425 True\n",
      "Only one candidate, we got the mask: 3814005975 True\n",
      "Only one candidate, we got the mask: 4238733795 True\n",
      "Only one candidate, we got the mask: 4168083955 True\n",
      "Only one candidate, we got the mask: 2761553491 True\n",
      "Only one candidate, we got the mask: 3411331906 True\n",
      "Only one candidate, we got the mask: 789326198 True\n",
      "Only one candidate, we got the mask: 2285458765 True\n",
      "Only one candidate, we got the mask: 774733453 True\n",
      "Only one candidate, we got the mask: 3283055299 True\n",
      "Only one candidate, we got the mask: 2838749009 True\n",
      "Only one candidate, we got the mask: 3276117366 True\n",
      "Only one candidate, we got the mask: 860570263 True\n",
      "Only one candidate, we got the mask: 981664829 True\n",
      "Only one candidate, we got the mask: 4134679928 True\n",
      "Only one candidate, we got the mask: 3988645114 True\n",
      "Only one candidate, we got the mask: 2280222647 True\n",
      "Only one candidate, we got the mask: 3577824626 True\n",
      "Only one candidate, we got the mask: 3679892960 True\n",
      "Only one candidate, we got the mask: 1774221601 True\n",
      "Only one candidate, we got the mask: 2623580147 True\n",
      "Only one candidate, we got the mask: 1702008059 True\n",
      "Only one candidate, we got the mask: 3933544069 True\n",
      "Only one candidate, we got the mask: 3999529804 True\n",
      "Only one candidate, we got the mask: 3480801033 True\n",
      "Only one candidate, we got the mask: 3813421579 True\n",
      "Only one candidate, we got the mask: 3417626666 True\n",
      "Only one candidate, we got the mask: 4170438661 True\n",
      "Only one candidate, we got the mask: 3502288943 True\n",
      "Only one candidate, we got the mask: 2786122643 True\n",
      "Only one candidate, we got the mask: 2127512283 True\n",
      "Only one candidate, we got the mask: 2531786983 True\n",
      "Only one candidate, we got the mask: 3983366093 True\n",
      "Only one candidate, we got the mask: 2379294079 True\n",
      "Only one candidate, we got the mask: 2930272946 True\n",
      "Only one candidate, we got the mask: 2500319501 True\n",
      "Only one candidate, we got the mask: 2947684105 True\n",
      "Only one candidate, we got the mask: 3995028346 True\n",
      "Only one candidate, we got the mask: 3421022802 True\n",
      "Only one candidate, we got the mask: 3953171129 True\n",
      "Only one candidate, we got the mask: 2982511438 True\n",
      "Only one candidate, we got the mask: 2830285508 True\n",
      "Only one candidate, we got the mask: 277646521 True\n",
      "Only one candidate, we got the mask: 3775642326 True\n",
      "Only one candidate, we got the mask: 3528903061 True\n",
      "Only one candidate, we got the mask: 2433595133 True\n",
      "Only one candidate, we got the mask: 2809043104 True\n",
      "Only one candidate, we got the mask: 3918854258 True\n",
      "Only one candidate, we got the mask: 2172498737 True\n",
      "Only one candidate, we got the mask: 2614989645 True\n",
      "Only one candidate, we got the mask: 3881796054 True\n",
      "Only one candidate, we got the mask: 3763131597 True\n",
      "Only one candidate, we got the mask: 3333059164 True\n",
      "Only one candidate, we got the mask: 781918702 True\n",
      "Only one candidate, we got the mask: 3898624034 True\n",
      "Only one candidate, we got the mask: 2695908441 True\n",
      "Only one candidate, we got the mask: 2428288661 True\n",
      "Only one candidate, we got the mask: 380310234 True\n",
      "Only one candidate, we got the mask: 3806008683 True\n",
      "Only one candidate, we got the mask: 1583055543 True\n",
      "Only one candidate, we got the mask: 3071695987 True\n",
      "Only one candidate, we got the mask: 2466420323 True\n",
      "Only one candidate, we got the mask: 3668827111 True\n",
      "Only one candidate, we got the mask: 3030308051 True\n",
      "Only one candidate, we got the mask: 4178268350 True\n",
      "Only one candidate, we got the mask: 601467334 True\n",
      "Only one candidate, we got the mask: 1756886305 True\n",
      "Only one candidate, we got the mask: 3789366239 True\n",
      "Only one candidate, we got the mask: 1709528826 True\n",
      "Only one candidate, we got the mask: 4179236943 True\n",
      "Only one candidate, we got the mask: 3246542896 True\n",
      "Only one candidate, we got the mask: 1036989838 True\n",
      "Only one candidate, we got the mask: 2843879303 True\n",
      "Only one candidate, we got the mask: 2943368159 True\n",
      "Only one candidate, we got the mask: 3694910341 True\n",
      "Only one candidate, we got the mask: 3509390042 True\n",
      "Only one candidate, we got the mask: 3797507269 True\n",
      "Only one candidate, we got the mask: 3948777957 True\n",
      "Only one candidate, we got the mask: 3649873740 True\n",
      "Only one candidate, we got the mask: 2612080546 True\n",
      "Only one candidate, we got the mask: 3671883118 True\n",
      "Only one candidate, we got the mask: 2973101524 True\n",
      "Only one candidate, we got the mask: 3724708289 True\n",
      "Only one candidate, we got the mask: 4134344947 True\n",
      "Only one candidate, we got the mask: 3489598522 True\n",
      "Only one candidate, we got the mask: 3984902078 True\n",
      "Only one candidate, we got the mask: 4259116327 True\n",
      "Only one candidate, we got the mask: 3389988907 True\n",
      "Only one candidate, we got the mask: 868217594 True\n",
      "Only one candidate, we got the mask: 3729431567 True\n",
      "Only one candidate, we got the mask: 3542657779 True\n",
      "Only one candidate, we got the mask: 2695799377 True\n",
      "Only one candidate, we got the mask: 3972219937 True\n",
      "Only one candidate, we got the mask: 2701111433 True\n",
      "Only one candidate, we got the mask: 3806713198 True\n",
      "Only one candidate, we got the mask: 3503145915 True\n",
      "Only one candidate, we got the mask: 2397268241 True\n",
      "Only one candidate, we got the mask: 2919558765 True\n",
      "Only one candidate, we got the mask: 543703399 True\n",
      "Total recovered masks: 104 out of 1000\n"
     ]
    }
   ],
   "source": [
    "candidate_amounts = []\n",
    "for t, blen, r, (primes, powers), result in zip(ts, blens, rs, facts, 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",
    "    #print(\"--\")\n",
    "print(f\"Total recovered masks: {len(list(filter(lambda a: a == 1, candidate_amounts)))} out of {tries}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 237,
   "id": "6274ff91-325f-4c6b-a4d7-d66b994d730f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "d557894c3fbd4543b9fd6c240b676212",
       "version_major": 2,
       "version_minor": 0
      },
      "image/png": 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ghQsXtHLlSh09etRRJQAAAFAHDguArq6uCgoK4mHPAAAAjYxD7wJ+8sknNX36dH3//ffXPNbixYsVEhIiDw8PRUdHa9euXVfsv27dOnXt2lUeHh4KCwvT+++/f1mf/fv3a+jQofLx8VHz5s3Vu3dv5efnX/NcAQAAGjOHBsBFixbp448/VkBAgLp06aJevXpVetXU2rVrlZycrFmzZik3N1fh4eGKj49XcXFxlf23b9+uESNGaMyYMdqzZ48SEhKUkJCgffv22focPnxY/fr1U9euXbV161Z9+eWXSk1N5fE0AADgumcxDMNw1OCzZ8++4vuzZs2q0TjR0dHq3bu3Fi1aJEmyWq0KDAzU+PHjNXXq1Mv6JyYmqqysTBs3brS13XrrrYqIiNDSpUslScOHD5erq6tWrlxZ0925TGlpqXx8fFRSUiJvb+8abRMydVOd68H+vp0zpKGnAACoZ3X5/L7eOOQu4ItqGvCu5Pz588rJydG0adNsbU5OToqLi1N2dnaV22RnZ9ueRXhRfHy80tPTJf0cIDdt2qQnnnhC8fHx2rNnjzp06KBp06YpISGh2rmUl5ervLzc9nNpaWnddwwAAKCBOPQU8EU5OTlatWqVVq1aZXtGYE2dOnVKFRUV8vPzq9Tu5+enwsLCKrcpLCy8Yv/i4mKdPXtWc+bM0aBBg/TRRx9p2LBhuvfee7Vt27Zq55KWliYfHx/bKzAwsFb7AgAA0Bg4dAWwuLhYw4cP19atW9WiRQtJ0unTp3XHHXdozZo1at26tSPLV8tqtUqS7rnnHk2ePFmSFBERoe3bt2vp0qXq379/ldtNmzat0spiaWkpIRAAADQ5Dl0BHD9+vM6cOaOvvvpK33//vb7//nvt27dPpaWlmjBhQo3G8PX1lbOzs4qKiiq1FxUVyd/fv8pt/P39r9jf19dXLi4u6tatW6U+N9988xXvAnZ3d5e3t3elFwAAQFPj0ACYkZGhP//5z7r55pttbd26ddPixYv1wQcf1GgMNzc3RUZGKisry9ZmtVqVlZWlmJiYKreJiYmp1F+SMjMzbf3d3NzUu3dvHThwoFKfgwcPKjg4uEbzAgAAaKocegrYarXK1dX1snZXV1fbadiaSE5OVlJSkqKiotSnTx8tWLBAZWVlGj16tCRp1KhRateundLS0iRJEydOVP/+/TVv3jwNGTJEa9as0e7du7Vs2TLbmCkpKUpMTNTtt9+uO+64QxkZGXrvvfe0devWa9tpAACARs6hK4B33nmnJk6cqBMnTtjajh8/rsmTJ2vAgAE1HicxMVEvvPCCZs6cqYiICOXl5SkjI8N2o0d+fr4KCgps/WNjY7V69WotW7ZM4eHh+tvf/qb09HT16NHD1mfYsGFaunSp5s6dq7CwML366qv6+9//rn79+tlhzwEAABovhz4H8NixYxo6dKi++uor280Sx44dU48ePbRhwwa1b9/eUaXrBc8BbPp4DiAAmA/PAXTwKeDAwEDl5uZq8+bN+uabbyT9fKNFXFycI8sCAADgChwaACXJYrHo17/+tX796187uhQAAABqwOEBMCsrSy+++KL2798v6ecVwEmTJrEKCFOoj1P+nMYGANSWQ28C+fOf/6xBgwbJy8tLEydO1MSJE+Xt7a277rpLixcvdmRpAAAAVMOhK4DPPvusXnzxRY0bN87WNmHCBPXt21fPPvusxo4d68jyAAAAqIJDVwBPnz6tQYMGXdY+cOBAlZSUOLI0AAAAquHQADh06FCtX7/+svZ3331X//Vf/+XI0gAAAKiGQ08Bd+vWTc8884y2bt1q+xq2HTt26LPPPtPjjz+ul19+2da3pt8NDAAAgGvj0AC4YsUK3Xjjjfr666/19ddf29pbtGihFStW2H62WCwEQAAAgHri0AB45MgRRw4PAACAOnDoNYC/ZBiGHPitcwAAAKghhwfAN998U2FhYfL09JSnp6d69uyplStXOrosAAAAquHQU8Dz589Xamqqxo0bp759+0qSPv30Uz322GM6deqUJk+e7MjyAAAAqIJDA+DChQu1ZMkSjRo1ytY2dOhQde/eXU899RQBEAAAoAE49BRwQUGBYmNjL2uPjY1VQUGBI0sDAACgGg4NgJ06ddLbb799WfvatWsVGhrqyNIAAACohkNPAc+ePVuJiYn6+OOPbdcAfvbZZ8rKyqoyGAIAAMDxHLoCeN9992nXrl3y9fVVenq60tPT5evrq127dmnYsGGOLA0AAIBqOGwF8MKFC/rf//1fpaamatWqVY4qAwAAgFpy2Aqgq6ur/v73vztqeAAAANSRQ08BJyQkKD093ZElAAAAUEsOvQkkNDRUf/zjH/XZZ58pMjJSzZs3r/T+hAkTHFkeAAAAVXBoAFyxYoVatGihnJwc5eTkVHrPYrEQAAEAABqAQwPgkSNHbH82DEPSz8EPAAAADceh1wBKP68C9ujRQx4eHvLw8FCPHj306quvOrosAAAAquHQFcCZM2dq/vz5Gj9+vGJiYiRJ2dnZmjx5svLz8/XHP/7RkeUBAABQBYcGwCVLlmj58uUaMWKErW3o0KHq2bOnxo8fTwAEAABoAA4NgBcuXFBUVNRl7ZGRkfrPf/7jyNJAjYRM3dTQUwAAoN459BrABx98UEuWLLmsfdmyZRo5cqQjSwMAAKAaDl0BlH6+CeSjjz7SrbfeKknauXOn8vPzNWrUKCUnJ9v6zZ8/39FTAQAAgBwcAPft26devXpJkg4fPixJ8vX1la+vr/bt22frx6NhAAAA6o9DA+CWLVscOTwAAADqwOHPAQQAAEDjQgAEAAAwGQIgAACAyRAAAQAATIYACAAAYDIEQAAAAJMhAAIAAJgMARAAAMBkCIAAAAAmQwAEAAAwGQIgAACAyRAAAQAATIYACAAAYDIEQAAAAJMhAAIAAJgMARAAAMBkCIAAAAAmQwAEAAAwmSYVABcvXqyQkBB5eHgoOjpau3btumL/devWqWvXrvLw8FBYWJjef//9avs+9thjslgsWrBggZ1nDQAA0Lg0mQC4du1aJScna9asWcrNzVV4eLji4+NVXFxcZf/t27drxIgRGjNmjPbs2aOEhAQlJCRo3759l/Vdv369duzYoYCAAEfvBgAAQINrMgFw/vz5evTRRzV69Gh169ZNS5cuVbNmzfTaa69V2f+ll17SoEGDlJKSoptvvll/+tOf1KtXLy1atKhSv+PHj2v8+PH661//KldX1/rYFQAAgAbVJALg+fPnlZOTo7i4OFubk5OT4uLilJ2dXeU22dnZlfpLUnx8fKX+VqtVDz74oFJSUtS9e3fHTB4AAKCRcWnoCdTEqVOnVFFRIT8/v0rtfn5++uabb6rcprCwsMr+hYWFtp+fe+45ubi4aMKECTWaR3l5ucrLy20/l5aW1nQXAAAAGo0msQLoCDk5OXrppZf0xhtvyGKx1GibtLQ0+fj42F6BgYEOniUAAID9NYkA6OvrK2dnZxUVFVVqLyoqkr+/f5Xb+Pv7X7H/J598ouLiYgUFBcnFxUUuLi46evSoHn/8cYWEhFQ55rRp01RSUmJ7HTt27Np3DgAAoJ41iQDo5uamyMhIZWVl2dqsVquysrIUExNT5TYxMTGV+ktSZmamrf+DDz6oL7/8Unl5ebZXQECAUlJS9OGHH1Y5pru7u7y9vSu9AAAAmpomcQ2gJCUnJyspKUlRUVHq06ePFixYoLKyMo0ePVqSNGrUKLVr105paWmSpIkTJ6p///6aN2+ehgwZojVr1mj37t1atmyZJKlVq1Zq1apVpRqurq7y9/dXly5d6nfnAAAA6lGTCYCJiYk6efKkZs6cqcLCQkVERCgjI8N2o0d+fr6cnP5/QTM2NlarV6/WjBkzNH36dIWGhio9PV09evRoqF0AAABoFCyGYRgNPYmmqrS0VD4+PiopKanx6eCQqZscPCuYzbdzhjT0FACgSanL5/f1psmsAAKoWn38o4KQCQDXlyZxEwgAAADshwAIAABgMgRAAAAAkyEAAgAAmAwBEAAAwGQIgAAAACZDAAQAADAZAiAAAIDJEAABAABMhgAIAABgMgRAAAAAkyEAAgAAmAwBEAAAwGQIgAAAACZDAAQAADAZAiAAAIDJEAABAABMhgAIAABgMgRAAAAAkyEAAgAAmIxLQ08AQOMXMnWTw2t8O2eIw2sAAH7GCiAAAIDJEAABAABMhgAIAABgMgRAAAAAkyEAAgAAmAwBEAAAwGQIgAAAACZDAAQAADAZAiAAAIDJ8E0gABoFvm0EAOoPK4AAAAAmQwAEAAAwGQIgAACAyRAAAQAATIYACAAAYDIEQAAAAJMhAAIAAJgMARAAAMBkCIAAAAAmQwAEAAAwGQIgAACAyRAAAQAATIYACAAAYDIEQAAAAJMhAAIAAJgMARAAAMBkCIAAAAAm49LQEwCA+hIydVNDT+GafTtnSENPAcB1gBVAAAAAk2lSAXDx4sUKCQmRh4eHoqOjtWvXriv2X7dunbp27SoPDw+FhYXp/ffft7134cIFTZkyRWFhYWrevLkCAgI0atQonThxwtG7AQAA0KCaTABcu3atkpOTNWvWLOXm5io8PFzx8fEqLi6usv/27ds1YsQIjRkzRnv27FFCQoISEhK0b98+SdK5c+eUm5ur1NRU5ebm6p133tGBAwc0dOjQ+twtAACAemcxDMNo6EnURHR0tHr37q1FixZJkqxWqwIDAzV+/HhNnTr1sv6JiYkqKyvTxo0bbW233nqrIiIitHTp0iprfP755+rTp4+OHj2qoKCgq86ptLRUPj4+Kikpkbe3d43243q4BglAw+EaQODa1eXz+3rTJFYAz58/r5ycHMXFxdnanJycFBcXp+zs7Cq3yc7OrtRfkuLj46vtL0klJSWyWCxq0aKFXeYNAADQGDWJu4BPnTqliooK+fn5VWr38/PTN998U+U2hYWFVfYvLCyssv9PP/2kKVOmaMSIEdX+a6C8vFzl5eW2n0tLS2uzGwCAeuToMy6sxqIpaxIrgI524cIFPfDAAzIMQ0uWLKm2X1pamnx8fGyvwMDAepwlAACAfTSJAOjr6ytnZ2cVFRVVai8qKpK/v3+V2/j7+9eo/8Xwd/ToUWVmZl7xWoBp06appKTE9jp27Fgd9wgAAKDhNIkA6ObmpsjISGVlZdnarFarsrKyFBMTU+U2MTExlfpLUmZmZqX+F8PfoUOHtHnzZrVq1eqK83B3d5e3t3elFwAAQFPTJK4BlKTk5GQlJSUpKipKffr00YIFC1RWVqbRo0dLkkaNGqV27dopLS1NkjRx4kT1799f8+bN05AhQ7RmzRrt3r1by5Ytk/Rz+PvNb36j3Nxcbdy4URUVFbbrA1u2bCk3N7eG2VEAAAAHazIBMDExUSdPntTMmTNVWFioiIgIZWRk2G70yM/Pl5PT/y9oxsbGavXq1ZoxY4amT5+u0NBQpaenq0ePHpKk48ePa8OGDZKkiIiISrW2bNmiX/3qV/WyXwAAAPWtyTwHsDHiOYAA6ht3ntYcdwGjOjwHsIlcAwgAAAD7IQACAACYDAEQAADAZAiAAAAAJkMABAAAMBkCIAAAgMk0mecAAgAAVKc2j/2xlp9z4EyaBlYAAQAATIYACAAAYDKcAgYAwMTq4xuq+NaUxocVQAAAAJMhAAIAAJgMp4ABAJVwShC4/rECCAAAYDIEQAAAAJPhFDAANCH1cXoWwPWPFUAAAACTIQACAACYDKeAAQCoA+6WrjkuXWh8WAEEAAAwGQIgAACAyRAAAQAATIYACAAAYDIEQAAAAJMhAAIAAJgMARAAAMBkCIAAAAAmQwAEAAAwGQIgAACAyRAAAQAATIbvAgYA1Du+G7Zm+D3BUVgBBAAAMBkCIAAAgMkQAAEAAEyGAAgAAGAyBEAAAACTIQACAACYDAEQAADAZAiAAAAAJkMABAAAMBkCIAAAgMkQAAEAAEyGAAgAAGAyBEAAAACTIQACAACYDAEQAADAZAiAAAAAJkMABAAAMBkCIAAAgMkQAAEAAEyGAAgAAGAyTSoALl68WCEhIfLw8FB0dLR27dp1xf7r1q1T165d5eHhobCwML3//vuV3jcMQzNnzlTbtm3l6empuLg4HTp0yJG7AAAA0OCaTABcu3atkpOTNWvWLOXm5io8PFzx8fEqLi6usv/27ds1YsQIjRkzRnv27FFCQoISEhK0b98+W5+5c+fq5Zdf1tKlS7Vz5041b95c8fHx+umnn+prtwAAAOqdxTAMo6EnURPR0dHq3bu3Fi1aJEmyWq0KDAzU+PHjNXXq1Mv6JyYmqqysTBs3brS13XrrrYqIiNDSpUtlGIYCAgL0+OOP6w9/+IMkqaSkRH5+fnrjjTc0fPjwq86ptLRUPj4+Kikpkbe3d432I2Tqphr1AwAAjmEtP6djCx6o1ef39caloSdQE+fPn1dOTo6mTZtma3NyclJcXJyys7Or3CY7O1vJycmV2uLj45Weni5JOnLkiAoLCxUXF2d738fHR9HR0crOzq4yAJaXl6u8vNz2c0lJiaSfg2BNWcvP1bgvAACwv4ufxU1kDcwhmkQAPHXqlCoqKuTn51ep3c/PT998802V2xQWFlbZv7Cw0Pb+xbbq+lwqLS1Ns2fPvqw9MDCwZjsCAAAaje+++04+Pj4NPY0G0SQCYGMxbdq0SquKp0+fVnBwsPLz8x32H1BpaakCAwN17Ngxhy1TO7rG9bAP1Gg841OjcdW4HvaBGo1n/PqqUVJSoqCgILVs2dIh4zcFTSIA+vr6ytnZWUVFRZXai4qK5O/vX+U2/v7+V+x/8X+LiorUtm3bSn0iIiKqHNPd3V3u7u6Xtfv4+Dj8GgJvb+8mX+N62AdqNJ7xqdG4alwP+0CNxjN+fdVwcmoy98LaXZPYczc3N0VGRiorK8vWZrValZWVpZiYmCq3iYmJqdRfkjIzM239O3ToIH9//0p9SktLtXPnzmrHBAAAuB40iRVASUpOTlZSUpKioqLUp08fLViwQGVlZRo9erQkadSoUWrXrp3S0tIkSRMnTlT//v01b948DRkyRGvWrNHu3bu1bNkySZLFYtGkSZP09NNPKzQ0VB06dFBqaqoCAgKUkJDQULsJAADgcE0mACYmJurkyZOaOXOmCgsLFRERoYyMDNtNHPn5+ZWWcmNjY7V69WrNmDFD06dPV2hoqNLT09WjRw9bnyeeeEJlZWX6n//5H50+fVr9+vVTRkaGPDw8ajQnd3d3zZo1q8rTwvZyPdS4HvaBGo1nfGo0rhrXwz5Qo/GMfz3VaOyazHMAAQAAYB9N4hpAAAAA2A8BEAAAwGQIgAAAACZDAAQAADAZAmAdLV68WCEhIfLw8FB0dLR27dpl1/E//vhj3X333QoICJDFYrF9h7G9pKWlqXfv3vLy8lKbNm2UkJCgAwcO2LXGkiVL1LNnT9vDPGNiYvTBBx/Ytcal5syZY3vEj7089dRTslgslV5du3a12/iSdPz4cf32t79Vq1at5OnpqbCwMO3evdtu44eEhFy2DxaLRWPHjrVbjYqKCqWmpqpDhw7y9PTUTTfdpD/96U92/67NM2fOaNKkSQoODpanp6diY2P1+eef13m8qx1rhmFo5syZatu2rTw9PRUXF6dDhw7Zbfx33nlHAwcOVKtWrWSxWJSXl2fXfbhw4YKmTJmisLAwNW/eXAEBARo1apROnDhhtxrSz8dJ165d1bx5c914442Ki4vTzp077Vrjlx577DFZLBYtWLDArjUeeuihy46TQYMG2XUf9u/fr6FDh8rHx0fNmzdX7969lZ+fb7caVR3rFotFzz//vN1qnD17VuPGjVP79u3l6empbt26aenSpTUevyY1ioqK9NBDDykgIEDNmjXToEGDanXs1eRz7qefftLYsWPVqlUr3XDDDbrvvvsu+xKJ6xUBsA7Wrl2r5ORkzZo1S7m5uQoPD1d8fLyKi4vtVqOsrEzh4eFavHix3cb8pW3btmns2LHasWOHMjMzdeHCBQ0cOFBlZWV2q9G+fXvNmTNHOTk52r17t+68807dc889+uqrr+xW45c+//xzvfLKK+rZs6fdx+7evbsKCgpsr08//dRuY//www/q27evXF1d9cEHH+jrr7/WvHnzdOONN9qtxueff15p/pmZmZKk+++/3241nnvuOS1ZskSLFi3S/v379dxzz2nu3LlauHCh3WpI0iOPPKLMzEytXLlSe/fu1cCBAxUXF6fjx4/XabyrHWtz587Vyy+/rKVLl2rnzp1q3ry54uPj9dNPP9ll/LKyMvXr10/PPfdcneZ/tRrnzp1Tbm6uUlNTlZubq3feeUcHDhzQ0KFD7VZDkjp37qxFixZp7969+vTTTxUSEqKBAwfq5MmTdqtx0fr167Vjxw4FBATUah9qWmPQoEGVjpe33nrLbuMfPnxY/fr1U9euXbV161Z9+eWXSk1NrfHjx2pS45dzLygo0GuvvSaLxaL77rvPbjWSk5OVkZGhVatWaf/+/Zo0aZLGjRunDRs22KWGYRhKSEjQv/71L7377rvas2ePgoODFRcXV+PPqZp8zk2ePFnvvfee1q1bp23btunEiRO69957a7wPTZqBWuvTp48xduxY288VFRVGQECAkZaW5pB6koz169c7ZOyLiouLDUnGtm3bHFrnxhtvNF599VW7j3vmzBkjNDTUyMzMNPr3729MnDjRbmPPmjXLCA8Pt9t4l5oyZYrRr18/h41flYkTJxo33XSTYbVa7TbmkCFDjIcffrhS27333muMHDnSbjXOnTtnODs7Gxs3bqzU3qtXL+PJJ5+85vEvPdasVqvh7+9vPP/887a206dPG+7u7sZbb711zeP/0pEjRwxJxp49e2o9bk1rXLRr1y5DknH06FGH1SgpKTEkGZs3b7ZrjX//+99Gu3btjH379hnBwcHGiy++WKfxq6uRlJRk3HPPPXUe82rjJyYmGr/97W/tMn51NS51zz33GHfeeadda3Tv3t344x//WKntWo7DS2scOHDAkGTs27fP1lZRUWG0bt3aWL58eZ1qXPo5d/r0acPV1dVYt26drc/+/fsNSUZ2dnadajQlrADW0vnz55WTk6O4uDhbm5OTk+Li4pSdnd2AM7s2JSUlkuSwL8auqKjQmjVrVFZW5pCv2hs7dqyGDBlS6f8Xezp06JACAgLUsWNHjRw5slana65mw4YNioqK0v333682bdrolltu0fLly+02/qXOnz+vVatW6eGHH5bFYrHbuLGxscrKytLBgwclSV988YU+/fRTDR482G41/vOf/6iiouKy1RJPT0+7rspedOTIERUWFlb678rHx0fR0dFN/ni3WCxq0aKFQ8Y/f/68li1bJh8fH4WHh9ttXKvVqgcffFApKSnq3r273ca91NatW9WmTRt16dJFv/vd7/Tdd9/ZZVyr1apNmzapc+fOio+PV5s2bRQdHW33S3x+qaioSJs2bdKYMWPsOm5sbKw2bNig48ePyzAMbdmyRQcPHtTAgQPtMn55ebkkVTrWnZyc5O7uXudj/dLPuZycHF24cKHS8d21a1cFBQU16eO7pgiAtXTq1ClVVFTYvoHkIj8/PxUWFjbQrK6N1WrVpEmT1Ldv30rflGIPe/fu1Q033CB3d3c99thjWr9+vbp162bXGmvWrFFubq7tawDtLTo6Wm+88YYyMjK0ZMkSHTlyRLfddpvOnDljl/H/9a9/acmSJQoNDdWHH36o3/3ud5owYYL+8pe/2GX8S6Wnp+v06dN66KGH7Dru1KlTNXz4cHXt2lWurq665ZZbNGnSJI0cOdJuNby8vBQTE6M//elPOnHihCoqKrRq1SplZ2eroKDAbnUuunhMX0/H+08//aQpU6ZoxIgR8vb2tuvYGzdu1A033CAPDw+9+OKLyszMlK+vr93Gf+655+Ti4qIJEybYbcxLDRo0SG+++aaysrL03HPPadu2bRo8eLAqKiqueezi4mKdPXtWc+bM0aBBg/TRRx9p2LBhuvfee7Vt2zY7zP5yf/nLX+Tl5WX305oLFy5Ut27d1L59e7m5uWnQoEFavHixbr/9druMfzGITZs2TT/88IPOnz+v5557Tv/+97/rdKxX9TlXWFgoNze3y/4h1JSP79poMl8FB8cZO3as9u3b55AVlC5duigvL08lJSX629/+pqSkJG3bts1uIfDYsWOaOHGiMjMza3UNTW38cgWrZ8+eio6OVnBwsN5++227/KvaarUqKipKzz77rCTplltu0b59+7R06VIlJSVd8/iXWrFihQYPHlyn66eu5O2339Zf//pXrV69Wt27d1deXp4mTZqkgIAAu+7HypUr9fDDD6tdu3ZydnZWr169NGLECOXk5NitxvXqwoULeuCBB2QYhpYsWWL38e+44w7l5eXp1KlTWr58uR544AHt3LlTbdq0ueaxc3Jy9NJLLyk3N9euK9eXGj58uO3PYWFh6tmzp2666SZt3bpVAwYMuKaxrVarJOmee+7R5MmTJUkRERHavn27li5dqv79+1/T+FV57bXXNHLkSLv//bhw4ULt2LFDGzZsUHBwsD7++GONHTtWAQEBdjkT4+rqqnfeeUdjxoxRy5Yt5ezsrLi4OA0ePLhON5Y58nOuqWIFsJZ8fX3l7Ox82V1CRUVF8vf3b6BZ1d24ceO0ceNGbdmyRe3bt7f7+G5uburUqZMiIyOVlpam8PBwvfTSS3YbPycnR8XFxerVq5dcXFzk4uKibdu26eWXX5aLi4td/tV+qRYtWqhz58765z//aZfx2rZte1kgvvnmm+16mvmio0ePavPmzXrkkUfsPnZKSoptFTAsLEwPPvigJk+ebPeV2Ztuuknbtm3T2bNndezYMe3atUsXLlxQx44d7VpHku2Yvh6O94vh7+jRo8rMzLT76p8kNW/eXJ06ddKtt96qFStWyMXFRStWrLDL2J988omKi4sVFBRkO9aPHj2qxx9/XCEhIXapUZWOHTvK19fXLse7r6+vXFxc6u14/+STT3TgwAG7H+8//vijpk+frvnz5+vuu+9Wz549NW7cOCUmJuqFF16wW53IyEjl5eXp9OnTKigoUEZGhr777rtaH+vVfc75+/vr/PnzOn36dKX+TfH4rgsCYC25ubkpMjJSWVlZtjar1aqsrCyHXNvmKIZhaNy4cVq/fr3+8Y9/qEOHDvVS12q12q7tsIcBAwZo7969ysvLs72ioqI0cuRI5eXlydnZ2W61Ljp79qwOHz6stm3b2mW8vn37XvZogoMHDyo4ONgu4//S66+/rjZt2mjIkCF2H/vcuXNycqr8V4qzs7Nt1cPemjdvrrZt2+qHH37Qhx9+qHvuucfuNTp06CB/f/9Kx3tpaal27tzZpI73i+Hv0KFD2rx5s1q1alUvde15vD/44IP68ssvKx3rAQEBSklJ0YcffmiXGlX597//re+++84ux7ubm5t69+5db8f7ihUrFBkZadfrMKWf/3u6cOFCvR3vPj4+at26tQ4dOqTdu3fX+Fi/2udcZGSkXF1dKx3fBw4cUH5+fpM6vuuKU8B1kJycrKSkJEVFRalPnz5asGCBysrKNHr0aLvVOHv2bKV/cR45ckR5eXlq2bKlgoKCrnn8sWPHavXq1Xr33Xfl5eVlu97Bx8dHnp6e1zy+JE2bNk2DBw9WUFCQzpw5o9WrV2vr1q12/cvay8vrsusWmzdvrlatWtntesY//OEPuvvuuxUcHKwTJ05o1qxZcnZ21ogRI+wy/uTJkxUbG6tnn31WDzzwgHbt2qVly5Zp2bJldhn/IqvVqtdff11JSUlycbH/oX/33XfrmWeeUVBQkLp37649e/Zo/vz5evjhh+1a58MPP5RhGOrSpYv++c9/KiUlRV27dq3z8Xe1Y23SpEl6+umnFRoaqg4dOig1NVUBAQFKSEiwy/jff/+98vPzbc/luxgO/P39a7wKcaUabdu21W9+8xvl5uZq48aNqqiosB3vLVu2lJub2zXXaNWqlZ555hkNHTpUbdu21alTp7R48WIdP368Vo8autrv6tLg6urqKn9/f3Xp0sUuNVq2bKnZs2frvvvuk7+/vw4fPqwnnnhCnTp1Unx8vF32ISUlRYmJibr99tt1xx13KCMjQ++99562bt1ql324+PlQWlqqdevWad68eTUetzY1+vfvr5SUFHl6eio4OFjbtm3Tm2++qfnz59utxrp169S6dWsFBQVp7969mjhxohISEmp8o8nVPud8fHw0ZswYJScnq2XLlvL29tb48eMVExOjW2+9tcb70WQ15C3ITdnChQuNoKAgw83NzejTp4+xY8cOu46/ZcsWQ9Jlr6SkJLuMX9XYkozXX3/dLuMbhmE8/PDDRnBwsOHm5ma0bt3aGDBggPHRRx/Zbfzq2PsxMImJiUbbtm0NNzc3o127dkZiYqLxz3/+027jG4ZhvPfee0aPHj0Md3d3o2vXrsayZcvsOr5hGMaHH35oSDIOHDhg97ENwzBKS0uNiRMnGkFBQYaHh4fRsWNH48knnzTKy8vtWmft2rVGx44dDTc3N8Pf398YO3ascfr06TqPd7VjzWq1GqmpqYafn5/h7u5uDBgwoFa/w6uN//rrr1f5/qxZs+xS4+LjZap6bdmyxS41fvzxR2PYsGFGQECA4ebmZrRt29YYOnSosWvXrhqPX5Pf1aXq8hiYK9U4d+6cMXDgQKN169aGq6urERwcbDz66KNGYWGhXfdhxYoVRqdOnQwPDw8jPDzcSE9Pt9s+XPTKK68Ynp6edT42rlajoKDAeOihh4yAgADDw8PD6NKlizFv3rxaPVrqajVeeuklo3379oarq6sRFBRkzJgxo1Z/n9Tkc+7HH380fv/73xs33nij0axZM2PYsGFGQUFBjWs0ZRbDsPNj+gEAANCocQ0gAACAyRAAAQAATIYACAAAYDIEQAAAAJMhAAIAAJgMARAAAMBkCIAAAAAmQwAE0GT86le/0qRJk2w/h4SEaMGCBVfcxmKxKD093aHzakxq8jsBAL4KDkCT9fnnn6t58+Z2HfOpp55Senq68vLy7DouADQmBEAATVbr1q0begoA0CRxChhArVmtVs2dO1edOnWSu7u7goKC9Mwzz9jenzJlijp37qxmzZqpY8eOSk1N1YULF2zvP/XUU4qIiNDKlSsVEhIiHx8fDR8+XGfOnLH1KSsr06hRo3TDDTeobdu2VX6p/aWnOw8dOqTbb79dHh4e6tatmzIzMy/b5kpze+ONNzR79mx98cUXslgsslgseuONNyRJp0+f1iOPPKLWrVvL29tbd955p7744otqf0fffvutLBaL3n77bd12223y9PRU7969dfDgQX3++eeKiorSDTfcoMGDB+vkyZO27T7//HP9+te/lq+vr3x8fNS/f3/l5uba3jcMQ0899ZSCgoLk7u6ugIAATZgwodp5vPrqq2rRooWysrIkSX/7298UFhYmT09PtWrVSnFxcSorK6t2ewDXJ1YAAdTatGnTtHz5cr344ovq16+fCgoK9M0339je9/Ly0htvvKGAgADt3btXjz76qLy8vPTEE0/Y+hw+fFjp6enauHGjfvjhBz3wwAOaM2eOLUimpKRo27Ztevfdd9WmTRtNnz5dubm5ioiIqHJOVqtV9957r/z8/LRz506VlJRUul6wJnNLTEzUvn37lJGRoc2bN0uSfHx8JEn333+/PD099cEHH8jHx0evvPKKBgwYoIMHD6ply5bV/q5mzZqlBQsWKCgoSA8//LD++7//W15eXnrppZfUrFkzPfDAA5o5c6aWLFkiSTpz5oySkpK0cOFCGYahefPm6a677tKhQ4fk5eWlv//973rxxRe1Zs0ade/eXYWFhdUG0blz52ru3Ln66KOP1KdPHxUUFGjEiBGaO3euhg0bpjNnzuiTTz4RXwkPmJABALVQWlpquLu7G8uXL6/xNs8//7wRGRlp+3nWrFlGs2bNjNLSUltbSkqKER0dbRiGYZw5c8Zwc3Mz3n77bdv73333neHp6WlMnDjR1hYcHGy8+OKLhmEYxocffmi4uLgYx48ft73/wQcfGJKM9evX12pu4eHhlfp88sknhre3t/HTTz9Var/pppuMV155pcpxjxw5YkgyXn31VVvbW2+9ZUgysrKybG1paWlGly5dqp1fRUWF4eXlZbz33nuGYRjGvHnzjM6dOxvnz5+vsv/F38kTTzxhtG3b1ti3b5/tvZycHEOS8e2331ZbD4A5sAIIoFb279+v8vJyDRgwoNo+a9eu1csvv6zDhw/r7Nmz+s9//iNvb+9KfUJCQuTl5WX7uW3btiouLpb08+rg+fPnFR0dbXu/ZcuW6tKlyxXnFRgYqICAAFtbTExMneZ2qS+++EJnz55Vq1atKrX/+OOPOnz48BW37dmzp+3Pfn5+kqSwsLBKbRf3W5KKioo0Y8YMbd26VcXFxaqoqNC5c+eUn58v6eeVyAULFqhjx44aNGiQ7rrrLt19991ycfn/v87nzZunsrIy7d69Wx07drS1h4eHa8CAAQoLC1N8fLwGDhyo3/zmN7rxxhuvuA8Arj9cAwigVjw9Pa/4fnZ2tkaOHKm77rpLGzdu1J49e/Tkk0/q/Pnzlfq5urpW+tlischqtdp9vnWZ26XOnj2rtm3bKi8vr9LrwIEDSklJueK2v9xPi8VSZdsv9zspKUl5eXl66aWXtH37duXl5alVq1a2OQYGBurAgQP685//LE9PT/3+97/X7bffXukay9tuu00VFRV6++23K83F2dlZmZmZ+uCDD9StWzctXLhQXbp00ZEjR67ymwNwvSEAAqiV0NBQeXp62m4quNT27dsVHBysJ598UlFRUQoNDdXRo0drVeOmm26Sq6urdu7caWv74YcfdPDgwWq3ufnmm3Xs2DEVFBTY2nbs2FHrubm5uamioqJSW69evVRYWCgXFxd16tSp0svX17dW+3Y1n332mSZMmKC77rpL3bt3l7u7u06dOlWpj6enp+6++269/PLL2rp1q7Kzs7V3717b+3369NEHH3ygZ599Vi+88EKlbS0Wi/r27avZs2drz549cnNz0/r16+26DwAaP04BA6gVDw8PTZkyRU888YTc3NzUt29fnTx5Ul999ZXGjBmj0NBQ5efna82aNerdu7c2bdpU64Bxww03aMyYMUpJSVGrVq3Upk0bPfnkk3Jyqv7frHFxcercubOSkpL0/PPPq7S0VE8++WSlPjWZW0hIiI4cOaK8vDy1b99eXl5eiouLU0xMjBISEjR37lx17txZJ06c0KZNmzRs2DBFRUXVav+uJDQ0VCtXrlRUVJRKS0uVkpJSadX1jTfeUEVFhaKjo9WsWTOtWrVKnp6eCg4OrjRObGys3n//fQ0ePFguLi6aNGmSdu7cqaysLA0cOFBt2rTRzp07dfLkSd188812mz+ApoEVQAC1lpqaqscff1wzZ87UzTffrMTERNt1bEOHDtXkyZM1btw4RUREaPv27UpNTa11jeeff1633Xab7r77bsXFxalfv36KjIystr+Tk5PWr1+vH3/8UX369NEjjzxS6dE0NZ3bfffdp0GDBumOO+5Q69at9dZbb8lisej999/X7bffrtGjR6tz584aPny4jh49aruuz15WrFihH374Qb169dKDDz6oCRMmqE2bNrb3W7RooeXLl6tv377q2bOnNm/erPfee++y6xMlqV+/ftq0aZNmzJihhQsXytvbWx9//LHuuusude7cWTNmzNC8efM0ePBgu+4DgMbPYhjc/w8AAGAmrAACAACYDAEQAADAZAiAAAAAJkMABAAAMBkCIAAAgMkQAAEAAEyGAAgAAGAyBEAAAACTIQACAACYDAEQAADAZAiAAAAAJkMABAAAMJn/A1cXgnhL9+q3AAAAAElFTkSuQmCC",
      "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": 238,
   "id": "9f22ca9d-bdc2-4ea5-b2bc-249a256bb8ad",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "c64139ea7ed94a1ea0e153e7728d34eb",
       "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()"
   ]
  }
 ],
 "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
}