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logos-blockchain-pocs/Stwo_wrapper/verifier_script.ipynb
thomaslavaur 5c7ff0fad5 Last update
2024-09-25 14:14:48 +02:00

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{
"cells": [
{
"cell_type": "code",
"execution_count": 1,
"id": "43f7fdc0",
"metadata": {},
"outputs": [],
"source": [
"import json\n",
"\n",
"with open(\"crates/prover/proof.json\", \"r\") as f:\n",
" proof = json.load(f)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "88f232b5",
"metadata": {},
"outputs": [],
"source": []
},
{
"cell_type": "code",
"execution_count": 2,
"id": "12ef3767",
"metadata": {},
"outputs": [],
"source": [
"F = FiniteField(52435875175126190479447740508185965837690552500527637822603658699938581184513)\n",
"\n",
"poseidon_comp_consts = [50207570499218320245539736680169582180207201335688461025883902752909290481781,\n",
"24448666467656506447555018649749346340705294023832615387641453784702583464707,\n",
"34092944507611308604157957266676007619644244199372265837364557849561670729974,\n",
"46954129210702959446093971191783182601726081775951103310666314834569091037713,\n",
"38612156878839717097806285947575477749087608521505464809942918879152074545066,\n",
"19752610610343814834081989345964253902282700341539483876504601969121084774539,\n",
"46567545048462867923299713424766325689670511126407629551256255807498976196546,\n",
"9520793415506326549109545537894287560752519598132096386048093015534488804808,\n",
"22814234098357034097599682726494820560934925862581927123816510593532324971186,\n",
"3277621627834606517208177071759088097855048183641615082769528872043050020787,\n",
"29230456498980145088774069819561206654397510279226264474986155631775387918911,\n",
"19087113294497892618475669593723876605785307026981218038380435259594863105240,\n",
"39932371919358015185769877859035474336011770016475087638554815294278664040916,\n",
"17645770319151120318035258350885823104235488352935695302274836429012504407725,\n",
"17990728141399065004015538797609951295983853332644474801890158217822768128628,\n",
"12607949331462269429981198199999740921418125994747028428126661151190418292729,\n",
"33067617079394435172767143524489677593390850035349407507374659268468278200906,\n",
"10025233623562179533044093426455032352895184661359005809314430689113735312874,\n",
"20398677688057466110325934731430812468657996794663167456321709689030080949228,\n",
"32085671199853825909918260218834827339732598508827083525700252644622592932757,\n",
"36451986593067827349794003109666944974266236856145879921902940325507228739480,\n",
"51835224419566813714481533481210630888564327175625175437244377303858990291964,\n",
"1944662263588038198375346521900053780907777056656211622999059135594196413076,\n",
"12995068374816903282074967132431954020410301768622808407703775963080983755183,\n",
"13278128079226679628648689279705910775020794457648431336050464485837924986341,\n",
"39207195481789228835625472428521288347432218258431761869689775532020546099642,\n",
"21081768833381902942114733002158882075348844281359283013642620389621494952015,\n",
"20751788049060260683191405008569080723662271828149227137187075968560831545739,\n",
"20820291785607398388900832350860967875629907105847554413318238165275470374689,\n",
"6971878585215744613467847324629115462668098071102846520957717612260531709386,\n",
"42421164250058173810994728364144776180689735894673627964404703973460802099146,\n",
"32890116643831560295329417521056875595733120141391587236744387068135440602102,\n",
"42670005614507618780436482775021159957307712089310941922452133588875084445464,\n",
"21120353743307986506720883740380468652053382764895882204680310593048134053982,\n",
"7853308243263055176258751393326645428041138029306706980470113526802326214700,\n",
"17545076036297840030021082424260289805456380863517895917265467158332801090765,\n",
"29526223376722400691172584788126610514669516909826971155598997488361793726636,\n",
"48421712782536172546302502401679048379568171245541707202282458591545347755349,\n",
"10740853637774754893036062076749871837371049036966225040269105665447180116170,\n",
"34042041521558704677804677569712674569738576001717295340556848855085089618161,\n",
"24290796201833228559129233924595614281891670608675107544294264860003803501509,\n",
"26722678647461522072509896114724736555938247563993442152746954157222882824350,\n",
"20252491387019425681551488261397157776479297799360691728406809731508542196845,\n",
"50322025264206689090790987370440439179141270613911973034521438238687587958097,\n",
"17070806525931584028449131949070191143344166668070820337429561524629464200550,\n",
"25856554324149146992239414502939942208580094928192925471532421030223074525051,\n",
"17714998974036855356530338446243137421735047395517260588250413348153258772076,\n",
"44833315250334176776685835079382312848180252180173884969157994737319426976437,\n",
"35603718839327251012037553292043899153393807438387129505923567878785822738162,\n",
"20515196301761603016197694845695272699608637099106794944737311528118558777570,\n",
"10100400556460905874275078234698187530913105549037797180493988678937053918124,\n",
"29943022708270799252522211109308629054849337552699067311814388215768905671554,\n",
"33400164627534996188947689774080657908147988421361870074239537729877153299092,\n",
"45574161704098228712016716221086232277248798839906622903502141601878895917316,\n",
"40623265267364613450776577487319920007897396936924051398790906883872334022964,\n",
"37929176440858430683261948300797278761072096845318183419284347376614069989808,\n",
"12242010394227909997626655999345208835040087302065045201635069094289920778463,\n",
"38947272924417356803622776795797899233194116520680026665045628837194239730633,\n",
"6838505804652359252670794375725267665530548946030641535297433541475260948424,\n",
"21345718918993308853491352363460625447157796362108157527364130872100101143328,\n",
"26397988737034501095129796920971941795766209722106383463197090306632188634870,\n",
"47092791129593573928369881528796435131623991381197863072979392492232678100884,\n",
"36850972241154890671857874025605504779963735054128436776319531005864791472123,\n",
"27893799443241349360688137159923920340185830261519093384488134540544971987330,\n",
"34031071010517479317003393843135868322188010660871691856659878788331169912272,\n",
"3102550735908358465878301372253437950829524988677083749179431098369388780259,\n",
"2963742902601529003553690631564645593518709846059084207036841793643477514707,\n",
"34538583661636382515652368664945657625216404085453317149263146639486246251503,\n",
"49179786922858759927440465310900376749726765337268308911471491527044937447403,\n",
"31668552784983283483593666924944066737680315058069542500069213700768949573692,\n",
"47303630019147536941220901582952982856517915740884282232588733470564849742080,\n",
"41561182787858915334837446901194440640033856888621022207410120224293681204923,\n",
"40208795410444394963490428737133513683110766973508056822474493355065333491217,\n",
"24620569969402072776192280888011017497854992833864712509770555543278833718751,\n",
"31418811028946653724823259636547682581071379929451162101915628592655152015310,\n",
"25964807298150242099204032696543021731332498792173212422070959505270506288817,\n",
"31766013031271106581980804902159064978010553325475976472264348555438361464655,\n",
"15107529391758643095716794813038523751713309080738989300826699946985294497278,\n",
"26149402682269665088314773514719203730233986608723938665192802061570851149320,\n",
"35053126320072620250684851851709987160095640397875384355477447570643983599564]\n",
"\n",
"def mix(state):\n",
" state[0] = state[0] + state[1] + state[2];\n",
" state[1] = state[0] + state[1];\n",
" state[2] = state[0] + state[2];\n",
" return state\n",
"\n",
"def round_comp(state, idx, full):\n",
" if full:\n",
" state[0] += poseidon_comp_consts[idx]\n",
" state[1] += poseidon_comp_consts[idx + 1]\n",
" state[2] += poseidon_comp_consts[idx + 2]\n",
" # Optimize multiplication\n",
" state[0] = state[0] * state[0] * state[0] * state[0] * state[0]\n",
" state[1] = state[1] * state[1] * state[1] * state[1] * state[1]\n",
" state[2] = state[2] * state[2] * state[2] * state[2] * state[2]\n",
" else:\n",
" state[0] += poseidon_comp_consts[idx]\n",
" state[2] = state[2] * state[2] * state[2] * state[2] * state[2]\n",
" state = mix(state)\n",
" return state\n",
"\n",
"def poseidon_permute_comp_bls(state):\n",
" idx = 0;\n",
" state = mix(state);\n",
"\n",
" # Full rounds\n",
" for i in range(4):\n",
" state = round_comp(state, idx, true);\n",
" idx += 3;\n",
"\n",
" # Partial rounds\n",
" for i in range(56):\n",
" state = round_comp(state, idx, false);\n",
" idx += 1;\n",
"\n",
" # Full rounds\n",
" for i in range(4):\n",
" state= round_comp(state, idx, true);\n",
" idx += 3;\n",
" return state\n",
"\n",
" \n",
"def poseidon_hash_bls(x, y):\n",
" state = [F(x), F(y), F(0)];\n",
" poseidon_permute_comp_bls(state);\n",
" return state[0] + F(x)\n",
"\n",
"\n",
"def poseidon_hash_many_bls(msgs):\n",
" state = [F(0), F(0), F(0)];\n",
" for i in range(len(msgs)//2):\n",
" state[0] += F(msgs[2*i])\n",
" state[1] += F(msgs[2*i + 1])\n",
" state = poseidon_permute_comp_bls(state)\n",
" if len(msgs) % 2 == 1:\n",
" state[0] += F(msgs[len(msgs)-1])\n",
" state[len(msgs)%2] += F(1);\n",
" state = poseidon_permute_comp_bls(state)\n",
" return state[0]"
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "67a09953",
"metadata": {},
"outputs": [],
"source": [
"def draw_felt252(digest, n_sent):\n",
" res = poseidon_hash_bls(digest, F(n_sent));\n",
" return res\n",
"\n",
"def draw_base_felts(digest, n_sent):\n",
" shift = 1 << 31\n",
" cur = int(draw_felt252(digest, n_sent))\n",
" n_sent += 1\n",
" u32s = []\n",
" quotient = 1\n",
" while(quotient != 0):\n",
" quotient = cur // shift\n",
" if quotient != 0:\n",
" remainder = cur % shift\n",
" cur = quotient\n",
" u32s.append(M31(remainder))\n",
" return u32s\n",
"\n",
"def draw_random_bytes(digest, n_sent):\n",
" shift = 1 << 8\n",
" cur = int(draw_felt252(digest, n_sent))\n",
" byte = []\n",
" for i in range(31):\n",
" quotient = cur // shift\n",
" remainder = cur % shift\n",
" cur = quotient\n",
" byte.append(remainder)\n",
" return byte\n",
"\n",
"def hash_node(has_children, left, right, column_values):\n",
" n_column_blocks = ceil(len(column_values) / 8.0);\n",
" values = []\n",
" if has_children:\n",
" values.append(left)\n",
" values.append(right)\n",
" padding_length = 8 * n_column_blocks - len(column_values)\n",
" padded_values = column_values + [F(0) for i in range(padding_length)]\n",
" for i in range(int(len(padded_values) / 8)):\n",
" word = F(0)\n",
" for j in range(8):\n",
" word = word * F(2**31) + F(padded_values[i*8+j])\n",
" values.append(word)\n",
" return poseidon_hash_many_bls(values)"
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "1bce81d8",
"metadata": {},
"outputs": [],
"source": [
"M31 = FiniteField(2**31 - 1)\n",
"Poly_1.<x> = PolynomialRing(M31)\n",
"poly_1 = x**2+1\n",
"CM31.<i> = M31.extension(poly_1)\n",
"CM31\n",
"\n",
"Poly_2.<y> = PolynomialRing(CM31)\n",
"poly_2 = y**2 - 2 - i\n",
"QM31.<u> = CM31.extension(poly_2)"
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "16aa579c",
"metadata": {},
"outputs": [],
"source": [
"# A class to represent points on a circle using fields.\n",
"class CirclePoint:\n",
" def __init__(self, x, y):\n",
" self.x = x\n",
" self.y = y\n",
"\n",
" @staticmethod\n",
" def zero():\n",
" \"\"\"Returns the identity element of the circle.\"\"\"\n",
" return CirclePoint(1, 0)\n",
"\n",
" def double(self):\n",
" \"\"\"Returns the point after doubling.\"\"\"\n",
" return self + self\n",
"\n",
" @staticmethod\n",
" def double_x(x):\n",
" \"\"\"Applies the x-coordinate doubling map.\"\"\"\n",
" sx = x**2\n",
" return sx + sx - 1\n",
"\n",
" def log_order(self):\n",
" \"\"\"Returns the log order of the point (as a power of 2).\"\"\"\n",
" res = 0\n",
" cur = self.x\n",
" while cur != 1:\n",
" cur = self.double_x(cur)\n",
" res += 1\n",
" return res\n",
"\n",
" def mul(self, scalar):\n",
" \"\"\"Multiplies the point by a scalar.\"\"\"\n",
" res = CirclePoint.zero()\n",
" cur = self\n",
" while scalar > 0:\n",
" if scalar & 1 == 1:\n",
" res = res + cur\n",
" cur = cur + cur\n",
" scalar = int(scalar / 2)\n",
" return res\n",
"\n",
" def repeated_double(self, n):\n",
" \"\"\"Returns the point after repeated doubling.\"\"\"\n",
" res = self\n",
" for _ in range(n):\n",
" res = res.double()\n",
" return res\n",
"\n",
" def conjugate(self):\n",
" \"\"\"Returns the conjugate of the point.\"\"\"\n",
" return CirclePoint(self.x, -self.y)\n",
"\n",
" def antipode(self):\n",
" \"\"\"Returns the antipode of the point.\"\"\"\n",
" return CirclePoint(-self.x, -self.y)\n",
"\n",
" def mul_signed(self, off):\n",
" \"\"\"Multiplies the point by a signed scalar.\"\"\"\n",
" if off > 0:\n",
" return self.mul(off)\n",
" else:\n",
" return self.conjugate().mul(-off)\n",
"\n",
" def __add__(self, other):\n",
" \"\"\"Adds two circle points.\"\"\"\n",
" return CirclePoint(self.x * other.x - self.y * other.y, self.x * other.y + self.y * other.x)\n",
"\n",
" def __repr__(self):\n",
" return f\"CirclePoint(x={self.x}, y={self.y})\""
]
},
{
"cell_type": "code",
"execution_count": 9,
"id": "d9371b02",
"metadata": {},
"outputs": [],
"source": [
"# First Fiat-Shamir\n",
"digest = poseidon_hash_bls(0,proof[\"commitments\"][0])\n",
"\n",
"# Draw first random coeff\n",
"random_coeff = QM31([draw_base_felts(digest,0)[:2],draw_base_felts(digest,0)[2:4]])\n",
"\n",
"# Second Fiat-Shamir\n",
"digest = poseidon_hash_bls(digest,proof[\"commitments\"][1])\n",
"\n",
"# Draw OODS sample point\n",
"t = QM31([draw_base_felts(digest,0)[:2],draw_base_felts(digest,0)[2:4]])\n",
"t_square = t**2\n",
"one_plus_tsquared_inv = (t_square + QM31(1)) ** (-1)\n",
"x = (1 - t_square) * one_plus_tsquared_inv\n",
"y = 2*t*one_plus_tsquared_inv\n",
"oods_point = CirclePoint(x,y)\n",
"\n",
"# Derive sample points for now offset is always 0 so sampled points are all equals oods_point\n",
"\n",
"# Load sampled_values_1 and verify that they are in QM31\n",
"\n",
"# Then we verify the composition polynomial evaluation:\n",
"point = oods_point\n",
"mask_values = proof[\"sampled_values_0\"] + proof[\"sampled_values_1\"]\n",
"\n",
"\n",
"\n",
"\n",
"# Evaluate random point on the vanishing polynomial of the coset\n",
"evaluation = point.x\n",
"for i in range(5):\n",
" evaluation = CirclePoint.double_x(evaluation)\n",
"evaluation_inverse = evaluation ** (-1)\n",
"\n",
"# Compute evaluation using the mask values\n",
"accumulator = QM31(0)\n",
"a = QM31([mask_values[0][:2],mask_values[0][2:4]])\n",
"b = QM31([mask_values[1][:2],mask_values[1][2:4]])\n",
"for i in range(len(proof[\"sampled_values_0\"])-2):\n",
" c = QM31([mask_values[2+i][:2],mask_values[2+i][2:4]])\n",
" accumulator = (c - (a**2 + b**2))*evaluation_inverse + accumulator * random_coeff\n",
" a = b\n",
" b = c\n",
"\n",
"expected_value = QM31([proof[\"sampled_values_1\"][0][:2],proof[\"sampled_values_1\"][0][2:4]]) + QM31([proof[\"sampled_values_1\"][1][:2],proof[\"sampled_values_1\"][1][2:4]]) * QM31([[0,1],[0,0]]) + QM31([proof[\"sampled_values_1\"][2][:2],proof[\"sampled_values_1\"][2][2:4]]) * QM31([[0,0],[1,0]]) + QM31([proof[\"sampled_values_1\"][3][:2],proof[\"sampled_values_1\"][3][2:4]]) * QM31([[0,0],[0,1]])\n",
"\n",
"assert(expected_value == accumulator)\n",
"\n",
"\n",
"####FRI part\n",
"# Fiat-Shamir\n",
"shift = F(1 << 31)\n",
"res = [digest]\n",
"for i in range(len(mask_values) / 2):\n",
" cur = F(0)\n",
" for j in range(2):\n",
" for k in range(4):\n",
" cur = cur * shift + F(mask_values[2*i+j][k])\n",
" res.append(cur)\n",
"digest = poseidon_hash_many_bls(res)\n",
"random_coeff = QM31([draw_base_felts(digest,0)[:2],draw_base_felts(digest,0)[2:4]])\n",
"\n",
"# Verify commitment stage\n",
"circle_poly_alpha = QM31([draw_base_felts(digest,1)[:2],draw_base_felts(digest,1)[2:4]])\n",
"folding_alpha = []\n",
"for i in range(6):\n",
" digest = poseidon_hash_bls(digest,proof[\"inner_commitment_\"+str(i)])\n",
" folding_alpha.append(QM31([draw_base_felts(digest,0)[:2],draw_base_felts(digest,0)[2:4]]))\n",
"last_layer_poly = QM31([proof[\"coeffs\"][:2],proof[\"coeffs\"][2:4]])\n",
"\n",
"#Check proof of work\n",
"res = [digest]\n",
"cur = F(0)\n",
"for i in range(4):\n",
" cur = cur * shift + F(proof[\"coeffs\"][i])\n",
"res.append(cur)\n",
"digest = poseidon_hash_many_bls(res)\n",
"digest = poseidon_hash_bls(digest, proof[\"proof of work\"])\n",
"# The first 5 bits following the first 1 must be 0 for proof of work\n",
"for i in range(5):\n",
" assert(bin(digest)[3+i] == '0')\n",
"\n",
"# Compute openings positions\n",
"querries = []\n",
"max_querry = (1<<8)-1\n",
"random_bytes = draw_random_bytes(digest,0)\n",
"# The number of required querries is 3 so we need 3*8 bytes wich is enough with the 31 bytes of random_bytes\n",
"for i in range(3):\n",
" querry_bits = int().from_bytes(random_bytes[i*4:i*4+4],byteorder=\"little\")\n",
" querries.append(querry_bits & max_querry)\n",
"positions = []\n",
"# For each collumn\n",
"for i in range(2):\n",
" # For each querry\n",
" for j in range(3):\n",
" positions.append(querries[j] >> (1+i))\n",
"positions_sav = positions.copy()\n",
"\n",
"\n",
"#Verify decommitments (this part must be simplified repeting duplicated inputs)\n",
"# First tree\n",
"nodes = {}\n",
"for i in range(6):\n",
" leaf = []\n",
" for j in range(100):\n",
" leaf.append(M31(proof[\"queried_values_0\"][j][i]))\n",
" nodes[\"level_0_node_\"+str(positions[i//2] // 2 * 2 + i % 2) ] = hash_node(false,0,0,leaf)\n",
"counter = 0\n",
"for level in range(7):\n",
" for i in range(3):\n",
" if \"level_\"+str(level)+\"_node_\"+str(positions[len(positions)-3] * 2) in nodes:\n",
" lhs = nodes[\"level_\"+str(level)+\"_node_\"+str(positions[len(positions)-3] * 2)]\n",
" else:\n",
" lhs = F(proof[\"decommitment_0\"][counter])\n",
" counter += 1\n",
" if \"level_\"+str(level)+\"_node_\"+str(positions[len(positions) - 3] * 2 + 1) in nodes:\n",
" rhs = nodes[\"level_\"+str(level)+\"_node_\"+str(positions[len(positions) - 3] * 2 + 1)]\n",
" else:\n",
" rhs = F(proof[\"decommitment_0\"][counter])\n",
" counter += 1\n",
" nodes[\"level_\"+str(level+1)+\"_node_\"+str(positions[len(positions)-3])] = hash_node(true,lhs,rhs,[])\n",
" positions.append(positions[len(positions)-3] // 2)\n",
"assert(nodes[\"level_7_node_0\"] == F(proof[\"commitments\"][0]))\n",
"\n",
"#Second tree\n",
"positions = positions_sav\n",
"for i in range(len(positions)):\n",
" positions[i] = positions[i] * 2\n",
"nodes = {}\n",
"for i in range(6):\n",
" leaf = []\n",
" for j in range(4):\n",
" leaf.append(M31(proof[\"queried_values_1\"][j][i]))\n",
" nodes[\"level_0_node_\"+str(positions[i//2] // 2 * 2 + i % 2) ] = hash_node(false,0,0,leaf)\n",
"counter = 0\n",
"for level in range(8):\n",
" for i in range(3):\n",
" if \"level_\"+str(level)+\"_node_\"+str(positions[len(positions)-3] * 2) in nodes:\n",
" lhs = nodes[\"level_\"+str(level)+\"_node_\"+str(positions[len(positions)-3] * 2)]\n",
" else:\n",
" lhs = F(proof[\"decommitment_1\"][counter])\n",
" counter += 1\n",
" if \"level_\"+str(level)+\"_node_\"+str(positions[len(positions) - 3] * 2 + 1) in nodes:\n",
" rhs = nodes[\"level_\"+str(level)+\"_node_\"+str(positions[len(positions) - 3] * 2 + 1)]\n",
" else:\n",
" rhs = F(proof[\"decommitment_1\"][counter])\n",
" counter += 1\n",
" nodes[\"level_\"+str(level+1)+\"_node_\"+str(positions[len(positions)-3])] = hash_node(true,lhs,rhs,[])\n",
" if level == 0:\n",
" nodes[\"level_1_node_\"+str(positions[len(positions)-3] + 1)] = hash_node(true,nodes[\"level_0_node_\"+str((positions[len(positions)-3] + 1)* 2)],nodes[\"level_0_node_\"+str((positions[len(positions)-3] + 1)* 2 + 1)],[])\n",
" positions.append(positions[len(positions)-3] // 2)\n",
"assert(nodes[\"level_8_node_0\"] == F(proof[\"commitments\"][1]))\n",
"\n",
"\n",
"# Check querries answer we have 104 samples each equal to x = oods_point and f(x) = sampled_value_i\n",
"# For each column:\n",
"for i in range(2):\n",
" evaluations = []\n",
" queried_values_per_column = proof[\"queried_values_\"+str(i)]"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "56e4c016",
"metadata": {},
"outputs": [],
"source": []
},
{
"cell_type": "code",
"execution_count": null,
"id": "6b89599c",
"metadata": {},
"outputs": [],
"source": []
},
{
"cell_type": "code",
"execution_count": 10,
"id": "e486cb3a",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[['316772341',\n",
" '1526280133',\n",
" '663010112',\n",
" '224983897',\n",
" '510598760',\n",
" '1109503351'],\n",
" ['754832207',\n",
" '435790299',\n",
" '883623752',\n",
" '553207508',\n",
" '154784232',\n",
" '199176676'],\n",
" ['689603315',\n",
" '1763523007',\n",
" '1720552945',\n",
" '1983603154',\n",
" '367841669',\n",
" '319325418'],\n",
" ['1290247052',\n",
" '1120744584',\n",
" '193500372',\n",
" '294491115',\n",
" '951360807',\n",
" '891034447']]"
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"queried_values_per_column"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "85dd52b3",
"metadata": {},
"outputs": [],
"source": []
}
],
"metadata": {
"kernelspec": {
"display_name": "SageMath 9.5",
"language": "sage",
"name": "sagemath"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.10.12"
}
},
"nbformat": 4,
"nbformat_minor": 5
}
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