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LICENSE IN LICENSE","homepage":"https://github.com/Yaffle/bigint-gcd#readme","keywords":["gcd","Euclid","Lehmer","Subquadratic","greatest common divisor","BigInt"],"repository":{"type":"git","url":"git+https://github.com/Yaffle/bigint-gcd.git"},"description":"greater common divisor (gcd) of two BigInt values using Lehmer's GCD algorithm","maintainers":[{"name":"yaffle","email":"vic99999@yandex.ru"}],"readme":"# bigint-gcd\r\nGreater common divisor (gcd) of two BigInt values using Lehmer's GCD algorithm.\r\nSee https://en.wikipedia.org/wiki/Greatest_common_divisor#Lehmer's_GCD_algorithm.\r\nOn my tests it is faster than Euclidean algorithm starting from 80-bit integers.\r\n\r\nA version 1.0.2 also has something similar to \"Subquadratic GCD\" (see https://gmplib.org/manual/Subquadratic-GCD ),\r\nwhich is faster for large bigints (> 65000 bits), it should has better time complexity in case \r\nthe multiplication is subquadratic, which is true in Chrome 93.\r\n\r\nInstallation\r\n============\r\n\r\n```cmd\r\n$ npm install bigint-gcd\r\n```\r\n\r\nUsage\r\n=====\r\n\r\n```\r\nimport gcd from './node_modules/bigint-gcd/gcd.js';\r\n\r\nconsole.log(gcd(120n, 18n));\r\n\r\n```\r\n\r\nThere is also an implementation of the Extended Euclidean algorithm, which is useful to find the multiplicative modular inverse:\r\n```\r\nconsole.log(gcd.gcdext(3n, 5n)); // [2n, -1n, 1n]\r\n```\r\n\r\nAnd \"Half GCD\" which is useful to do the [Rational reconstruction](https://en.wikipedia.org/wiki/Rational_reconstruction_(mathematics)):\r\nIt returns the transformation matrix and the transformed values after applying about half of the Euclidean steps.\r\n```\r\nconsole.log(gcd.halfgcd(1000000n, 1234567n)); // [-16n, 13n, 21n, -17n, 49371n, 12361n]\r\n```\r\n\r\n\r\n\r\nPerformance:\r\n============\r\n\r\nThe benchmark (see [benchmark.html](benchmark.html)) resutls under Chrome 131:\r\n\r\n|           bit size |                 gcd |          gmpy2 gcd |             invmod |       gmpy2 invert |\r\n| ------------------ | ------------------- | ------------------ | ------------------ | ------------------ |\r\n|                 64 |          0.000270ms |          0.00030ms |         0.000310ms |          0.00066ms |\r\n|                128 |          0.001270ms |          0.00047ms |         0.001720ms |          0.00137ms |\r\n|                256 |          0.002660ms |          0.00153ms |         0.003650ms |          0.00224ms |\r\n|                512 |          0.005460ms |          0.00321ms |         0.007630ms |          0.00391ms |\r\n|               1024 |          0.012080ms |          0.00653ms |         0.018250ms |          0.00806ms |\r\n|               2048 |          0.031130ms |          0.01429ms |         0.048220ms |          0.01587ms |\r\n|               4096 |          0.067870ms |          0.02979ms |         0.137700ms |          0.03590ms |\r\n|               8192 |          0.174320ms |          0.06837ms |         0.341310ms |          0.09035ms |\r\n|              16384 |          0.503910ms |          0.17093ms |         0.867190ms |          0.24908ms |\r\n|              32768 |          1.677730ms |          0.49816ms |         2.281250ms |          0.75801ms |\r\n|              65536 |          4.406250ms |          1.43795ms |         6.152340ms |          1.94962ms |\r\n|             131072 |         11.828130ms |          3.98527ms |        16.937500ms |          4.98559ms |\r\n|             262144 |         32.296880ms |         10.52619ms |        47.203130ms |         14.05025ms |\r\n|             524288 |         86.625000ms |         28.16362ms |       123.500000ms |         38.94622ms |\r\n|            1048576 |        213.312500ms |         70.89262ms |       310.062500ms |        103.71075ms |\r\n|            2097152 |        519.250000ms |        177.16650ms |       773.875000ms |        269.43650ms |\r\n|            4194304 |       1255.750000ms |        433.85675ms |      1870.500000ms |        658.39875ms |\r\n|            8388608 |       2988.500000ms |       1069.74050ms |      4548.000000ms |       1673.88250ms |\r\n\r\nBenchmark:\r\n==========\r\n\r\n```javascript\r\nimport {default as LehmersGCD} from './gcd.js';\r\n\r\nfunction EuclideanGCD(a, b) {\r\n  while (b !== 0n) {\r\n    const r = a % b;\r\n    a = b;\r\n    b = r;\r\n  }\r\n  return a;\r\n}\r\n\r\nfunction ctz4(n) {\r\n  return 31 - Math.clz32(n & -n);\r\n}\r\nconst BigIntCache = new Array(32).fill(0n).map((x, i) => BigInt(i));\r\nfunction ctz1(bigint) {\r\n  return BigIntCache[ctz4(Number(BigInt.asUintN(32, bigint)))];\r\n}\r\nfunction BinaryGCD(a, b) {\r\n  if (a === 0n) {\r\n    return b;\r\n  }\r\n  if (b === 0n) {\r\n    return a;\r\n  }\r\n  const k = ctz1(a | b);\r\n  a >>= k;\r\n  b >>= k;\r\n  while (b !== 0n) {\r\n    b >>= ctz1(b);\r\n    if (a > b) {\r\n      const t = b;\r\n      b = a;\r\n      a = t;\r\n    }\r\n    b -= a;\r\n  }\r\n  return k === 0n ? a : a << k;\r\n}\r\n\r\nfunction FibonacciNumber(n) {\r\n  console.assert(n > 0);\r\n  var a = 0n;\r\n  var b = 1n;\r\n  for (var i = 1; i < n; i += 1) {\r\n    var c = a + b;\r\n    a = b;\r\n    b = c;\r\n  }\r\n  return b;\r\n}\r\n\r\nfunction RandomBigInt(size) {\r\n  if (size <= 32) {\r\n    return BigInt(Math.floor(Math.random() * 2**size));\r\n  }\r\n  const q = Math.floor(size / 2);\r\n  return (RandomBigInt(size - q) << BigInt(q)) | RandomBigInt(q);\r\n}\r\n\r\nfunction test(a, b, f) {\r\n  const g = EuclideanGCD(a, b);\r\n  const count = 100000;\r\n  console.time();\r\n  for (let i = 0; i < count; i++) {\r\n    const I = BigInt(i);\r\n    if (f(a * I, b * I) !== g * I) {\r\n      throw new Error();\r\n    }\r\n  }\r\n  console.timeEnd();\r\n}\r\n\r\nconst a1 = RandomBigInt(128);\r\nconst b1 = RandomBigInt(128);\r\n\r\ntest(a1, b1, LehmersGCD);\r\n// default: 426.200927734375 ms\r\ntest(a1, b1, EuclideanGCD);\r\n// default: 1136.77294921875 ms\r\ntest(a1, b1, BinaryGCD);\r\n// default: 1456.793212890625 ms\r\n\r\nconst a = FibonacciNumber(186n);\r\nconst b = FibonacciNumber(186n - 1n);\r\n\r\ntest(a, b, LehmersGCD);\r\n// default: 459.796875 ms\r\ntest(a, b, EuclideanGCD);\r\n// default: 2565.871826171875 ms\r\ntest(a, b, BinaryGCD);\r\n// default: 1478.333984375 ms\r\n\r\n```\r\n","readmeFilename":"README.md"}