commit 1ede4f712c7ae9bbb3cbe6e127d7e3b4320901b8 Author: Juan Hernández Serrano Date: Sun Mar 17 09:40:35 2019 +0100 first commit diff --git a/.eslintrc.json b/.eslintrc.json new file mode 100644 index 0000000..eaed3bc --- /dev/null +++ b/.eslintrc.json @@ -0,0 +1,27 @@ +{ + "env": { + "node": true, + "browser": true, + "mocha": true, + "es6": true + }, + "extends": "eslint:recommended", + "rules": { + "indent": [ + "error", + 4 + ], + "linebreak-style": [ + "error", + "unix" + ], + "quotes": [ + "error", + "single" + ], + "semi": [ + "error", + "always" + ] + } +} \ No newline at end of file diff --git a/.gitignore b/.gitignore new file mode 100644 index 0000000..19349b6 --- /dev/null +++ b/.gitignore @@ -0,0 +1,65 @@ +# Logs +logs +*.log +npm-debug.log* +yarn-debug.log* +yarn-error.log* + +# Runtime data +pids +*.pid +*.seed +*.pid.lock + +# Directory for instrumented libs generated by jscoverage/JSCover +lib-cov + +# Coverage directory used by tools like istanbul +coverage + +# nyc test coverage +.nyc_output + +# Grunt intermediate storage (http://gruntjs.com/creating-plugins#storing-task-files) +.grunt + +# Bower dependency directory (https://bower.io/) +bower_components + +# node-waf configuration +.lock-wscript + +# Compiled binary addons (https://nodejs.org/api/addons.html) +build/Release + +# Dependency directories +node_modules/ +jspm_packages/ + +# TypeScript v1 declaration files +typings/ + +# Optional npm cache directory +.npm + +# Optional eslint cache +.eslintcache + +# Optional REPL history +.node_repl_history + +# Output of 'npm pack' +*.tgz + +# Yarn Integrity file +.yarn-integrity + +# dotenv environment variables file +.env + +# next.js build output +.next + +# Visual Studio Code +.vscode + diff --git a/LICENSE b/LICENSE new file mode 100644 index 0000000..b76dbea --- /dev/null +++ b/LICENSE @@ -0,0 +1,21 @@ +MIT License + +Copyright (c) 2018 Juan Hernández Serrano + +Permission is hereby granted, free of charge, to any person obtaining a copy +of this software and associated documentation files (the "Software"), to deal +in the Software without restriction, including without limitation the rights +to use, copy, modify, merge, publish, distribute, sublicense, and/or sell +copies of the Software, and to permit persons to whom the Software is +furnished to do so, subject to the following conditions: + +The above copyright notice and this permission notice shall be included in all +copies or substantial portions of the Software. + +THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR +IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, +FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE +AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER +LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, +OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE +SOFTWARE. diff --git a/README.hbs b/README.hbs new file mode 100644 index 0000000..9bbff90 --- /dev/null +++ b/README.hbs @@ -0,0 +1,23 @@ +# bigint-mod-arith + +Some extra functions to work with modular arithmetics using native JS (stage 3) implementation of BigInt. + +## Usage examples + +```javascript +const modArith = require('bigint-mod-arith'); + +let a = 5n; +let b = 2n; +let n = 19n; + +console.log(modArith.modPow(a, b, n)); // prints 13 + +console.log(modArith.modInv(2n, 5n)); // prints 3 + +console.log(modArith.modInv(3n, 5n)); // prints 2 +``` + +{{>main}} + +* * * diff --git a/README.md b/README.md new file mode 100644 index 0000000..ebf85c1 --- /dev/null +++ b/README.md @@ -0,0 +1,162 @@ +# bigint-mod-arith + +Some extra functions to work with modular arithmetics using native JS (stage 3) implementation of BigInt. + +## Usage examples + +```javascript +const modArith = require('bigint-mod-arith'); + +let a = 5n; +let b = 2n; +let n = 19n; + +console.log(modArith.modPow(a, b, n)); // prints 13 + +console.log(modArith.modInv(2n, 5n)); // prints 3 + +console.log(modArith.modInv(3n, 5n)); // prints 2 +``` + +## Functions + +
+
abs(a) ⇒ bigint
+

Absolute value. abs(a)==a if a>=0. abs(a)==-a if a<0

+
+
gcd(a, b) ⇒ bigint
+

Greatest-common divisor of two integers based on the iterative binary algorithm.

+
+
lcm(a, b) ⇒ bigint
+

The least common multiple computed as abs(a*b)/gcd(a,b)

+
+
toZn(a, n) ⇒ bigint
+

Finds the smallest positive element that is congruent to a in modulo n

+
+
eGcd(a, b) ⇒ egcdReturn
+

An iterative implementation of the extended euclidean algorithm or extended greatest common divisor algorithm. +Take positive integers a, b as input, and return a triple (g, x, y), such that ax + by = g = gcd(a, b).

+
+
modInv(a, n) ⇒ bigint
+

Modular inverse.

+
+
modPow(a, b, n) ⇒ bigint
+

Modular exponentiation a**b mod n

+
+
+ +## Typedefs + +
+
egcdReturn : Object
+

A triple (g, x, y), such that ax + by = g = gcd(a, b).

+
+
+ + + +## abs(a) ⇒ bigint +Absolute value. abs(a)==a if a>=0. abs(a)==-a if a<0 + +**Kind**: global function +**Returns**: bigint - the absolute value of a + +| Param | Type | +| --- | --- | +| a | number \| bigint | + + + +## gcd(a, b) ⇒ bigint +Greatest-common divisor of two integers based on the iterative binary algorithm. + +**Kind**: global function +**Returns**: bigint - The greatest common divisor of a and b + +| Param | Type | +| --- | --- | +| a | number \| bigint | +| b | number \| bigint | + + + +## lcm(a, b) ⇒ bigint +The least common multiple computed as abs(a*b)/gcd(a,b) + +**Kind**: global function +**Returns**: bigint - The least common multiple of a and b + +| Param | Type | +| --- | --- | +| a | number \| bigint | +| b | number \| bigint | + + + +## toZn(a, n) ⇒ bigint +Finds the smallest positive element that is congruent to a in modulo n + +**Kind**: global function +**Returns**: bigint - The smallest positive representation of a in modulo n + +| Param | Type | Description | +| --- | --- | --- | +| a | number \| bigint | An integer | +| n | number \| bigint | The modulo | + + + +## eGcd(a, b) ⇒ [egcdReturn](#egcdReturn) +An iterative implementation of the extended euclidean algorithm or extended greatest common divisor algorithm. +Take positive integers a, b as input, and return a triple (g, x, y), such that ax + by = g = gcd(a, b). + +**Kind**: global function + +| Param | Type | +| --- | --- | +| a | number \| bigint | +| b | number \| bigint | + + + +## modInv(a, n) ⇒ bigint +Modular inverse. + +**Kind**: global function +**Returns**: bigint - the inverse modulo n + +| Param | Type | Description | +| --- | --- | --- | +| a | number \| bigint | The number to find an inverse for | +| n | number \| bigint | The modulo | + + + +## modPow(a, b, n) ⇒ bigint +Modular exponentiation a**b mod n + +**Kind**: global function +**Returns**: bigint - a**b mod n + +| Param | Type | Description | +| --- | --- | --- | +| a | number \| bigint | base | +| b | number \| bigint | exponent | +| n | number \| bigint | modulo | + + + +## egcdReturn : Object +A triple (g, x, y), such that ax + by = g = gcd(a, b). + +**Kind**: global typedef +**Properties** + +| Name | Type | +| --- | --- | +| g | bigint | +| x | bigint | +| y | bigint | + + +* * * diff --git a/package-lock.json b/package-lock.json new file mode 100644 index 0000000..ffcf228 --- /dev/null +++ b/package-lock.json @@ -0,0 +1,741 @@ +{ + "name": "bigint-mod-arith", + "version": "1.0.0", + "lockfileVersion": 1, + "requires": true, + "dependencies": { + "ansi-escape-sequences": { + "version": "4.1.0", + "resolved": "https://registry.npmjs.org/ansi-escape-sequences/-/ansi-escape-sequences-4.1.0.tgz", + "integrity": "sha512-dzW9kHxH011uBsidTXd14JXgzye/YLb2LzeKZ4bsgl/Knwx8AtbSFkkGxagdNOoh0DlqHCmfiEjWKBaqjOanVw==", + "dev": true, + "requires": { + "array-back": "^3.0.1" + }, + "dependencies": { + "array-back": { + "version": "3.0.1", + "resolved": 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arithmetics", + "BigInt", + "lcm", + "gcd", + "egcd", + "modular inverse", + "modular exponentiation" + ], + "license": "MIT", + "author": { + "name": "Juan Hernández Serrano", + "email": "jserrano@entel.upc.edu", + "url": "https://github.com/juanelas" + }, + "repository": "github:juanelas/bigint-mod-arith", + "main": "./src/main.js", + "directories": { + "src": "./src" + }, + "scripts": { + "docs:build": "jsdoc2md --template=README.hbs --files ./src/main.js > README.md" + }, + "devDependencies": { + "jsdoc-to-markdown": "^4.0.1" + } +} diff --git a/src/main.js b/src/main.js new file mode 100644 index 0000000..35d77d2 --- /dev/null +++ b/src/main.js @@ -0,0 +1,169 @@ +'use strict'; + +/** + * Absolute value. abs(a)==a if a>=0. abs(a)==-a if a<0 + * + * @param {number|bigint} a + * + * @returns {bigint} the absolute value of a + */ +const abs = function (a) { + a = BigInt(a); + return (a >= 0n) ? a : -a; +}; + +/** + * Greatest-common divisor of two integers based on the iterative binary algorithm. + * + * @param {number|bigint} a + * @param {number|bigint} b + * + * @returns {bigint} The greatest common divisor of a and b + */ +const gcd = function (a, b) { + a = abs(a); + b = abs(b); + let shift = 0n; + while (!((a | b) & 1n)) { + a >>= 1n; + b >>= 1n; + shift++; + } + while (!(a & 1n)) a >>= 1n; + do { + while (!(b & 1n)) b >>= 1n; + if (a > b) { + let x = a; + a = b; + b = x; + } + b -= a; + } while (b); + + // rescale + return a << shift; +}; + +/** + * The least common multiple computed as abs(a*b)/gcd(a,b) + * @param {number|bigint} a + * @param {number|bigint} b + * + * @returns {bigint} The least common multiple of a and b + */ +const lcm = function (a, b) { + a = BigInt(a); + b = BigInt(b); + return abs(a * b) / gcd(a, b); +}; + +/** + * Finds the smallest positive element that is congruent to a in modulo n + * @param {number|bigint} a An integer + * @param {number|bigint} n The modulo + * + * @returns {bigint} The smallest positive representation of a in modulo n + */ +const toZn = function (a, n) { + n = BigInt(n); + a = BigInt(a) % n; + return (a < 0) ? a + n : a; +}; + +/** + * @typedef {Object} egcdReturn A triple (g, x, y), such that ax + by = g = gcd(a, b). + * @property {bigint} g + * @property {bigint} x + * @property {bigint} y + */ +/** + * An iterative implementation of the extended euclidean algorithm or extended greatest common divisor algorithm. + * Take positive integers a, b as input, and return a triple (g, x, y), such that ax + by = g = gcd(a, b). + * + * @param {number|bigint} a + * @param {number|bigint} b + * + * @returns {egcdReturn} + */ +const eGcd = function (a, b) { + a = BigInt(a); + b = BigInt(b); + let x = 0n; + let y = 1n; + let u = 1n; + let v = 0n; + + while (a !== 0n) { + let q = b / a; + let r = b % a; + let m = x - (u * q); + let n = y - (v * q); + b = a; + a = r; + x = u; + y = v; + u = m; + v = n; + } + return { + b: b, + x: x, + y: y + } +}; + +/** + * Modular inverse. + * + * @param {number|bigint} a The number to find an inverse for + * @param {number|bigint} n The modulo + * + * @returns {bigint} the inverse modulo n + */ +const modInv = function (a, n) { + let egcd = eGcd(a, n); + if (egcd.b !== 1n) { + return null; // modular inverse does not exist + } else { + return toZn(egcd.x, n); + } +}; + +/** + * Modular exponentiation a**b mod n + * @param {number|bigint} a base + * @param {number|bigint} b exponent + * @param {number|bigint} n modulo + * + * @returns {bigint} a**b mod n + */ +const modPow = function (a, b, n) { + // See Knuth, volume 2, section 4.6.3. + n = BigInt(n); + a = toZn(a, n); + b = BigInt(b); + if (b < 0n) { + return modInv(modPow(a, abs(b), n), n); + } + let result = 1n; + let x = a; + while (b > 0) { + var leastSignificantBit = b % 2n; + b = b / 2n; + if (leastSignificantBit == 1n) { + result = result * x; + result = result % n; + } + x = x * x; + x = x % n; + } + return result; +}; + +module.exports = { + abs: abs, + gcd: gcd, + lcm: lcm, + modInv: modInv, + modPow: modPow +}; \ No newline at end of file