first commit

2019-03-17 09:40:35 +01:00 · 2019-03-17 09:40:35 +01:00 · 1ede4f712c
commit 1ede4f712c
8 changed files with 1239 additions and 0 deletions
--- a/.eslintrc.json
+++ 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"
+        ]
+    }
+}
--- a/.gitignore
+++ 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
+
--- a/21
+++ b/21
@ -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.
--- a/README.hbs
+++ 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}}
+
+* * *
--- a/README.md
+++ 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
+
+<dl>
+<dt><a href="#abs">abs(a)</a> ⇒ <code>bigint</code></dt>
+<dd><p>Absolute value. abs(a)==a if a&gt;=0. abs(a)==-a if a&lt;0</p>
+</dd>
+<dt><a href="#gcd">gcd(a, b)</a> ⇒ <code>bigint</code></dt>
+<dd><p>Greatest-common divisor of two integers based on the iterative binary algorithm.</p>
+</dd>
+<dt><a href="#lcm">lcm(a, b)</a> ⇒ <code>bigint</code></dt>
+<dd><p>The least common multiple computed as abs(a*b)/gcd(a,b)</p>
+</dd>
+<dt><a href="#toZn">toZn(a, n)</a> ⇒ <code>bigint</code></dt>
+<dd><p>Finds the smallest positive element that is congruent to a in modulo n</p>
+</dd>
+<dt><a href="#eGcd">eGcd(a, b)</a> ⇒ <code><a href="#egcdReturn">egcdReturn</a></code></dt>
+<dd><p>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).</p>
+</dd>
+<dt><a href="#modInv">modInv(a, n)</a> ⇒ <code>bigint</code></dt>
+<dd><p>Modular inverse.</p>
+</dd>
+<dt><a href="#modPow">modPow(a, b, n)</a> ⇒ <code>bigint</code></dt>
+<dd><p>Modular exponentiation a**b mod n</p>
+</dd>
+</dl>
+
+## Typedefs
+
+<dl>
+<dt><a href="#egcdReturn">egcdReturn</a> : <code>Object</code></dt>
+<dd><p>A triple (g, x, y), such that ax + by = g = gcd(a, b).</p>
+</dd>
+</dl>
+
+<a name="abs"></a>
+
+## abs(a) ⇒ <code>bigint</code>
+Absolute value. abs(a)==a if a>=0. abs(a)==-a if a<0
+
+**Kind**: global function  
+**Returns**: <code>bigint</code> - the absolute value of a  
+
+| Param | Type |
+| --- | --- |
+| a | <code>number</code> \| <code>bigint</code> | 
+
+<a name="gcd"></a>
+
+## gcd(a, b) ⇒ <code>bigint</code>
+Greatest-common divisor of two integers based on the iterative binary algorithm.
+
+**Kind**: global function  
+**Returns**: <code>bigint</code> - The greatest common divisor of a and b  
+
+| Param | Type |
+| --- | --- |
+| a | <code>number</code> \| <code>bigint</code> | 
+| b | <code>number</code> \| <code>bigint</code> | 
+
+<a name="lcm"></a>
+
+## lcm(a, b) ⇒ <code>bigint</code>
+The least common multiple computed as abs(a*b)/gcd(a,b)
+
+**Kind**: global function  
+**Returns**: <code>bigint</code> - The least common multiple of a and b  
+
+| Param | Type |
+| --- | --- |
+| a | <code>number</code> \| <code>bigint</code> | 
+| b | <code>number</code> \| <code>bigint</code> | 
+
+<a name="toZn"></a>
+
+## toZn(a, n) ⇒ <code>bigint</code>
+Finds the smallest positive element that is congruent to a in modulo n
+
+**Kind**: global function  
+**Returns**: <code>bigint</code> - The smallest positive representation of a in modulo n  
+
+| Param | Type | Description |
+| --- | --- | --- |
+| a | <code>number</code> \| <code>bigint</code> | An integer |
+| n | <code>number</code> \| <code>bigint</code> | The modulo |
+
+<a name="eGcd"></a>
+
+## eGcd(a, b) ⇒ [<code>egcdReturn</code>](#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 | <code>number</code> \| <code>bigint</code> | 
+| b | <code>number</code> \| <code>bigint</code> | 
+
+<a name="modInv"></a>
+
+## modInv(a, n) ⇒ <code>bigint</code>
+Modular inverse.
+
+**Kind**: global function  
+**Returns**: <code>bigint</code> - the inverse modulo n  
+
+| Param | Type | Description |
+| --- | --- | --- |
+| a | <code>number</code> \| <code>bigint</code> | The number to find an inverse for |
+| n | <code>number</code> \| <code>bigint</code> | The modulo |
+
+<a name="modPow"></a>
+
+## modPow(a, b, n) ⇒ <code>bigint</code>
+Modular exponentiation a**b mod n
+
+**Kind**: global function  
+**Returns**: <code>bigint</code> - a**b mod n  
+
+| Param | Type | Description |
+| --- | --- | --- |
+| a | <code>number</code> \| <code>bigint</code> | base |
+| b | <code>number</code> \| <code>bigint</code> | exponent |
+| n | <code>number</code> \| <code>bigint</code> | modulo |
+
+<a name="egcdReturn"></a>
+
+## egcdReturn : <code>Object</code>
+A triple (g, x, y), such that ax + by = g = gcd(a, b).
+
+**Kind**: global typedef  
+**Properties**
+
+| Name | Type |
+| --- | --- |
+| g | <code>bigint</code> | 
+| x | <code>bigint</code> | 
+| y | <code>bigint</code> | 
+
+
+* * *
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+      "resolved": "https://registry.npmjs.org/wrappy/-/wrappy-1.0.2.tgz",
+      "integrity": "sha1-tSQ9jz7BqjXxNkYFvA0QNuMKtp8=",
+      "dev": true
+    },
+    "xmlcreate": {
+      "version": "1.0.2",
+      "resolved": "https://registry.npmjs.org/xmlcreate/-/xmlcreate-1.0.2.tgz",
+      "integrity": "sha1-+mv3YqYKQT+z3Y9LA8WyaSONMI8=",
+      "dev": true
+    }
+  }
+}
--- a/package.json
+++ b/package.json
@ -0,0 +1,31 @@
+{
+  "name": "bigint-mod-arith",
+  "version": "1.0.0",
+  "description": "Some additional common functions for modular arithmetics using native JS (stage 3) implementation of BigInt",
+  "keywords": [
+    "modular 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"
+  }
+}
--- a/src/main.js
+++ 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
+};