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Juan Hernández Serrano 2019-03-17 09:40:35 +01:00
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{
"env": {
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# Logs
logs
*.log
npm-debug.log*
yarn-debug.log*
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# Runtime data
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*.pid
*.seed
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# Directory for instrumented libs generated by jscoverage/JSCover
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.next
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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.

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# 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}}
* * *

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# 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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31
package.json Normal file
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@ -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"
}
}

169
src/main.js Normal file
View File

@ -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
};