bn.js

What is bn.js?

The bn.js package is a library that provides support for arbitrary-precision arithmetic operations on big numbers in JavaScript. It is commonly used in cryptography, financial calculations, and anywhere precise arithmetic with large integers is required.

What are bn.js's main functionalities?

Big Number Arithmetic

Performing basic arithmetic operations such as addition, subtraction, multiplication, division, and modulo on big numbers.

const BN = require('bn.js');
let a = new BN('123456789');
let b = new BN('987654321');
let sum = a.add(b); // Addition
let diff = a.sub(b); // Subtraction
let product = a.mul(b); // Multiplication
let quotient = a.div(b); // Division
let remainder = a.mod(b); // Modulo

Comparison Operations

Comparing big numbers to determine if they are equal, less than, or greater than each other.

const BN = require('bn.js');
let a = new BN('12345');
let b = new BN('12345');
let c = new BN('54321');
a.eq(b); // true, a equals b
a.lt(c); // true, a is less than c
a.gt(c); // false, a is not greater than c

Bitwise Operations

Performing bitwise operations such as AND, OR, XOR, and NOT on big numbers.

const BN = require('bn.js');
let a = new BN('1011', 2);
let b = new BN('1101', 2);
let and = a.and(b); // AND operation
let or = a.or(b); // OR operation
let xor = a.xor(b); // XOR operation
let not = a.not(); // NOT operation

Reduction Contexts

Using reduction contexts to perform modular arithmetic efficiently, particularly useful in modular exponentiation and cryptographic operations.

const BN = require('bn.js');
let p = new BN('fffffffffffffffffffffffffffffffebaaedce6af48a03bbfd25e8cd0364141', 16);
let a = new BN('7fffffffffffffffffffffff7ffffe17ff', 16);
let red = BN.red(p);
let b = a.toRed(red); // Convert to Montgomery representation
let c = b.redSqr().redMul(b); // Perform operations in reduction context
let d = c.fromRed(); // Convert out of Montgomery representation

Other packages similar to bn.js

big.js

big.js is a small, fast JavaScript library for arbitrary-precision decimal arithmetic. It focuses on decimal numbers and provides a simpler API but does not support bitwise operations like bn.js.

bignumber.js

bignumber.js is another arbitrary-precision arithmetic library for JavaScript and Node.js. It supports decimal and non-decimal arithmetic and has a more extensive API than bn.js, but it might be slower for some operations due to its focus on decimal precision.

decimal.js

decimal.js is a library for arbitrary-precision decimal arithmetic. It is similar to bignumber.js in its focus on decimal numbers and provides a comprehensive set of features for decimal arithmetic, but it does not provide the same level of support for non-decimal and bitwise operations as bn.js.

README

BigNum in pure javascript

Install

npm install --save bn.js

Usage

const BN = require('bn.js');

var a = new BN('dead', 16);
var b = new BN('101010', 2);

var res = a.add(b);
console.log(res.toString(10));  // 57047

Note: decimals are not supported in this library.

Notation

Prefixes

There are several prefixes to instructions that affect the way the work. Here is the list of them in the order of appearance in the function name:

• i - perform operation in-place, storing the result in the host object (on which the method was invoked). Might be used to avoid number allocation costs
• u - unsigned, ignore the sign of operands when performing operation, or always return positive value. Second case applies to reduction operations like mod(). In such cases if the result will be negative - modulo will be added to the result to make it positive

Postfixes

• n - the argument of the function must be a plain JavaScript Number. Decimals are not supported.
• rn - both argument and return value of the function are plain JavaScript Numbers. Decimals are not supported.

Examples

• a.iadd(b) - perform addition on a and b, storing the result in a
• a.umod(b) - reduce a modulo b, returning positive value
• a.iushln(13) - shift bits of a left by 13

Instructions

Prefixes/postfixes are put in parens at the of the line. endian - could be either le (little-endian) or be (big-endian).

Utilities

• a.clone() - clone number
• a.toString(base, length) - convert to base-string and pad with zeroes
• a.toNumber() - convert to Javascript Number (limited to 53 bits)
• a.toJSON() - convert to JSON compatible hex string (alias of toString(16))
• a.toArray(endian, length) - convert to byte Array, and optionally zero pad to length, throwing if already exceeding
• a.toArrayLike(type, endian, length) - convert to an instance of type, which must behave like an Array
• a.toBuffer(endian, length) - convert to Node.js Buffer (if available). For compatibility with browserify and similar tools, use this instead: a.toArrayLike(Buffer, endian, length)
• a.bitLength() - get number of bits occupied
• a.zeroBits() - return number of less-significant consequent zero bits (example: 1010000 has 4 zero bits)
• a.byteLength() - return number of bytes occupied
• a.isNeg() - true if the number is negative
• a.isEven() - no comments
• a.isOdd() - no comments
• a.isZero() - no comments
• a.cmp(b) - compare numbers and return -1 (a < b), 0 (a == b), or 1 (a > b) depending on the comparison result (ucmp, cmpn)
• a.lt(b) - a less than b (n)
• a.lte(b) - a less than or equals b (n)
• a.gt(b) - a greater than b (n)
• a.gte(b) - a greater than or equals b (n)
• a.eq(b) - a equals b (n)
• a.toTwos(width) - convert to two's complement representation, where width is bit width
• a.fromTwos(width) - convert from two's complement representation, where width is the bit width
• BN.isBN(object) - returns true if the supplied object is a BN.js instance
• BN.max(a, b) - return a if a bigger than b
• BN.min(a, b) - return a if a less than b

Arithmetics

• a.neg() - negate sign (i)
• a.abs() - absolute value (i)
• a.add(b) - addition (i, n, in)
• a.sub(b) - subtraction (i, n, in)
• a.mul(b) - multiply (i, n, in)
• a.sqr() - square (i)
• a.pow(b) - raise a to the power of b
• a.div(b) - divide (divn, idivn)
• a.mod(b) - reduct (u, n) (but no umodn)
• a.divRound(b) - rounded division

Bit operations

• a.or(b) - or (i, u, iu)
• a.and(b) - and (i, u, iu, andln) (NOTE: andln is going to be replaced with andn in future)
• a.xor(b) - xor (i, u, iu)
• a.setn(b) - set specified bit to 1
• a.shln(b) - shift left (i, u, iu)
• a.shrn(b) - shift right (i, u, iu)
• a.testn(b) - test if specified bit is set
• a.maskn(b) - clear bits with indexes higher or equal to b (i)
• a.bincn(b) - add 1 << b to the number
• a.notn(w) - not (for the width specified by w) (i)

Reduction

• a.gcd(b) - GCD
• a.egcd(b) - Extended GCD results ({ a: ..., b: ..., gcd: ... })
• a.invm(b) - inverse a modulo b

Fast reduction

When doing lots of reductions using the same modulo, it might be beneficial to use some tricks: like Montgomery multiplication, or using special algorithm for Mersenne Prime.

Reduction context

To enable this tricks one should create a reduction context:

var red = BN.red(num);

where num is just a BN instance.

Or:

var red = BN.red(primeName);

Where primeName is either of these Mersenne Primes:

• 'k256'
• 'p224'
• 'p192'
• 'p25519'

Or:

var red = BN.mont(num);

To reduce numbers with Montgomery trick. .mont() is generally faster than .red(num), but slower than BN.red(primeName).

Converting numbers

Before performing anything in reduction context - numbers should be converted to it. Usually, this means that one should:

• Convert inputs to reducted ones
• Operate on them in reduction context
• Convert outputs back from the reduction context

Here is how one may convert numbers to red:

var redA = a.toRed(red);

Where red is a reduction context created using instructions above

Here is how to convert them back:

var a = redA.fromRed();

Red instructions

Most of the instructions from the very start of this readme have their counterparts in red context:

• a.redAdd(b), a.redIAdd(b)
• a.redSub(b), a.redISub(b)
• a.redShl(num)
• a.redMul(b), a.redIMul(b)
• a.redSqr(), a.redISqr()
• a.redSqrt() - square root modulo reduction context's prime
• a.redInvm() - modular inverse of the number
• a.redNeg()
• a.redPow(b) - modular exponentiation

Number Size

Optimized for elliptic curves that work with 256-bit numbers. There is no limitation on the size of the numbers.

LICENSE

This software is licensed under the MIT License.

Copyright Fedor Indutny, 2015.

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.