# emilbayes / secure-random-uniform

Generate secure, random, uniform integers, compensating for modulo bias

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# `secure-random-uniform`

Generate secure, random, uniform integers, compensating for modulo bias

## Usage

```var secureRandom = require('secure-random-uniform')

// Numbers from [0, 2000)
secureRandom(2000)

// Numbers from [100, 110)
secureRandom(10) + 100

// Numbers from [-10, 10]
secureRandom(21) - 10```

### BigInt support (Experimental!)

```var secureRandom = require('secure-random-uniform/bigint')

// Numbers from [0, 2^64)
secureRandom(2n ** 64n)

// Numbers from [0, googol)
secureRandom(10n ** 100n)```

## API

#### `var num = secureRandomUniform(limit)`

Returns a number from the uniform distribution `[0, limit)` (limit exclusive). Note that limit must not be larger than `2^53 - 1` (`Number.MAX_SAFE_INTEGER`).

## Background

### Modulo reduction: Bytes to integers

A naive implementation might look like:

```function insecureRandom (limit) {
return secureRandomSource() % limit
}```

However this will only yield a uniform distribution if `limit` is a divisor of whatever is the maximum value of `secureRandomSource()`. Consider `limit = 3` and the maximum value returned by `secureRandomSource()` being `5`. Then in the long run the frequency of numbers returned will be `[0 = 2/5, 1 = 2/5, 2 = 1/5]`, causing the distribution to be skewed (ie. not uniform). This is called "Modulo Bias".

This module borrows from `arc4random_uniform` and keeps generating a new random number until it hits a range that's congruent to `limit`. This is not as bad as it sounds. The worst case is if `limit ≈ (2^48 - 1) / 2`, in which case it will have a ~ 0.5 chance of doing a redraw. The number of redraws required can be modelled by as `0.5^(redraws)` which quickly converges towards zero. In practise only one draw is required on average.

See `verify-modulo-reduction.js` for a deterministic test of the algorithm

### Random bytes to integers

The next issue is transforming random bytes into unsigned numbers. We can efficiently transform bytes into signed 32-bit integers in JS with:

`(byte << 24) | (byte << 16) | (byte << 8) | (byte)`

To make the number unsigned we can do a zero-fill right shift, which will cause the sign bit to become 0:

`((byte << 24) | (byte << 16) | (byte << 8) | (byte)) >>> 0`

To go beyond 32-bit integers, to the maximum of 53-bit integers representable in Javascript `Number`s (IEEE 754), we can construct the remaining 21 bits and move them up using a floating point multiplication.

```((((buf & 0b00011111) << 16) | (buf << 8) | (buf)) >>> 0) * 0x100000000
+ (((byte << 24) | (byte << 16) | (byte << 8) | (byte)) >>> 0)```

Note that the bitwise operations have been wrapped in parenthesis, otherwise the add and multiplication operation will become 32-bit operations, reducing the number modulo 2^32

See `verify-readle.js` for verification against a known implementation of converting bytes to unsigned integers.

ISC

Generate secure, random, uniform integers, compensating for modulo bias