Pseudo random number generation

This module provides pseudo random number generation and implements a number of base generator algorithms. Most are provided through a plug-in framework that adds features to the base generators.

At the end of this module documentation there are some niche algorithms that don't use this module's normal plug-in framework. They may be useful for special purposes like short generation time when quality is not essential, for seeding other generators, and such.

Plug-in framework

The plug-in framework implements a common API to, and enhancements of the base generators:

• Operating on a generator state in the process dictionary.
• Automatic seeding.
• Manual seeding support to avoid common pitfalls.
• Generating integers in any range, with uniform distribution, without noticable bias.
• Generating integers in any range, larger than the base generator's, with uniform distribution.
• Generating floating-point numbers with uniform distribution.
• Generating floating-point numbers with normal distribution.
• Generating any number of bytes.

The base generator algorithms implements the Xoroshiro and Xorshift algorithms by Sebastiano Vigna. During an iteration they generate a large integer (at least 58-bit) and operate on a state of several large integers.

To create numbers with normal distribution the Ziggurat Method by Marsaglia and Tsang is used on the output from a base generator.

For most algorithms, jump functions are provided for generating non-overlapping sequences. A jump function perform a calculation equivalent to a large number of repeated state iterations, but execute in a time roughly equivalent to one regular iteration per generator bit.

The following algorithms are provided:

• exsss, the default algorithm (Since OTP 22.0)
  Xorshift116**, 58 bits precision and period of 2^116-1

  Jump function: equivalent to 2^64 calls

  This is the Xorshift116 generator combined with the StarStar scrambler from the 2018 paper by David Blackman and Sebastiano Vigna: Scrambled Linear Pseudorandom Number Generators

  The generator doesn't use 58-bit rotates so it is faster than the Xoroshiro116 generator, and when combined with the StarStar scrambler it doesn't have any weak low bits like exrop (Xoroshiro116+).

  Alas, this combination is about 10% slower than exrop, but despite that it is the default algorithm thanks to its statistical qualities.

• exro928ss (Since OTP 22.0)
  Xoroshiro928**, 58 bits precision and a period of 2^928-1

  Jump function: equivalent to 2^512 calls

  This is a 58 bit version of Xoroshiro1024**, from the 2018 paper by David Blackman and Sebastiano Vigna: Scrambled Linear Pseudorandom Number Generators that on a 64 bit Erlang system executes only about 40% slower than the default exsss algorithm but with much longer period and better statistical properties, but on the flip side a larger state.

  Many thanks to Sebastiano Vigna for his help with the 58 bit adaption.

• exrop (Since OTP 20.0)
  Xoroshiro116+, 58 bits precision and period of 2^116-1

  Jump function: equivalent to 2^64 calls

• exs1024s (Since OTP 20.0)
  Xorshift1024*, 64 bits precision and a period of 2^1024-1

  Jump function: equivalent to 2^512 calls

• exsp (Since OTP 20.0)
  Xorshift116+, 58 bits precision and period of 2^116-1

  Jump function: equivalent to 2^64 calls

  This is a corrected version of a previous default algorithm (exsplus, deprecated), that was superseded by Xoroshiro116+ (exrop). Since this algorithm doesn't use rotate it executes a little (say < 15%) faster than exrop (that has to do a 58 bit rotate, for which there is no native instruction). See the algorithms' homepage.

Default Algorithm

The current default algorithm is exsss (Xorshift116**). If a specific algorithm is required, ensure to always use seed/1 to initialize the state.

Which algorithm that is the default may change between Erlang/OTP releases, and is selected to be one with high speed, small state and "good enough" statistical properties.

Old Algorithms

Undocumented (old) algorithms are deprecated but still implemented so old code relying on them will produce the same pseudo random sequences as before.

Note

There were a number of problems in the implementation of the now undocumented algorithms, which is why they are deprecated. The new algorithms are a bit slower but do not have these problems:

Uniform integer ranges had a skew in the probability distribution that was not noticable for small ranges but for large ranges less than the generator's precision the probability to produce a low number could be twice the probability for a high.

Uniform integer ranges larger than or equal to the generator's precision used a floating point fallback that only calculated with 52 bits which is smaller than the requested range and therefore all numbers in the requested range weren't even possible to produce.

Uniform floats had a non-uniform density so small values for example less than 0.5 had got smaller intervals decreasing as the generated value approached 0.0 although still uniformly distributed for sufficiently large subranges. The new algorithms produces uniformly distributed floats on the form N * 2.0^(-53) hence they are equally spaced.

Generator State

Every time a random number is generated, a state is used to calculate it, producing a new state. The state can either be implicit or be an explicit argument and return value.

The functions with implicit state operates on a state stored in the process dictionary under the key rand_seed. If that key doesn't exist when the function is called, seed/1 is called automatically with the default algorithm and creates a reasonably unpredictable seed.

The functions with explicit state don't use the process dictionary.

Examples

Simple use; create and seed the default algorithm with a non-fixed seed, if not already done, and generate two uniformly distibuted floating point numbers.

R0 = rand:uniform(),
R1 = rand:uniform(),

Use a specified algorithm:

_ = rand:seed(exs928ss),
R2 = rand:uniform(),

Use a specified algorithm with a fixed seed:

_ = rand:seed(exs928ss, {123, 123534, 345345}),
R3 = rand:uniform(),

Use the functional API with a non-fixed seed:

S0 = rand:seed_s(exsss),
{R4, S1} = rand:uniform_s(S0),

Generate a textbook basic form Box-Muller standard normal distribution number:

R5 = rand:uniform_real(),
R6 = rand:uniform(),
SND0 = math:sqrt(-2 * math:log(R5)) * math:cos(math:pi() * R6)

Generate a standard normal distribution number:

{SND1, S2} = rand:normal_s(S1),

Generate a normal distribution number with with mean -3 and variance 0.5:

{ND0, S3} = rand:normal_s(-3, 0.5, S2),

Quality of the Generated Numbers

Note

The builtin random number generator algorithms are not cryptographically strong. If a cryptographically strong random number generator is needed, use something like crypto:rand_seed/0.

For all these generators except exro928ss and exsss the lowest bit(s) have got a slightly less random behaviour than all other bits. 1 bit for exrop (and exsp), and 3 bits for exs1024s. See for example this explanation in the Xoroshiro128+ generator source code:

Beside passing BigCrush, this generator passes the PractRand test suite up to (and included) 16TB, with the exception of binary rank tests, which fail due to the lowest bit being an LFSR; all other bits pass all tests. We suggest to use a sign test to extract a random Boolean value.

If this is a problem; to generate a boolean with these algorithms, use something like this:

(rand:uniform(256) > 128) % -> boolean()

((rand:uniform(256) - 1) bsr 7) % -> 0 | 1

For a general range, with N = 1 for exrop, and N = 3 for exs1024s:

(((rand:uniform(Range bsl N) - 1) bsr N) + 1)

The floating point generating functions in this module waste the lowest bits when converting from an integer so they avoid this snag.

Niche algorithms

The niche algorithms API contains special purpose algorithms that don't use the plug-in framework, mainly for performance reasons.

Since these algorithms lack the plug-in framework support, generating numbers in a range other than the base generator's range may become a problem.

There are at least four ways to do this, assuming the Range is less than the generator's range:

• Modulo
  To generate a number V in the range 0..Range-1:

  Generate a number X.
  Use V = X rem Range as your value.

  This method uses rem, that is, the remainder of an integer division, which is a slow operation.

  Low bits from the generator propagate straight through to the generated value, so if the generator has got weaknesses in the low bits this method propagates them too.

  If Range is not a divisor of the generator range, the generated numbers have a bias. Example:

  Say the generator generates a byte, that is, the generator range is 0..255, and the desired range is 0..99 (Range = 100). Then there are 3 generator outputs that produce the value 0, these are; 0, 100 and 200. But there are only 2 generator outputs that produce the value 99, which are; 99 and 199. So the probability for a value V in 0..55 is 3/2 times the probability for the other values 56..99.

  If Range is much smaller than the generator range, then this bias gets hard to detect. The rule of thumb is that if Range is smaller than the square root of the generator range, the bias is small enough. Example:

  A byte generator when Range = 20. There are 12 (256 div 20) possibilities to generate the highest numbers and one more to generate a number V < 16 (256 rem 20). So the probability is 13/12 for a low number versus a high. To detect that difference with some confidence you would need to generate a lot more numbers than the generator range, 256 in this small example.

• Truncated multiplication
  To generate a number V in the range 0..Range-1, when you have a generator with a power of 2 range (0..2^Bits-1):

  Generate a number X.
  Use V = X * Range bsr Bits as your value.

  If the multiplication X * Range creates a bignum this method becomes very slow.

  High bits from the generator propagate through to the generated value, so if the generator has got weaknesses in the high bits this method propagates them too.

  If Range is not a divisor of the generator range, the generated numbers have a bias, pretty much as for the Modulo method above.

• Shift or mask
  To generate a number in a power of 2 range (0..2^RBits-1), when you have a generator with a power of 2 range (0..2^Bits):

  Generate a number X.
  Use V = X band ((1 bsl RBits)-1) or V = X bsr (Bits-RBits) as your value.

  Masking with band preserves the low bits, and right shifting with bsr preserves the high, so if the generator has got weaknesses in high or low bits; choose the right operator.

  If the generator has got a range that is not a power of 2 and this method is used anyway, it introduces bias in the same way as for the Modulo method above.

• Rejection

  Generate a number X.
  If X is in the range, use it as your value, otherwise reject it and repeat.

  In theory it is not certain that this method will ever complete, but in practice you ensure that the probability of rejection is low. Then the probability for yet another iteration decreases exponentially so the expected mean number of iterations will often be between 1 and 2. Also, since the base generator is a full length generator, a value that will break the loop must eventually be generated.

  These methods can be combined, such as using the Modulo method and only if the generator value would create bias use Rejection. Or using Shift or mask to reduce the size of a generator value so that Truncated multiplication will not create a bignum.

  The recommended way to generate a floating point number (IEEE 745 Double, that has got a 53-bit mantissa) in the range 0..1, that is 0.0 =< V < 1.0 is to generate a 53-bit number X and then use V = X * (1.0/((1 bsl 53))) as your value. This will create a value on the form N*2^-53 with equal probability for every possible N for the range.