# Random Number Generator Library Concepts

## Introduction

Random numbers are required in a number of different problem domains, such as

- numerics (simulation, Monte-Carlo integration)
- games (non-deterministic enemy behavior)
- security (key generation)
- testing (random coverage in white-box tests)

The Boost Random Number Generator Library provides a framework for random number generators with well-defined properties so that the generators can be used in the demanding numerics and security domains. For a general introduction to random numbers in numerics, see

> "Numerical Recipes in C: The art of scientific computing", William H. Press, Saul A. Teukolsky, William A. Vetterling, Brian P. Flannery, 2nd ed., 1992, pp. 274-328

Depending on the requirements of the problem domain, different variations of random number generators are appropriate:

- non-deterministic random number generator
- pseudo-random number generator
- quasi-random number generator

All variations have some properties in common, these concepts (in the STL sense) are called NumberGenerator and UniformRandomNumberGenerator. Each concept will be defined in a subsequent section.

The goals for this library are the following:

- allow easy integration of third-party random-number generators
- define a validation interface for the generators
- provide easy-to-use front-end classes which model popular distributions
- provide maximum efficiency
- allow control on quantization effects in front-end processing (not yet done)

## Number Generator

A number generator is a function object (std:20.3 [lib.function.objects]) that takes zero arguments. Each call to `operator()` returns a number. In the following table, `X` denotes a number generator class returning objects of type `T`, and `u` is a value of `X`.

`NumberGenerator` requirements
expression | return type | pre/post-condition
`X::result_type` | T | `std::numeric_limits<T>::is_specialized` is true, `T` is `LessThanComparable`
`u.operator()()` | T | -

Note: The NumberGenerator requirements do not impose any restrictions on the characteristics of the returned numbers.

## Uniform Random Number Generator

A uniform random number generator is a NumberGenerator that provides a sequence of random numbers uniformly distributed on a given range. The range can be compile-time fixed or available (only) after run-time construction of the object.

The tight lower bound of some (finite) set S is the (unique) member l in S, so that for all v in S, l <= v holds. Likewise, the tight upper bound of some (finite) set S is the (unique) member u in S, so that for all v in S, v <= u holds.

In the following table, `X` denotes a number generator class returning objects of type `T`, and `v` is a const value of `X`.

`UniformRandomNumberGenerator` requirements
expression | return type | pre/post-condition
`X::has_fixed_range` | `bool` | compile-time constant; if `true`, the range on which the random numbers are uniformly distributed is known at compile-time and members `min_value` and `max_value` exist. Note: This flag may also be `false` due to compiler limitations.
`X::min_value` | `T` | compile-time constant; `min_value` is equal to `v.min()`
`X::max_value` | `T` | compile-time constant; `max_value` is equal to `v.max()`
`v.min()` | `T` | tight lower bound on the set of all values returned by `operator()`. The return value of this function shall not change during the lifetime of the object.
`v.max()` | `T` | if `std::numeric_limits<T>::is_integer`, tight upper bound on the set of all values returned by `operator()`, otherwise, the smallest representable number larger than the tight upper bound on the set of all values returned by `operator()`. In any case, the return value of this function shall not change during the lifetime of the object.

The member functions `min`, `max`, and `operator()` shall have amortized constant time complexity.

Note: For integer generators (i.e. integer `T`), the generated values `x` fulfill `min() <= x <= max()`, for non-integer generators (i.e. non-integer `T`), the generated values `x` fulfill `min() <= x < max()`.
Rationale: The range description with `min` and `max` serves two purposes. First, it allows scaling of the values to some canonical range, such as [0..1). Second, it describes the significant bits of the values, which may be relevant for further processing.
The range is a closed interval [min,max] for integers, because the underlying type may not be able to represent the half-open interval [min,max+1). It is a half-open interval [min, max) for non-integers, because this is much more practical for borderline cases of continuous distributions.

Note: The UniformRandomNumberGenerator concept does not require `operator()(long)` and thus it does not fulfill the RandomNumberGenerator (std:25.2.11 [lib.alg.random.shuffle]) requirements. Use the `random_number_generator` adapter for that.
Rationale: `operator()(long)` is not provided, because mapping the output of some generator with integer range to a different integer range is not trivial.

## Non-deterministic Uniform Random Number Generator

A non-deterministic uniform random number generator is a UniformRandomNumberGenerator that is based on some stochastic process. Thus, it provides a sequence of truly-random numbers. Examples for such processes are nuclear decay, noise of a Zehner diode, tunneling of quantum particles, rolling a die, drawing from an urn, and tossing a coin. Depending on the environment, inter-arrival times of network packets or keyboard events may be close approximations of stochastic processes.

The class `random_device` is a model for a non-deterministic random number generator.

Note: This type of random-number generator is useful for security applications, where it is important to prevent that an outside attacker guesses the numbers and thus obtains your encryption or authentication key. Thus, models of this concept should be cautious not to leak any information, to the extent possible by the environment. For example, it might be advisable to explicitly clear any temporary storage as soon as it is no longer needed.

## Pseudo-Random Number Generator

A pseudo-random number generator is a UniformRandomNumberGenerator which provides a deterministic sequence of pseudo-random numbers, based on some algorithm and internal state. Linear congruential and inversive congruential generators are examples of such pseudo-random number generators. Often, these generators are very sensitive to their parameters. In order to prevent wrong implementations from being used, an external testsuite should check that the generated sequence and the validation value provided do indeed match.

Donald E. Knuth gives an extensive overview on pseudo-random number generation in his book "The Art of Computer Programming, Vol. 2, 3rd edition, Addison-Wesley, 1997". The descriptions for the specific generators contain additional references.

Note: Because the state of a pseudo-random number generator is necessarily finite, the sequence of numbers returned by the generator will loop eventually.

In addition to the UniformRandomNumberGenerator requirements, a pseudo-random number generator has some additional requirements. In the following table, `X` denotes a pseudo-random number generator class returning objects of type `T`, `x` is a value of `T`, `u` is a value of `X`, and `v` is a `const` value of `X`.

`PseudoRandomNumberGenerator` requirements
expression | return type | pre/post-condition
`X()` | - | creates a generator in some implementation-defined state. Note: Several generators thusly created may possibly produce dependent or identical sequences of random numbers.
`explicit X(...)` | - | creates a generator with user-provided state; the implementation shall specify the constructor argument(s)
`u.seed(...)` | void | sets the current state according to the argument(s); at least functions with the same signature as the non-default constructor(s) shall be provided.
`X::validation(x)` | `bool` | compares the pre-computed and hardcoded 10001th element in the generator's random number sequence with `x`. The generator must have been constructed by its default constructor and `seed` must not have been called for the validation to be meaningful.

Note: The `seed` member function is similar to the `assign` member function in STL containers. However, the naming did not seem appropriate.

Classes which model a pseudo-random number generator shall also model EqualityComparable, i.e. implement `operator==`. Two pseudo-random number generators are defined to be equivalent if they both return an identical sequence of numbers starting from a given state.

Classes which model a pseudo-random number generator should also model the Streamable concept, i.e. implement `operator<<` and `operator>>`. If so, `operator<<` writes all current state of the pseudo-random number generator to the given `ostream` so that `operator>>` can restore the state at a later time. The state shall be written in a platform-independent manner, but it is assumed that the `locale`s used for writing and reading be the same. The pseudo-random number generator with the restored state and the original at the just-written state shall be equivalent.

Classes which model a pseudo-random number generator may also model the CopyConstructible and Assignable concepts. However, note that the sequences of the original and the copy are strongly correlated (in fact, they are identical), which may make them unsuitable for some problem domains. Thus, copying pseudo-random number generators is discouraged; they should always be passed by (non-`const`) reference.

The classes `rand48`, `minstd_rand`, and `mt19937` are models for a pseudo-random number generator.

Note: This type of random-number generator is useful for numerics, games and testing. The non-zero arguments constructor(s) and the `seed()` member function(s) allow for a user-provided state to be installed in the generator. This is useful for debugging Monte-Carlo algorithms and analyzing particular test scenarios. The Streamable concept allows to save/restore the state of the generator, for example to re-run a test suite at a later time.

## Random Distribution

A radom distribution produces random numbers distributed according to some distribution, given uniformly distributed random values as input. In the following table, `X` denotes a random distribution class returning objects of type `T`, `u` is a value of `X`, `x` is a (possibly const) value of `X`, and `e` is an lvalue of an arbitrary type that meets the requirements of a uniform random number generator, returning values of type `U`.

Random distribution requirements (in addition to number generator, `CopyConstructible`, and `Assignable`)
expression | return type | pre/post-condition | complexity
`X::input_type` | U | - | compile-time
`u.reset()` | `void` | subsequent uses of `u` do not depend on values produced by `e` prior to invoking `reset`. | constant
`u(e)` | `T` | the sequence of numbers returned by successive invocations with the same object `e` is randomly distributed with some probability density function p(x) | amortized constant number of invocations of `e`
`os << x` | `std::ostream&` | writes a textual representation for the parameters and additional internal data of the distribution `x` to `os`. post: The `os.fmtflags` and fill character are unchanged. | O(size of state)
`is >> u` | `std::istream&` | restores the parameters and additional internal data of the distribution `u`. pre: `is` provides a textual representation that was previously written by `operator<<` post: The `is.fmtflags` are unchanged. | O(size of state)

Additional requirements: The sequence of numbers produced by repeated invocations of `x(e)` does not change whether or not `os << x` is invoked between any of the invocations `x(e)`. If a textual representation is written using `os << x` and that representation is restored into the same or a different object `y` of the same type using `is >> y`, repeated invocations of `y(e)` produce the same sequence of random numbers as would repeated invocations of `x(e)`.

## Quasi-Random Number Generators

A quasi-random number generator is a Number Generator which provides a deterministic sequence of numbers, based on some algorithm and internal state. The numbers do not have any statistical properties (such as uniform distribution or independence of successive values).

Note: Quasi-random number generators are useful for Monte-Carlo integrations where specially crafted sequences of random numbers will make the approximation converge faster.

[Does anyone have a model?]

Revised 05 December, 2006

Copyright © 2000-2003 Jens Maurer

Distributed under the Boost Software License, Version 1.0. (See accompanying file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)