In a pseudorandom number generator (PRNG), a full cycle or full period is the behavior of a PRNG over its set of valid states. In particular, a PRNG is said to have a full cycle if, for any valid seed state, the PRNG traverses every valid state before returning to the seed state, i.e., the period is equal to the cardinality of the state space.
The restrictions on the parameters of a PRNG for it to possess a full cycle are known only for certain types of PRNGs, such as linear congruential generators and linear-feedback shift registers. There is no general method to determine whether a PRNG algorithm is full-cycle short of exhausting the state space, which may be exponentially large compared to the size of the algorithm's internal state.
Example 1 (in C/C++)
Given a random number seed that is greater or equal to zero, a total sample size greater than 1, and an increment coprime to the total sample size, a full cycle can be generated with the following logic. Each nonnegative number smaller than the sample size occurs exactly once.
# Generator that goes through a full cycledefcycle(seed:int,sample_size:int,increment:int):nb=seedforiinrange(sample_size):nb=(nb+increment)%sample_sizeyieldnb# Example valuesseed=17sample_size=100increment=13# Print all the numbersprint(list(cycle(seed,sample_size,increment)))# Verify that all numbers were generated correctlyassertset(cycle(seed,sample_size,increment))==set(range(sample_size))
A pseudorandom number generator (PRNG), also known as a deterministic random bit generator (DRBG), is an algorithm for generating a sequence of numbers whose properties approximate the properties of sequences of random numbers. The PRNG-generated sequence is not truly random, because it is completely determined by an initial value, called the PRNG's seed. Although sequences that are closer to truly random can be generated using hardware random number generators, pseudorandom number generators are important in practice for their speed in number generation and their reproducibility.
A linear congruential generator (LCG) is an algorithm that yields a sequence of pseudo-randomized numbers calculated with a discontinuous piecewise linear equation. The method represents one of the oldest and best-known pseudorandom number generator algorithms. The theory behind them is relatively easy to understand, and they are easily implemented and fast, especially on computer hardware which can provide modular arithmetic by storage-bit truncation.
The Mersenne Twister is a general-purpose pseudorandom number generator (PRNG) developed in 1997 by Makoto Matsumoto and Takuji Nishimura. Its name derives from the fact that its period length is chosen to be a Mersenne prime.
The Yarrow algorithm is a family of cryptographic pseudorandom number generators (CPRNG) devised by John Kelsey, Bruce Schneier, and Niels Ferguson and published in 1999. The Yarrow algorithm is explicitly unpatented, royalty-free, and open source; no license is required to use it. An improved design from Ferguson and Schneier, Fortuna, is described in their book, Practical Cryptography
In graph theory, a cycle in a graph is a non-empty trail in which only the first and last vertices are equal. A directed cycle in a directed graph is a non-empty directed trail in which only the first and last vertices are equal.
A cryptographically secure pseudorandom number generator (CSPRNG) or cryptographic pseudorandom number generator (CPRNG) is a pseudorandom number generator (PRNG) with properties that make it suitable for use in cryptography. It is also loosely known as a cryptographic random number generator (CRNG).
In mathematics, the middle-square method is a method of generating pseudorandom numbers. In practice it is not a good method, since its period is usually very short and it has some severe weaknesses; repeated enough times, the middle-square method will either begin repeatedly generating the same number or cycle to a previous number in the sequence and loop indefinitely.
A random password generator is software program or hardware device that takes input from a random or pseudo-random number generator and automatically generates a password. Random passwords can be generated manually, using simple sources of randomness such as dice or coins, or they can be generated using a computer.
Fortuna is a cryptographically secure pseudorandom number generator (PRNG) devised by Bruce Schneier and Niels Ferguson and published in 2003. It is named after Fortuna, the Roman goddess of chance. FreeBSD uses Fortuna for /dev/random and /dev/urandom is symbolically linked to it since FreeBSD 11. Apple OSes have switched to Fortuna since 2020 Q1.
Random number generation is a process by which, often by means of a random number generator (RNG), a sequence of numbers or symbols that cannot be reasonably predicted better than by random chance is generated. This means that the particular outcome sequence will contain some patterns detectable in hindsight but unpredictable to foresight. True random number generators can be hardware random-number generators (HRNGS) that generate random numbers, wherein each generation is a function of the current value of a physical environment's attribute that is constantly changing in a manner that is practically impossible to model. This would be in contrast to so-called "random number generations" done by pseudorandom number generators (PRNGs) that generate numbers that only look random but are in fact pre-determined—these generations can be reproduced simply by knowing the state of the PRNG.
In computer science, recursion is a method of solving a computational problem where the solution depends on solutions to smaller instances of the same problem. Recursion solves such recursive problems by using functions that call themselves from within their own code. The approach can be applied to many types of problems, and recursion is one of the central ideas of computer science.
The power of recursion evidently lies in the possibility of defining an infinite set of objects by a finite statement. In the same manner, an infinite number of computations can be described by a finite recursive program, even if this program contains no explicit repetitions.
A randomness test, in data evaluation, is a test used to analyze the distribution of a set of data to see if it can be described as random (patternless). In stochastic modeling, as in some computer simulations, the hoped-for randomness of potential input data can be verified, by a formal test for randomness, to show that the data are valid for use in simulation runs. In some cases, data reveals an obvious non-random pattern, as with so-called "runs in the data". If a selected set of data fails the tests, then parameters can be changed or other randomized data can be used which does pass the tests for randomness.
In computer science, multiply-with-carry (MWC) is a method invented by George Marsaglia for generating sequences of random integers based on an initial set from two to many thousands of randomly chosen seed values. The main advantages of the MWC method are that it invokes simple computer integer arithmetic and leads to very fast generation of sequences of random numbers with immense periods, ranging from around to .
The Fisher–Yates shuffle is an algorithm for generating a random permutation of a finite sequence—in plain terms, the algorithm shuffles the sequence. The algorithm effectively puts all the elements into a hat; it continually determines the next element by randomly drawing an element from the hat until no elements remain. The algorithm produces an unbiased permutation: every permutation is equally likely. The modern version of the algorithm is efficient: it takes time proportional to the number of items being shuffled and shuffles them in place.
The Lehmer random number generator, sometimes also referred to as the Park–Miller random number generator, is a type of linear congruential generator (LCG) that operates in multiplicative group of integers modulo n. The general formula is:
Wichmann–Hill is a pseudorandom number generator proposed in 1982 by Brian Wichmann and David Hill. It consists of three linear congruential generators with different prime moduli, each of which is used to produce a uniformly distributed number between 0 and 1. These are summed, modulo 1, to produce the result.
A permuted congruential generator (PCG) is a pseudorandom number generation algorithm developed in 2014 which applies an output permutation function to improve the statistical properties of a modulo-2n linear congruential generator. It achieves excellent statistical performance with small and fast code, and small state size.
The ACORN or ″Additive Congruential Random Number″ generators are a robust family of PRNGs for sequences of uniformly distributed pseudo-random numbers, introduced in 1989 and still valid in 2019, thirty years later.
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