A computer is a programmable machine designed to sequentially and automatically carry out a sequence of arithmetic or logical operations. The particular sequence of operations can be changed readily, allowing the computer to solve more than one kind of problem...
or physical device designed to generate a sequence of number
Number
A number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....
s or symbols that lack any pattern, i.e. appear random.
Randomness has many uses in gambling, statistics, cryptography, art, etc.These uses have different randomness requirements, which leads to the use of different randomization methods...
have led to the development of several different methods for generating random
Randomness
Randomness has somewhat differing meanings as used in various fields. It also has common meanings which are connected to the notion of predictability of events....
data. Many of these have existed since ancient times, including dice
Dice
A die is a small throwable object with multiple resting positions, used for generating random numbers...
Coin flipping or coin tossing or heads or tails is the practice of throwing a coin in the air to choose between two alternatives, sometimes to resolve a dispute between two parties...
A playing card is a piece of specially prepared heavy paper, thin cardboard, plastic-coated paper, cotton-paper blend, or thin plastic, marked with distinguishing motifs and used as one of a set for playing card games...
Achillea millefolium or yarrow is a flowering plant in the family Asteraceae, native to the Northern Hemisphere. In New Mexico and southern Colorado, it is called plumajillo, or "little feather", for the shape of the leaves. In antiquity, yarrow was known as herbal militaris, for its use in...
The I Ching or "Yì Jīng" , also known as the Classic of Changes, Book of Changes and Zhouyi, is one of the oldest of the Chinese classic texts...
, and many other techniques. Because of the mechanical nature of these techniques, generating large amounts of sufficiently random numbers (important in statistics) required a lot of work and/or time. Thus, results would sometimes be collected and distributed as random number table
Random number table
Random number tables have been used in statistics for tasks such as selected random samples. This was much more effective than manually selecting the random samples...
s. Nowadays, after the advent of computational random number generators, a growing number of government-run lotteries
Lottery
A lottery is a form of gambling which involves the drawing of lots for a prize.Lottery is outlawed by some governments, while others endorse it to the extent of organizing a national or state lottery. It is common to find some degree of regulation of lottery by governments...
, and lottery games, are using RNGs instead of more traditional drawing methods. RNGs are also used today to determine the odds of modern slot machine
Slot machine
A slot machine , informally fruit machine , the slots , poker machine or "pokies" or simply slot is a casino gambling machine with three or more reels which spin when a button is pushed...
s.
Several computational methods for random number generation exist, but often fall short of the goal of true randomness — though they may meet, with varying success, some of the statistical tests for randomness
Statistical randomness
A numeric sequence is said to be statistically random when it contains no recognizable patterns or regularities; sequences such as the results of an ideal dice roll, or the digits of π exhibit statistical randomness....
intended to measure how unpredictable their results are (that is, to what degree their patterns are discernible).
Practical applications and uses
Random number generators have applications in gambling
Gambling
Gambling is the wagering of money or something of material value on an event with an uncertain outcome with the primary intent of winning additional money and/or material goods...
A computer simulation, a computer model, or a computational model is a computer program, or network of computers, that attempts to simulate an abstract model of a particular system...
In the design of experiments, completely randomized designs are for studying the effects of one primary factor without the need to take other nuisance variables into account. This article describes completely randomized designs that have one primary factor. The experiment compares the values of a...
, and other areas where producing an unpredictable result is desirable.
Note that, in general, where unpredictability is paramount — such as in security applications — hardware generators are generally preferred (where feasible) over pseudo-random algorithms.
Monte Carlo methods are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used in computer simulations of physical and mathematical systems...
Debugging is a methodical process of finding and reducing the number of bugs, or defects, in a computer program or a piece of electronic hardware, thus making it behave as expected. Debugging tends to be harder when various subsystems are tightly coupled, as changes in one may cause bugs to emerge...
is facilitated by the ability to run the same sequence of random numbers again by starting from the same random seed
Random seed
A random seed is a number used to initialize a pseudorandom number generator.The choice of a good random seed is crucial in the field of computer security...
Cryptography is the practice and study of techniques for secure communication in the presence of third parties...
- so long as the seed is secret. Sender and receiver can generate the same set of numbers automatically to use as keys.
The generation of pseudo-random numbers is an important and common task in computer programming. While cryptography and certain numerical algorithms require a very high degree of apparent randomness, many other operations only need a modest amount of unpredictability. Some simple examples might be presenting a user with a "Random Quote of the Day", or determining which way a computer-controlled adversary might move in a computer game. Weaker forms of randomness also feature in hash algorithms and in creating amortized
Amortization
Amortization is the process of decreasing, or accounting for, an amount over a period. The word comes from Middle English amortisen to kill, alienate in mortmain, from Anglo-French amorteser, alteration of amortir, from Vulgar Latin admortire to kill, from Latin ad- + mort-, mors death.When used...
In computer science, a search algorithm is an algorithm for finding an item with specified properties among a collection of items. The items may be stored individually as records in a database; or may be elements of a search space defined by a mathematical formula or procedure, such as the roots...
In computer science, a sorting algorithm is an algorithm that puts elements of a list in a certain order. The most-used orders are numerical order and lexicographical order...
s.
Some applications which appear at first sight to be suitable for randomization are in fact not quite so simple. For instance, a system that "randomly" selects music tracks for a background music system must only appear random, and may even have ways to control the selection of music: a true random system would have no restriction on the same item appearing two or three times in succession.
"True" random numbers vs. pseudorandom numbers
There are two principal methods used to generate random numbers. One measures some physical phenomenon that is expected to be random and then compensates for possible biases in the measurement process. The other uses computational algorithm
Algorithm
In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...
s that produce long sequences of apparently random results, which are in fact completely determined by a shorter initial value, known as a seed or key
Key (cryptography)
In cryptography, a key is a piece of information that determines the functional output of a cryptographic algorithm or cipher. Without a key, the algorithm would produce no useful result. In encryption, a key specifies the particular transformation of plaintext into ciphertext, or vice versa...
A pseudorandom number generator , also known as a deterministic random bit generator , is an algorithm for generating a sequence of numbers that approximates the properties of random numbers...
s.
A "random number generator" based solely on deterministic computation cannot be regarded as a "true" random number generator, since its output is inherently predictable. How to distinguish a "true" random number from the output of a pseudo-random number generator is a very difficult problem. However, carefully chosen pseudo-random number generators can be used instead of true random numbers in many applications. Rigorous statistical analysis of the output is often needed to have confidence in the algorithm.
Physical methods
The earliest methods for generating random numbers — dice
Dice
A die is a small throwable object with multiple resting positions, used for generating random numbers...
Coin flipping or coin tossing or heads or tails is the practice of throwing a coin in the air to choose between two alternatives, sometimes to resolve a dispute between two parties...
Roulette is a casino game named after a French diminutive for little wheel. In the game, players may choose to place bets on either a single number or a range of numbers, the colors red or black, or whether the number is odd or even....
A game is structured playing, usually undertaken for enjoyment and sometimes used as an educational tool. Games are distinct from work, which is usually carried out for remuneration, and from art, which is more often an expression of aesthetic or ideological elements...
s and gambling as they tend to be too slow for most applications in statistics and cryptography.
A physical random number generator can be based on an essentially random atomic or subatomic physical phenomenon whose unpredictability can be traced to the laws of quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
Radioactive decay is the process by which an atomic nucleus of an unstable atom loses energy by emitting ionizing particles . The emission is spontaneous, in that the atom decays without any physical interaction with another particle from outside the atom...
Johnson–Nyquist noise is the electronic noise generated by the thermal agitation of the charge carriers inside an electrical conductor at equilibrium, which happens regardless of any applied voltage...
Shot noise is a type of electronic noise that may be dominant when the finite number of particles that carry energy is sufficiently small so that uncertainties due to the Poisson distribution, which describes the occurrence of independent random events, are of significance...
A Zener diode is a special kind of diode which allows current to flow in the forward direction in the same manner as an ideal diode, but will also permit it to flow in the reverse direction when the voltage is above a certain value known as the breakdown voltage, "Zener knee voltage" or "Zener...
s, clock drift, the timing of actual movements of a hard disk
Hard disk
A hard disk drive is a non-volatile, random access digital magnetic data storage device. It features rotating rigid platters on a motor-driven spindle within a protective enclosure. Data is magnetically read from and written to the platter by read/write heads that float on a film of air above the...
In radio reception, noise is the superposition of white noise and other disturbing influences on the signal, caused either by thermal noise and other electronic noise from receiver input circuits or by interference from radiated electromagnetic noise picked up by the receiver's antenna...
. However, physical phenomena and tools used to measure them generally feature asymmetries and systematic biases that make their outcomes not uniformly random. A randomness extractor
Randomness extractor
A randomness extractor, often simply called "an extractor," is a function which, when applied to a high-entropy source , generates a random output that is shorter, but uniformly distributed...
A cryptographic hash function is a deterministic procedure that takes an arbitrary block of data and returns a fixed-size bit string, the hash value, such that an accidental or intentional change to the data will change the hash value...
, can be used to approach a uniform distribution of bits from a non-uniformly random source, though at a lower bit rate.
In 2010 a team at Bar-Ilan University in Israel was able to create a physical random bit generator at a 300 Gbit/s rate, making it the fastest ever.
Various imaginative ways of collecting this entropic information have been devised. One technique is to run a hash function against a frame of a video stream from an unpredictable source. Lavarand
Lavarand
Lavarand was Silicon Graphics' name for its hardware random number generator that worked by taking pictures of the patterns made by the floating material in lava lamps, extracting random data from the pictures, and using the result to seed a pseudo-random number generator...
used this technique with images of a number of lava lamp
Lava lamp
A lava lamp is a decorative novelty item that contains blobs of colored wax inside a glass vessel filled with clear liquid; the wax rises and falls as its density changes due to heating from a incandescent light bulb underneath the vessel. The appearance of the wax is suggestive of pāhoehoe lava,...
A Geiger–Müller tube is the sensing element of a Geiger counter instrument that can detect a single particle of ionizing radiation, and typically produce an audible click for each. It was named for Hans Geiger who invented the device in 1908, and Walther Müller who collaborated with Geiger in...
Random.org is a website that produces "true" random numbers based on atmospheric noise.In addition to generating random numbers, it has free tools to do things such as flip coins, shuffle cards, and roll dice...
uses variations in the amplitude of atmospheric noise recorded with a normal radio.
Another common entropy source is the behavior of human users of the system. While people are not considered good randomness generators upon request, they generate random behavior quite well in the context of playing mixed strategy games. Some security-related computer software requires the user to make a lengthy series of mouse movements or keyboard inputs to create sufficient entropy needed to generate random keys
Key (cryptography)
In cryptography, a key is a piece of information that determines the functional output of a cryptographic algorithm or cipher. Without a key, the algorithm would produce no useful result. In encryption, a key specifies the particular transformation of plaintext into ciphertext, or vice versa...
or to initialize pseudorandom number generators.
Computational methods
Pseudo-random number generators (PRNGs) are algorithm
Algorithm
In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...
s that can automatically create long runs of numbers with good random properties but eventually the sequence repeats (or the memory usage grows without bound). The string of values generated by such algorithms is generally determined by a fixed number called a seed. One of the most common PRNG is the linear congruential generator
Linear congruential generator
A Linear Congruential Generator represents one of the oldest and best-known pseudorandom number generator algorithms. The theory behind them is easy to understand, and they are easily implemented and fast....
, which uses the recurrence
to generate numbers. The maximum number of numbers the formula can produce is the modulus
Modulus (algebraic number theory)
In mathematics, in the field of algebraic number theory, a modulus is a formal product of places of a global field...
, m. To avoid certain non-random properties of a single linear congruential generator, several such random number generators with slightly different values of the multiplier coefficient a can be used in parallel, with a "master" random number generator that selects from among the several different generators.
A simple pen-and-paper method for generating random numbers is the so-called middle square method suggested by John Von Neumann
John von Neumann
John von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...
. While simple to implement, its output is of poor quality.
Most computer programming languages include functions or library routines that purport to be random number generators. They are often designed to provide a random byte or word, or a floating point
Floating point
In computing, floating point describes a method of representing real numbers in a way that can support a wide range of values. Numbers are, in general, represented approximately to a fixed number of significant digits and scaled using an exponent. The base for the scaling is normally 2, 10 or 16...
In probability theory and statistics, the continuous uniform distribution or rectangular distribution is a family of probability distributions such that for each member of the family, all intervals of the same length on the distribution's support are equally probable. The support is defined by...
between 0 and 1.
Such library functions often have poor statistical properties and some will repeat patterns after only tens of thousands of trials. They are often initialized using a computer's real time clock as the seed, since such a clock generally measures in milliseconds, far beyond the person's precision
Accuracy and precision
In the fields of science, engineering, industry and statistics, the accuracy of a measurement system is the degree of closeness of measurements of a quantity to that quantity's actual value. The precision of a measurement system, also called reproducibility or repeatability, is the degree to which...
. These functions may provide enough randomness for certain tasks (for example video games) but are unsuitable where high-quality randomness is required, such as in cryptographic applications, statistics or numerical analysis. Better pseudo-random number generators such as the Mersenne Twister are widely available. Much higher quality random number sources are available on most operating systems; for example /dev/random
/dev/random
In Unix-like operating systems, /dev/random is a special file that serves as a random number generator or as a pseudorandom number generator. It allows access to environmental noise collected from device drivers and other sources. Not all operating systems implement the same semantics for /dev/random...
on various BSD flavors, Linux, Mac OS X, IRIX, and Solaris, or CryptGenRandom
CryptGenRandom
CryptGenRandom is a cryptographically secure pseudorandom number generator function that is included in Microsoft's Cryptographic Application Programming Interface. In Win32 programs, Microsoft recommends its use anywhere random number generation is needed...
for Microsoft Windows.
An example of a simple pseudo-random number generator is the Multiply-with-carry
Multiply-with-carry
In computer science, multiply-with-carry is a method invented by George Marsaglia for generating sequences of random integers based on an initial set of from two to many thousands of randomly chosen seed values...
George Marsaglia was an American mathematician and computer scientist. He established the lattice structure of congruential random number generators in the paper "Random numbers fall mainly in the planes". This phenomenon is sometimes called the Marsaglia effect...
. It is computationally fast and has good (albeit not cryptographically strong) randomness properties (note that this example is not thread safe):
m_w = ; /* must not be zero */
m_z = ; /* must not be zero */
In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...
. These methods involve transforming a uniform random number in some way. Because of this, these methods work equally well in generating both pseudo-random and true random numbers. One method, called the inversion method, involves integrating up to an area greater than or equal to the random number (which should be generated between 0 and 1 for proper distributions). A second method, called the acceptance-rejection method
Rejection sampling
In mathematics, rejection sampling is a basic pseudo-random number sampling technique used to generate observations from a distribution. It is also commonly called the acceptance-rejection method or "accept-reject algorithm"....
, involves choosing an x and y value and testing whether the function of x is greater than the y value. If it is, the x value is accepted. Otherwise, the x value is rejected and the algorithm tries again.
By humans
Random number generation may also be done by humans directly.
However, most studies find that human subjects have some degree of nonrandomness when generating a random sequence of, e.g., digits or letters.
They may alternate too much between choices compared to a good random generator.
Post-processing and statistical checks
Even given a source of plausible random numbers (perhaps from a quantum mechanically based hardware generator), obtaining numbers which are completely unbiased takes care. In addition, behavior of these generators often changes with temperature, power supply voltage, the age of the device, or other outside interference. And a software bug in a pseudo-random number routine, or a hardware bug in the hardware it runs on, may be similarly difficult to detect.
Generated random numbers are sometimes subjected to statistical tests before use to ensure that the underlying source is still working, and then post-processed to improve their statistical properties.
A numeric sequence is said to be statistically random when it contains no recognizable patterns or regularities; sequences such as the results of an ideal dice roll, or the digits of π exhibit statistical randomness....
Other considerations
Random numbers uniformly distributed between 0 and 1 can be used to generate random numbers of any desired distribution by passing them through the inverse cumulative distribution function
Cumulative distribution function
In probability theory and statistics, the cumulative distribution function , or just distribution function, describes the probability that a real-valued random variable X with a given probability distribution will be found at a value less than or equal to x. Intuitively, it is the "area so far"...
(CDF) of the desired distribution. Inverse CDFs are also called quantile function
Quantile function
In probability and statistics, the quantile function of the probability distribution of a random variable specifies, for a given probability, the value which the random variable will be at, or below, with that probability...
In probability theory, to say that two events are independent intuitively means that the occurrence of one event makes it neither more nor less probable that the other occurs...
In probability theory and statistics, the continuous uniform distribution or rectangular distribution is a family of probability distributions such that for each member of the family, all intervals of the same length on the distribution's support are equally probable. The support is defined by...
(see Box–Muller transform).
Some 0 to 1 RNGs include 0 but exclude 1, while others include or exclude both.
The outputs of multiple independent RNGs can be combined (for example, using a bit-wise XOR operation) to provide a combined RNG at least as good as the best RNG used. This is referred to as software whitening.
Computational and hardware random number generators are sometimes combined to reflect the benefits of both kinds. Computational random number generators can typically generate pseudo-random numbers much faster than physical generators, while physical generators can generate "true randomness."
Low-discrepancy sequences as an alternative
Some computations making use of a random number generator can be summarized as the computation of a total or average value, such as the computation of integrals by the Monte Carlo method
Monte Carlo method
Monte Carlo methods are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used in computer simulations of physical and mathematical systems...
. For such problems, it may be possible to find a more accurate solution by the use of so-called low-discrepancy sequence
Low-discrepancy sequence
In mathematics, a low-discrepancy sequence is a sequence with the property that for all values of N, its subsequence x1, ..., xN has a low discrepancy....
s, also called quasirandom numbers. Such sequences have a definite pattern that fills in gaps evenly, qualitatively speaking; a truly random sequence may, and usually does, leave larger gaps.
The Statistics Online Computational Resource is a suite of online tools and interactive aids for hands-on learning and teaching concepts in statistical analysis and probability theory developed at the University of California, Los Angeles...
Flipism, sometimes written as "Flippism," is a pseudophilosophy under which all decisions are made by flipping a coin. It originally appeared in the Disney comic "Flip Decision" by Carl Barks, published in 1953...
In computing, a hardware random number generator is an apparatus that generates random numbers from a physical process. Such devices are often based on microscopic phenomena that generate a low-level, statistically random "noise" signal, such as thermal noise or the photoelectric effect or other...
In complexity theory, PP is the class of decision problems solvable by a probabilistic Turing machine in polynomial time, with an error probability of less than 1/2 for all instances. The abbreviation PP refers to probabilistic polynomial time...
Procedural generation is a widely used term in the production of media; it refers to content generated algorithmically rather than manually. Often, this means creating content on the fly rather than prior to distribution...
Randomization is the process of making something random; this means:* Generating a random permutation of a sequence .* Selecting a random sample of a population ....
A randomized algorithm is an algorithm which employs a degree of randomness as part of its logic. The algorithm typically uses uniformly random bits as an auxiliary input to guide its behavior, in the hope of achieving good performance in the "average case" over all possible choices of random bits...
The security of cryptographic systems depends on some secret data that is known to authorized persons but unknown and unpredictable to others. To achieve this unpredictability, some randomization is typically employed...
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...
Randomness has somewhat differing meanings as used in various fields. It also has common meanings which are connected to the notion of predictability of events....
A slot machine , informally fruit machine , the slots , poker machine or "pokies" or simply slot is a casino gambling machine with three or more reels which spin when a button is pushed...
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