A pseudorandom process is a process that appears to be 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....
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....
while being generated by an entirely deterministic causal process. Such a process is easier to produce than a genuinely random one, and has the benefit that it can be used again and again to produce exactly the same numbers - useful for testing and fixing software.
To generate truly random numbers requires precise, accurate, and repeatable system measurements of absolutely non-deterministic processes. Linux
Linux
Linux is a Unix-like computer operating system assembled under the model of free and open source software development and distribution. The defining component of any Linux system is the Linux kernel, an operating system kernel first released October 5, 1991 by Linus Torvalds...
uses, for example, various system timings (like user keystrokes, I/O, or least-significant digit voltage measurements) to produce a pool of random numbers. It attempts to constantly replenish the pool, depending on the level of importance, and so will issue a random number. This system is an example, and similar to those of dedicated hardware random number generators.
History
The generation of random numbers has many uses (mostly in statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
In statistics and survey methodology, sampling is concerned with the selection of a subset of individuals from within a population to estimate characteristics of the whole population....
Simulation is the imitation of some real thing available, state of affairs, or process. The act of simulating something generally entails representing certain key characteristics or behaviours of a selected physical or abstract system....
). Before modern computing, researchers requiring random numbers would either generate them through various means (dice
Dice
A die is a small throwable object with multiple resting positions, used for generating random numbers...
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....
, etc.) or use existing random number tables.
The first attempt to provide researchers with a ready supply of random digits was in 1927, when the Cambridge University Press published a table of 41,600 digits developed by Leonard H.C. Tippet. In 1947, the RAND Corporation
RAND
RAND Corporation is a nonprofit global policy think tank first formed to offer research and analysis to the United States armed forces by Douglas Aircraft Company. It is currently financed by the U.S. government and private endowment, corporations including the healthcare industry, universities...
A Million Random Digits with 100,000 Normal Deviates
A Million Random Digits with 100,000 Normal Deviates is a 1955 book by the RAND Corporation. The book, consisting primarily of a random number table, was an important 20th century work in the field of statistics and random numbers...
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,...
was a pioneer in computer-based random number generators. A notable contributor in the field of pseudorandom number generation in practical use is a Pakistani mathematician Dr. Arif Zaman
Arif Zaman
Arif Zaman, Ph.D., is a Pakistani mathematician and an academic scientist. He is the Professor of Statistics and Mathematics and as well as Chairman of the Department of Mathematics at the School of Science and Engineering of the Lahore University of Management Sciences...
Derrick Henry "Dick" Lehmer was an American mathematician who refined Édouard Lucas' work in the 1930s and devised the Lucas–Lehmer test for Mersenne primes...
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....
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 today. With the spread of the use of computers, algorithmic pseudorandom number generators replaced random number tables, and "true" random number generators (hardware random number generator
Hardware random number generator
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...
s) are only used in a few cases.
Almost random
A pseudorandom variable is a variable which is created by a deterministic procedure (often a computer program or subroutine) which (generally) takes random bits as input. The pseudorandom string will typically be longer than the original random string, but less random (less entropic
Information entropy
In information theory, entropy is a measure of the uncertainty associated with a random variable. In this context, the term usually refers to the Shannon entropy, which quantifies the expected value of the information contained in a message, usually in units such as bits...
Information theory is a branch of applied mathematics and electrical engineering involving the quantification of information. Information theory was developed by Claude E. Shannon to find fundamental limits on signal processing operations such as compressing data and on reliably storing and...
sense). This can be useful for randomized algorithms.
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 are widely used in such applications as computer modeling (e.g., Markov chain
Markov chain
A Markov chain, named after Andrey Markov, is a mathematical system that undergoes transitions from one state to another, between a finite or countable number of possible states. It is a random process characterized as memoryless: the next state depends only on the current state and not on the...
Theoretical computer science is a division or subset of general computer science and mathematics which focuses on more abstract or mathematical aspects of computing....
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
is pseudorandom against a class of adversaries if no adversary from the class can distinguish it from the uniform distribution with significant advantage.
This notion of pseudorandomness is studied in computational complexity theory
Computational complexity theory
Computational complexity theory is a branch of the theory of computation in theoretical computer science and mathematics that focuses on classifying computational problems according to their inherent difficulty, and relating those classes to each other...
Cryptography is the practice and study of techniques for secure communication in the presence of third parties...
.
Formally, let S and T be finite sets and let F = {f: S → T} be a class of functions. A distribution
Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
D over S is ε-pseudorandom against F if for every f in F, the statistical distance between the distributions f(X), where X is sampled from D, and f(Y), where Y is sampled from the uniform distribution on S, is at most ε.
In typical applications, the class F describes a model of computation with bounded resources
and one is interested in designing distributions D with certain properties that are pseudorandom against F. The distribution D is often specified as the output of a pseudorandom generator
Pseudorandom generator
In theoretical computer science, a pseudorandom generator is a deterministic procedure that produces a pseudorandom distribution from a short uniform input, known as a random seed.-Definition:...
Cryptography is the practice and study of techniques for secure communication in the presence of third parties...
, the use of pseudorandom number generators (whether hardware or software or some combination) is insecure. When random values are required in cryptography, the goal is to make a message as hard to crack as possible, by eliminating or obscuring the parameters used to encrypt the message (the 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...
) from the message itself or from the context in which it is carried. Pseudorandom sequences are deterministic and reproducible; all that is required in order to discover and reproduce a pseudorandom sequence is the algorithm used to generate it and the initial seed. So the entire sequence of numbers is only as powerful as the randomly chosen parts - sometimes the algorithm and the seed, but usually only the seed.
There are many examples in cryptographic history of cyphers, otherwise excellent, in which random choices were not random enough and security was lost as a direct consequence. The World War II
World War II
World War II, or the Second World War , was a global conflict lasting from 1939 to 1945, involving most of the world's nations—including all of the great powers—eventually forming two opposing military alliances: the Allies and the Axis...
Japan is an island nation in East Asia. Located in the Pacific Ocean, it lies to the east of the Sea of Japan, China, North Korea, South Korea and Russia, stretching from the Sea of Okhotsk in the north to the East China Sea and Taiwan in the south...
In the history of cryptography, 97-shiki ōbun inji-ki or Angōki Taipu-B , codenamed Purple by the United States, was a diplomatic cryptographic machine used by the Japanese Foreign Office just before and during World War II...
cypher machine used for diplomatic communications is a good example. It was consistently broken throughout WWII, mostly because the "key values" used were insufficiently random. They had patterns, and those patterns made any intercepted traffic readily decryptable. Had the keys (i.e. the initial settings of the stepping switches in the machine) been made unpredictably (i.e. randomly), that traffic would have been much harder to break, and perhaps even secure in practice.
Users and designers of cryptography are strongly cautioned to treat their randomness needs with the utmost care. Absolutely nothing has changed with the era of computerized cryptography, except that patterns in pseudorandom data are easier to discover than ever before. Randomness is, if anything, more important than ever.
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...
simulation is defined as any method that utilizes sequences of random numbers to perform the simulation. Monte Carlo simulations are applied to many topics including quantum chromodynamics
Quantum chromodynamics
In theoretical physics, quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons . It is the study of the SU Yang–Mills theory of color-charged fermions...
Stellar evolution is the process by which a star undergoes a sequence of radical changes during its lifetime. Depending on the mass of the star, this lifetime ranges from only a few million years to trillions of years .Stellar evolution is not studied by observing the life of a single...
and VLSI design. All these simulations require the use of random numbers and therefore pseudorandom number generator
Pseudorandom number generator
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, which makes creating random-like numbers very important.
A simple example of how a computer would perform a Monte Carlo simulation is the calculation of π
Pi
' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...
. If a square enclosed a circle and a point were randomly chosen inside the square the point would either lie inside the circle or outside it. If the process were repeated many times, the ratio of the random points that lie inside the circle to the total number of random points in the square would approximate the ratio of the area of the circle to the area of the square. From this we can estimate pi, as shown in the Python
Python (programming language)
Python is a general-purpose, high-level programming language whose design philosophy emphasizes code readability. Python claims to "[combine] remarkable power with very clear syntax", and its standard library is large and comprehensive...
SciPy is an open source library of algorithms and mathematical tools for the Python programming language.SciPy contains modules for optimization, linear algebra, integration, interpolation, special functions, FFT, signal and image processing, ODE solvers and other tasks common in science and...
package to generate pseudorandom numbers with the MT19937 algorithm. Note that this method is a computationally inefficient way to numerically approximate π
Numerical approximations of π
This page is about the history of approximations for the mathematical constant pi . There is a table summarizing the chronology of computation of π. See also the history of pi for other aspects of the evolution of our knowledge about mathematical properties of pi...
A disperser is a one-sided extractor. Where an extractor requires that every event gets the same probability under the uniform distribution and the extracted distribution, only the latter is required for a disperser...
In combinatorics, an expander graph is a sparse graph that has strong connectivity properties, quantified using vertex, edge or spectral expansion as described below...
In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...
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...
The Art of Computer Programming is a comprehensive monograph written by Donald Knuth that covers many kinds of programming algorithms and their analysis....
