# Pseudorandomness

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A pseudorandom sequence of numbers is one that appears to be statistically random, despite having been produced by a completely deterministic and repeatable process. [1]

## Background

The generation of random numbers has many uses, such as for random sampling, Monte Carlo methods, board games, or gambling. In physics, however, most processes, such as gravitational acceleration, are deterministic, meaning that they always produce the same outcome from the same starting point. Some notable exceptions are radioactive decay and quantum measurement, which are both modeled as being truly random processes in the underlying physics. Since these processes are not practical sources of random numbers, people use pseudorandom numbers, which ideally have the unpredictability of a truly random sequence, despite being generated by a deterministic process. [2]

In many applications, the deterministic process is a computer algorithm called a pseudorandom number generator, which must first be provided with a number called a random seed. Since the same seed will yield the same sequence every time, it is important that the seed be well chosen and kept hidden, especially in security applications, where the pattern's unpredictability is a critical feature. [3]

In some cases where it is important for the sequence to be demonstrably unpredictable, people have used physical sources of random numbers, such as radioactive decay, atmospheric electromagnetic noise harvested from a radio tuned between stations, or intermixed timings of people's keystrokes. [1] [4] The time investment needed to obtain these numbers leads to a compromise: using some of these physics readings as a seed for a pseudorandom number generator.

## History

Before modern computing, researchers requiring random numbers would either generate them through various means (dice, cards, roulette wheels, [5] 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 L.H.C. Tippett. In 1947, the RAND Corporation generated numbers by the electronic simulation of a roulette wheel; [5] the results were eventually published in 1955 as A Million Random Digits with 100,000 Normal Deviates .

## In computational complexity

In theoretical computer science, a distribution is pseudorandom against a class of adversaries if no adversary from the class can distinguish it from the uniform distribution with significant advantage. [6] This notion of pseudorandomness is studied in computational complexity theory and has applications to cryptography.

Formally, let S and T be finite sets and let F = {f: ST} be a class of functions. A distribution D over S is ε-pseudorandom against F if for every f in F, the statistical distance between the distributions and ${\displaystyle f(X)}$, where and ${\displaystyle X}$ is sampled from D, and ${\displaystyle f(Y)}$, where and ${\displaystyle Y}$ 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. [7]

## Related Research Articles

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.

In computing, a hardware random number generator (HRNG) or true random number generator (TRNG) is a device that generates random numbers from a physical process, rather than by means of an algorithm. Such devices are often based on microscopic phenomena that generate low-level, statistically random "noise" signals, such as thermal noise, the photoelectric effect, involving a beam splitter, and other quantum phenomena. These stochastic processes are, in theory, completely unpredictable for as long as an equation governing such phenomena is unknown or uncomputable, and the theory's assertions of unpredictability are subject to experimental test. This is in contrast to the paradigm of pseudo-random number generation commonly implemented in computer programs.

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, computer science and physics, a deterministic system is a system in which no randomness is involved in the development of future states of the system. A deterministic model will thus always produce the same output from a given starting condition or initial state.

In computer science, a deterministic algorithm is an algorithm that, given a particular input, will always produce the same output, with the underlying machine always passing through the same sequence of states. Deterministic algorithms are by far the most studied and familiar kind of algorithm, as well as one of the most practical, since they can be run on real machines efficiently.

In cryptography, a hard-core predicate of a one-way function f is a predicate b which is easy to compute but is hard to compute given f(x). In formal terms, there is no probabilistic polynomial-time (PPT) algorithm that computes b(x) from f(x) with probability significantly greater than one half over random choice of x. In other words, if x is drawn uniformly at random, then given f(x), any PPT adversary can only distinguish the hard-core bit b(x) and a uniformly random bit with negligible advantage over the length of x.

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. Modern cryptographic protocols often require frequent generation of random quantities. Cryptographic attacks that subvert or exploit weaknesses in this process are known as random number generator attacks.

In computational complexity and cryptography, two families of distributions are computationally indistinguishable if no efficient algorithm can tell the difference between them except with small probability.

In theoretical computer science and cryptography, a pseudorandom generator (PRG) for a class of statistical tests is a deterministic procedure that maps a random seed to a longer pseudorandom string such that no statistical test in the class can distinguish between the output of the generator and the uniform distribution. The random seed itself is typically a short binary string drawn from the uniform distribution.

In cryptography, a verifiable random function (VRF) is a pseudo-random function that provides publicly verifiable proofs of its outputs' correctness. Given an input value x, the owner of the secret key SK can compute the function value y = FSK(x) and the proof pSK(x). Using the proof and the public key PK, everyone can check that the value y = FSK(x) was indeed computed correctly, yet this information cannot be used to find the secret key. A verifiable random function can be viewed as a public-key analogue of a keyed cryptographic hash and as a cryptographic commitment to an exponentisally large number of seemingly random bits. The concept of a verifiable random function is closely related to that of a verifiable unpredictable function (VUF), whose outputs are hard to predict but do not necessarily seem random.

A random seed is a number used to initialize a pseudorandom number generator.

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.

Random number generation is a process which, often by means of a random number generator (RNG), generates a sequence of numbers or symbols that cannot be reasonably predicted better than by a random chance. This means that the particular outcome sequence will contain some patterns, that are detectable in hindsight, however not predictable with foresight. Random number generators can be truly random hardware random-number generators (HRNGS), which generate random numbers as a function of current value of some physical environment attribute that is constantly changing in a manner that is practically impossible to model, or pseudorandom number generators (PRNGS), which generate numbers that look random, but are actually deterministic, and can be reproduced if the state of the PRNG is known.

Oded Goldreich is a professor of Computer Science at the Faculty of Mathematics and Computer Science of Weizmann Institute of Science, Israel. His research interests lie within the theory of computation and are, specifically, the interplay of randomness and computation, the foundations of cryptography, and computational complexity theory. He won the Knuth Prize in 2017.

In cryptography, a pseudorandom function family, abbreviated PRF, is a collection of efficiently-computable functions which emulate a random oracle in the following way: no efficient algorithm can distinguish between a function chosen randomly from the PRF family and a random oracle. Pseudorandom functions are vital tools in the construction of cryptographic primitives, especially secure encryption schemes.

A randomness extractor, often simply called an "extractor", is a function, which being applied to output from a weakly random entropy source, together with a short, uniformly random seed, generates a highly random output that appears independent from the source and uniformly distributed. Examples of weakly random sources include radioactive decay or thermal noise; the only restriction on possible sources is that there is no way they can be fully controlled, calculated or predicted, and that a lower bound on their entropy rate can be established. For a given source, a randomness extractor can even be considered to be a true random number generator (TRNG); but there is no single extractor that has been proven to produce truly random output from any type of weakly random source.

In common parlance, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination. Individual random events are, by definition, unpredictable, but if the probability distribution is known, the frequency of different outcomes over repeated events is predictable. For example, when throwing two dice, the outcome of any particular roll is unpredictable, but a sum of 7 will tend to occur twice as often as 4. In this view, randomness is not haphazardness; it is a measure of uncertainty of an outcome. Randomness applies to concepts of chance, probability, and information entropy.

Salil Vadhan is an American computer scientist. He is Vicky Joseph Professor of Computer Science and Applied Mathematics at Harvard University. After completing his undergraduate degree in Mathematics and Computer Science at Harvard in 1995, he obtained his PhD in Applied Mathematics from Massachusetts Institute of Technology in 1999, where his advisor was Shafi Goldwasser. His research centers around the interface between computational complexity theory and cryptography. He focuses on the topics of pseudorandomness and zero-knowledge proofs. His work on the zig-zag product, with Omer Reingold and Avi Wigderson, was awarded the 2009 Gödel Prize.

In 1997, Moni Naor and Omer Reingold described efficient constructions for various cryptographic primitives in private key as well as public-key cryptography. Their result is the construction of an efficient pseudorandom function. Let p and l be prime numbers with l |p−1. Select an element g of multiplicative order l. Then for each (n+1)-dimensional vector a = ∈ they define the function

## References

1. George Johnson (June 12, 2001). "Connoisseurs of Chaos Offer A Valuable Product: Randomness". The New York Times .
2. S. P. Vadhan (2012). Pseudorandomness. pseudorandomness, the theory of efficiently generating objects that “look random” despite being constructed using little or no randomness
3. Mark Ward (August 9, 2015). "Web's random numbers are too weak, researchers warn". BBC.
4. Jonathan Knudson (January 1998). "Javatalk: Horseshoes, hand grenades and random numbers". Sun Server. pp. 16–17.
5. "A Million Random Digits". RAND Corporation. Retrieved March 30, 2017.
6. Oded Goldreich. Computational Complexity: A Conceptual Perspective. Cambridge University Press. 2008.
7. "Pseudorandomness" (PDF).