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Randomness test
Randomness test
Randomness test
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A randomness test (or test for randomness), in data evaluation, is a test used to analyze the distribution of a set of data to see whether 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" (such as expecting random 0–9 but finding "4 3 2 1 0 4 3 2 1..." and rarely going above 4). 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.

Background

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The issue of randomness is an important philosophical and theoretical question. Tests for randomness can be used to determine whether a data set has a recognisable pattern, which would indicate that the process that generated it is significantly non-random. For the most part, statistical analysis has, in practice, been much more concerned with finding regularities in data as opposed to testing for randomness. Many "random number generators" in use today are defined by algorithms, and so are actually pseudo-random number generators. The sequences they produce are called pseudo-random sequences. These generators do not always generate sequences which are sufficiently random, but instead can produce sequences which contain patterns. For example, the infamous RANDU routine fails many randomness tests dramatically, including the spectral test.

Stephen Wolfram used randomness tests on the output of Rule 30 to examine its potential for generating random numbers,[1] though it was shown to have an effective key size far smaller than its actual size[2] and to perform poorly on a chi-squared test.[3] The use of an ill-conceived random number generator can put the validity of an experiment in doubt by violating statistical assumptions. Though there are commonly used statistical testing techniques such as NIST standards, Yongge Wang showed that NIST standards are not sufficient. Furthermore, Yongge Wang[4] designed statistical–distance–based and law–of–the–iterated–logarithm–based testing techniques. Using this technique, Yongge Wang and Tony Nicol[5] detected the weakness in commonly used pseudorandom generators such as the well known Debian version of OpenSSL pseudorandom generator which was fixed in 2008.

Specific tests for randomness

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There have been a fairly small number of different types of (pseudo-)random number generators used in practice. They can be found in the list of random number generators, and have included:

These different generators have varying degrees of success in passing the accepted test suites. Several widely used generators fail the tests more or less badly, while other 'better' and prior generators (in the sense that they passed all current batteries of tests and they already existed) have been largely ignored.

There are many practical measures of randomness for a binary sequence. These include measures based on statistical tests, transforms, and complexity or a mixture of these. A well-known and widely used collection of tests was the Diehard Battery of Tests, introduced by Marsaglia; this was extended to the TestU01 suite by L'Ecuyer and Simard. The use of Hadamard transform to measure randomness was proposed by S. Kak and developed further by Phillips, Yuen, Hopkins, Beth and Dai, Mund, and Marsaglia and Zaman.[6]

Several of these tests, which are of linear complexity, provide spectral measures of randomness. T. Beth and Z-D. Dai purported to show that Kolmogorov complexity and linear complexity are practically the same,[7] although Y. Wang later showed their claims are incorrect.[8] Nevertheless, Wang also demonstrated that for Martin-Löf random sequences, the Kolmogorov complexity is essentially the same as linear complexity.

These practical tests make it possible to compare the randomness of strings. On probabilistic grounds, all strings of a given length have the same randomness. However different strings have a different Kolmogorov complexity. For example, consider the following two strings.

String 1: 0101010101010101010101010101010101010101010101010101010101010101
String 2: 1100100001100001110111101110110011111010010000100101011110010110

String 1 admits a short linguistic description: "32 repetitions of '01'". This description has 22 characters, and it can be efficiently constructed out of some basis sequences.[clarification needed] String 2 has no obvious simple description other than writing down the string itself, which has 64 characters,[clarification needed] and it has no comparably efficient basis function representation. Using linear Hadamard spectral tests (see Hadamard transform), the first of these sequences will be found to be of much less randomness than the second one, which agrees with intuition.

Notable software implementations

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See also

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Notes

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Randomness test

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A randomness test, also known as a test for , is a statistical procedure designed to evaluate whether a of , such as bits or observations, conforms to the properties expected of a truly random process, where each outcome is independent and uniformly distributed. These tests typically formulate a assuming randomness and compute a —such as the number of runs or distributions—compared against a reference distribution (e.g., chi-squared or normal) to derive a ; if the falls below a significance level like 0.01, the is deemed non-random. Randomness tests play a critical role in validating random number generators (RNGs) and pseudorandom number generators (PRNGs), which produce sequences purporting to be unpredictable and uniformly distributed, essential for cryptographic applications like and protocol challenges. Beyond cryptography, they are applied in to detect trends or cycles in production data, in to ensure unbiased outcomes, and in scientific research to assess experimental variability against systematic errors. While no suite of tests can fully certify —serving only as a preliminary check against or deeper analysis—their outputs help identify deviations such as clustering, periodicity, or bias that could compromise security or reliability. Prominent examples include the run test, which counts sequences of consecutive similar values (e.g., above or below the median) to detect trends or oscillations, and more advanced suites like the NIST Statistical Test Suite, comprising 15 distinct tests such as frequency (monobit), block frequency, longest runs of ones, spectral (discrete Fourier transform), and linear complexity tests. These tests operate on binary sequences of sufficient length, typically 1,000,000 bits or more for the NIST suite, and aggregate P-values across multiple sequences to assess overall generator performance, with passing criteria often requiring at least 98% of P-values to exceed the 0.01 threshold. Developments in randomness testing continue to evolve, incorporating computational advances to handle larger datasets and more sophisticated patterns in modern applications.

Introduction

Definition and Purpose

A randomness test is a statistical procedure that evaluates the distribution and patterns within a , such as a binary or numerical , to determine whether it deviates significantly from the properties expected under true random behavior. These tests typically operate under a that the sequence is random, computing a that measures adherence to theoretical expectations, such as uniformity or . If the resulting falls below a predetermined significance level, the is rejected, indicating potential non-randomness. The primary purpose of randomness tests is to detect flaws like biases, serial correlations, or periodicities in outputs from pseudorandom number generators (PRNGs), true random number generators (TRNGs), or empirical data sources, ensuring their suitability for applications such as , simulations, and scientific modeling. PRNGs, being deterministic algorithms, aim to mimic statistically, while TRNGs draw from physical sources like thermal noise; tests validate both by probing for detectable patterns that could compromise reliability. For instance, in cryptographic contexts, failing tests might reveal vulnerabilities exploitable by adversaries. A key distinction exists between statistical randomness, which is verifiable through these tests (e.g., passing or runs tests indicates no detectable ), and true unpredictability, an idealized property of physical processes that defies deterministic prediction even with full knowledge of the system. While statistical tests assess observable properties like uniformity, they cannot confirm intrinsic unpredictability, as PRNGs may pass all known tests yet remain reproducible from their . In practice, the process involves inputting a —typically at least bits for robust evaluation—into one or more tests, deriving a from the test statistic's comparison to a reference distribution (e.g., chi-squared), and deciding at a significance level like α = 0.01, where p ≥ α accepts the sequence as sufficiently random. This threshold balances false positives and ensures high confidence in results for practical use.

Historical Overview

The foundations of randomness testing trace back to the early , when statisticians developed methods to assess uniformity in data distributions as a proxy for . In 1900, introduced the chi-squared goodness-of-fit test, which evaluates whether observed frequencies deviate significantly from expected uniform probabilities, providing an early tool for detecting non-random patterns in categorical data. This test became a cornerstone for frequency-based randomness assessments, influencing subsequent adaptations for sequential and binary data analysis. By the mid-20th century, attention shifted toward testing and ordering in sequences, leading to the development of the runs test. In 1940, Alexander M. Mood published work on the distribution of runs, establishing a statistical framework to detect clustering or excessive alternation in binary sequences, thereby assessing deviations from random . Concurrently, and Jacob Wolfowitz proposed a runs test specifically for comparing two samples to determine if they arise from the same random process, further refining tools for sequence . These contributions marked a pivotal advancement in nonparametric methods for empirical evaluation. The late saw the emergence of comprehensive test batteries tailored for evaluating generators (RNGs), particularly in . In 1995, George Marsaglia released the Diehard battery of tests, a suite of 15 rigorous statistical procedures designed to probe RNG output for subtle non-random artifacts, such as correlations in high-dimensional spaces, and distributed via for widespread use. Entering the , the National Institute of Standards and Technology (NIST) published Special Publication 800-22 in 2001, introducing a standardized suite of 16 tests for validating RNGs in cryptographic applications, which was revised in to 15 core tests by removing the Lempel-Ziv Compression test due to theoretical issues, with improved assessments. Complementing this, Pierre L'Ecuyer and Richard Simard developed the library in 2007, an open-source C implementation offering over 160 empirical tests organized into batteries like SmallCrush and BigCrush for scalable RNG scrutiny. Subsequent extensions and adaptations have addressed evolving RNG technologies. In 2004, Robert G. Brown extended Marsaglia's Diehard into Dieharder, a modular framework incorporating NIST tests and enhancing portability for modern systems. Post-2015, refinements have focused on quantum RNGs (QRNGs), with NIST's 2015 Bell test experiments confirming quantum sources' inherent unpredictability and prompting adaptations of suites like SP 800-22 to verify post-processing in QRNG outputs against classical biases. As of 2022, NIST announced plans to further revise SP 800-22, incorporating advances such as stochastic models, though no new version has been published as of November 2025.

Theoretical Foundations

Properties of Random Sequences

Ideal random sequences exhibit several key mathematical and statistical properties that distinguish them from deterministic or patterned data. These properties form the theoretical basis for evaluating the quality of random number generators and sequences used in simulations, , and statistical modeling. Uniformity requires that each possible outcome in the sequence occurs with equal probability. For a binary sequence over the alphabet {0,1}, this means each bit appears with probability 1/2, ensuring no bias toward any particular value. This property is fundamental to applications where balanced distribution is essential, such as methods. Independence implies that the occurrence of any element in does not influence the others, with no correlations between successive or non-adjacent elements. This is quantified by the autocorrelation function, which approaches zero for all non-zero lags in an ideal , confirming the absence of predictable dependencies. Incompressibility characterizes a sequence as one that cannot be significantly compressed without loss of information, as measured by —the length of the shortest program that generates . A truly has Kolmogorov complexity approximately equal to its length, resisting algorithmic description or pattern extraction. In pseudorandom generators like linear congruential generators, ideal sequences avoid short repeating cycles by achieving the full possible period, preventing detectable periodicity that would undermine randomness. For a binary sequence of length nn, the ideal Shannon entropy is H=nH = n bits, reflecting maximum uncertainty; any deviation below this value signals structure or non-randomness in the sequence.

Statistical Framework for Testing

The statistical framework for testing randomness in sequences, such as binary strings from generators, relies on classical hypothesis testing to assess whether the data conform to the expectations of true . The H0H_0 posits that the sequence is random, meaning it consists of independent and uniformly distributed bits (or symbols). In contrast, the H1H_1 asserts that the sequence exhibits non-random characteristics, such as toward certain values or dependencies between bits. This setup allows tests to quantify deviations from ideal empirically. Central to this framework is the , a numerical measure derived from the sequence that summarizes its adherence to H0H_0. For instance, in frequency-based assessments, the chi-squared statistic χ2=∑(Oi−Ei)2Ei\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i} is computed, where OiO_i are observed frequencies and EiE_i are expected frequencies under uniformity. The distribution of the under H0H_0 is typically known asymptotically (e.g., chi-squared with equal to the number of categories minus one), enabling the calculation of a . The represents the probability of obtaining a at least as extreme as the observed one, assuming H0H_0 is true; if the falls below a pre-specified significance level α\alpha (commonly 0.01), H0H_0 is rejected in favor of H1H_1. This threshold α\alpha controls the Type I error rate, the risk of falsely rejecting a truly random sequence. When applying a suite of multiple tests, the multiplicity problem arises, increasing the chance of false positives across the battery. A widely adopted correction is the Bonferroni adjustment, which divides the overall significance level by the number of tests (e.g., α′=α/k\alpha' = \alpha / k for kk tests) to maintain control over the . This conservative approach is particularly relevant in cryptographic randomness validation, where test batteries like those in NIST SP 800-22 are used, though some suites assess overall performance via uniformity or pass proportions instead. The power of a randomness test—the probability of correctly rejecting H0H_0 when H1H_1 is true—depends on factors like the specific non-random deviation, the test's sensitivity, and the sequence length nn. Seminal suites recommend large sample sizes, often n≥106n \geq 10^6 bits, to achieve adequate power against subtle patterns, as smaller sequences may fail to detect deviations reliably. For example, many NIST tests require at least 100 sequences of 10610^6 bits each to evaluate distributions robustly.

Categories of Randomness Tests

Frequency-Based Tests

Frequency-based tests evaluate the uniformity of symbol occurrences within a , serving as a fundamental check for global or local biases in the distribution of elements, such as in . These tests assume that in a truly random sequence, each should appear with equal probability, and deviations from this balance may indicate non-randomness due to generator flaws or deterministic patterns. By focusing on marginal frequencies rather than sequential dependencies, they provide an initial assessment of balance before more complex analyses. The monobit test, also known as the frequency test, counts the number of ones (or zeros) in a binary of length nn and assesses whether this proportion approximates 1/21/2. The is computed as z=∣∑(2ϵi−1)∣nz = \frac{|\sum (2\epsilon_i - 1)|}{\sqrt{n}}
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