1. Introduction

Ever since the antiquity, random number generation has played an important role both in common everyday life activities, such as leisure games, as well as in the advancement of science. One of the oldest ways in which humans generated random numbers has been through the use of dice. It does not seem to be known when dice were invented, but they have been employed since ancient times, alongside with coin flipping, for predicting the future, decision-making, fortune-telling, gambling, dispute resolution, and leisure games. However, coin tosses are known to have a certain bias, which has been studied extensively^[1]. In addition, coins could even rarely land on the edge, rendering the result useless^[2]. Modern usages of random numbers include Monte Carlo experiments, game decisions, and even Cryptography (see also Cryptographically-Secure Pseudorandom Number Generator).

Random Number Generators are often abbreviated and referred to as RNGs.

2. Desirable Properties

A RNG should have four desirable properties:

1. Uniformity: Each individual output must be equally probable.
2. Independence: The occurrence of one output must be independent of the occurrence of another output.
3. Long Period: If the RNG becomes deterministic after some time, the period should be as long as possible.
4. Practicability: Numbers should be generated efficiently and the RNG should be supported on different architectures and systems.

The coin toss mentioned above does not exactly fulfil these characteristics, but can still provide sufficient random numbers for everyday use.

3. Types of Random Number Generators

The two main types of RNGs are called True RNGs (TRNGs) and Pseudo-RNGs (PRNGs).

3.1. True Random Number Generators

A TRNG is able to generate random numbers that can only be predicted (correctly) with negligible probability, even if the initial value and conditions are known. TRNGs are typically slower than PRNGs and may additionally be biased. For debiasing, most often von-Neumann-correction is deployed^[3].

3.2. Pseudorandom Number Generators

A PRNG can generate sequences of numbers that appear to be random, but are in fact completely predictable when the initial value (being referred to as the seed) and conditions used for the number generation process are known. PRNGs are usually algorithms or simple mathematical formulae, making them faster than TRNGs at the cost of indeterminism.

4. Generation Methods

This section lists some commonly used and newly proposed RNGs.

4.1. TRNGs

• Radioactive decay: The fact that it is impossible to predict when an atom will decay can be used to generate random numbers^[4].
• Radio frequency noise: FM broadcasting and other means of wireless communication are susceptible to noise, which can be extracted to obtain random numbers^[5].
• Thermal noise: Electronic components such as resistors emit thermal noise, which can be measured to extract random numbers^[6].
• Memory timings: The write and read latency of memories is usually prone to noise which can be extracted to generate random numbers^[7].
• Ring oscillators: Extracting the jitter of multiple free running oscillators by comparing them can lead to random numbers^[8].

4.2. PRNGs

• Middle-Square Method: To generate a new random number, the previous one is squared and the middle digits are taken as the next random numbers^[9].
• Congruential Generators: The next random number is generated using a simple congruence relation^[10][11].
• Mersenne Twister: Uses an algorithm to generate 624 random 32-bit numbers in a single iteration^[12].
• Blum-Blum-Shub generator: Uses quadratic residues in a semiprime quotient ring to generate the next random number^[13].
• Block ciphers (ANSI X9.17): Appendix C of the standard includes a random generator using a block cipher^[14].

5. Attacks

Most RNGs are prone to some kind of attack. Attacks on PRNGs include, but are not limited to^[15]:

• Direct Cryptanalytic Attack: An attacker can directly distinguish between the output of the PRNG and random numbers (cryptanalyze the PRNG).
• Input-based Attack: An attacker is able to use knowledge and control of the PRNG inputs to cryptanalyze it.
• State Compromise Extension Attack: The attacker can guess state information due to an earlier breach of security.

6. Modern Applications

• Random noise (e.g., Gaussian noise)
• Key generation (TLS, SSH)
• Salts for Hashes
• RSA prime factors
• Nonces (TLS)
• Initialization Vectors (AES/Rijndael)