Pseudorandom Number Generator

A pseudorandom number generator (PRNG) is an algorithm that produces a long sequence of numbers that looks random but is entirely determined by its starting value, the seed. Computers are deterministic machines, so they can't produce randomness by calculation alone. A PRNG instead keeps some internal state, scrambles it with each call, and outputs part of the result. The same seed always gives the same sequence, which is a feature: it makes simulations, games, and tests reproducible, as described under Seeded Randomness.

John von Neumann proposed an early method in 1946, squaring a number and taking its middle digits, while remarking that anyone considering arithmetical methods of producing random digits is "in a state of sin." Common designs since:

• Linear congruential generators. Multiply the state by a constant, add another, and take the remainder. Fast and tiny, but with visible patterns; the infamous RANDU generator of the 1960s put all its output points onto just 15 planes in three dimensions.
• Mersenne Twister. Developed by Makoto Matsumoto and Takuji Nishimura in 1997, with a period of 2¹⁹⁹³⁷ − 1. It became the default in Python, Ruby, and many other languages.
• Xorshift and PCG. Small, fast modern generators. Xorshift uses only bit shifts and XORs; PCG scrambles a linear congruential generator's output with shifts and rotations. Both do far better on statistical test suites than older designs.

Quality is judged statistically: outputs should be evenly distributed, uncorrelated, and slow to repeat. Weak generators produce subtle patterns that can bias a Procedural Generation system, a Monte Carlo simulation, or a shuffle.

Security needs something stricter. A cryptographically secure PRNG (CSPRNG) must also be unpredictable: seeing past outputs shouldn't help anyone guess future ones. Ordinary PRNGs fail this badly, since a few hundred Mersenne Twister outputs reveal its entire state. CSPRNGs are built from ciphers or a Hash Function and seeded from true entropy, unpredictable physical events such as timing jitter, gathered by the operating system. Keys for Public-Key Cryptography, session tokens, and random identifiers such as a Universally Unique Identifier should come from one. Failures are severe: in 2008 it emerged that a Debian change had crippled OpenSSL's seeding for two years, leaving only about 32,000 possible keys per key type.

Games show both sides. Players exploit predictable generators through RNG manipulation, timing inputs to get a desired outcome, a staple of Speedrun routes, and reloading to reroll results is Save Scumming. Designers often bend randomness on purpose, as with a Pity Timer, because truly random streaks feel unfair. Multiplayer games that share a seed across machines rely on every client producing identical sequences, part of the Determinism that Rollback Netcode needs.

In testing, randomness drives Property-Based Testing and Fuzzing, and recording the seed turns a random failure into a reproducible one.