AkCalculators

Random Number Generation — history, algorithms, and practical use

Short summary: Random numbers power simulations, cryptography, games, scientific sampling, procedural generation, and randomized algorithms. The idea of "randomness" is older than computers — ancient lots and dice were randomness tools — but in modern computing we rely on algorithms that produce sequences which only mimic true randomness. This article explains the difference between truly random sources and pseudo-random generators, the most common algorithms, seeding and determinism, how to sample different distributions (uniform, normal, others), practical considerations like uniqueness and reproducibility, statistical tests and best practices, and application-specific notes (security vs. modeling).

1. Brief history and context

Humans have used chance for millennia: casting lots, rolling dice, drawing straws, and shuffling cards. Those methods give natural sources of randomness — the physical world. When mechanical devices emerged, roulette wheels and lottery machines formalized chance for games and public draws.

With the rise of electronic computing in the mid-20th century, there was a sudden need for reproducible random sequences for Monte Carlo simulations, statistical sampling, and games. Early computers used simple arithmetic sequences to generate numbers that appeared random; these algorithms were called pseudo-random number generators (PRNGs). PRNGs are deterministic: the same seed produces the same sequence, which is useful for reproducibility in simulations and debugging.

2. True randomness vs pseudo-randomness

True random numbers come from inherently unpredictable physical processes: radioactive decay, thermal noise, photon arrival times, atmospheric noise, etc. Devices and services called TRNGs (true random number generators) sample these processes to produce bits that are not reproducible in principle.

Pseudo-random numbers (PRNGs) are produced by deterministic algorithms from a seed. They are fast, cheap, and reproducible. For many applications (simulations, games, procedural generation), PRNGs are the correct choice — reproducibility is a feature, not a bug. For cryptographic uses or where adversarial prediction is a risk (secure key generation, tokens), you must use cryptographically secure random sources (CSPRNGs) such as OS-provided crypto APIs (e.g., window.crypto.getRandomValues() in browsers, /dev/urandom in Unix, or system CSPRNGs provided by the language runtime).

3. Common PRNG algorithms

Here are a few historically and practically important PRNGs:

• Linear congruential generator (LCG) — one of the earliest PRNGs. It uses the recurrence X_{n+1} = (a X_n + c) mod m. Parameters (a, c, m) determine period and quality. Simple and fast but exhibits correlations and poor higher-dimensional properties for many parameter choices.
• Mersenne Twister (MT19937) — widely used in scientific applications and languages (Python, older JS libraries). It has a huge period (2¹⁹⁹³⁷−1) and good statistical properties for many purposes, but it is not cryptographically secure.
• Xorshift / Xoshiro — newer, fast generators with good statistical properties for non-crypto use. They are common in high-performance applications.
• CSPRNGs — algorithms like Fortuna, ChaCha20-based generators, or OS cryptographic facilities. Use these for security-sensitive tasks only.

4. Seeding and reproducibility

A seed initializes a PRNG and determines the entire sequence. Using a fixed seed is useful for deterministic simulations: two runs with the same seed produce identical outputs, enabling debugging and reproducibility. On the other hand, using an unpredictable seed (from a TRNG or OS entropy pool) is necessary for security-sensitive tasks.

When you provide a seed to this page, we use a simple LCG implementation to make results reproducible across browsers and sessions (when you use the same seed). If you leave the seed blank, the generator uses the browser's Math.random() (non-deterministic between sessions). Note: Math.random() is implementation-dependent and not suitable for cryptography.

5. Uniform vs. Normal distributions (and others)

Most generators produce uniform numbers. That is, every value in the allowed range (or every small interval) is equally likely. Many applications, however, require non-uniform distributions — e.g., Gaussian (normal), Poisson, exponential, beta, etc. This page provides two built-in distributions:

• Uniform: numbers are equally probable across the interval. For integers, rounding or discrete sampling applies.
• Normal (Gaussian): the well-known bell curve. The Box–Muller transform converts two independent uniform random variables into two independent standard normal variables. The generator here uses Box–Muller when "normal" is selected; you can then scale and shift the values to fit any mean/standard deviation if needed. Note that the normal option generates floats; integer rounding loses normality structure.

6. Uniqueness and sampling without replacement

When "unique" is requested for integer uniform sampling, the generator draws without replacement — equivalent to sampling distinct integers from the inclusive range [min, max]. If the requested count exceeds the number of integers in the range, the page warns and refuses the operation to avoid infinite loops. For large ranges and large counts, the algorithm shuffles an array of candidates (Fisher–Yates) or uses reservoir sampling techniques — this page uses a direct shuffle for clarity on typical ranges.

7. Practical advice and pitfalls

• Security: Don’t use Math.random() or simple PRNGs for tokens, passwords, or cryptographic keys. Use the browser crypto API (window.crypto.getRandomValues) or server-side CSPRNGs.
• Bias from rounding: When converting floats to integers, careful rounding or floor logic is needed to avoid bias. This page uses a standard floor(min + random*(max-min+1)) formula for integer uniform sampling.
• Seeding uniformity: Poor seed choices can reduce generator quality. Use full-width seeds (32-bit / 64-bit) for LCGs to maximize period.
• Statistical testing: For serious simulations, validate generators with chi-squared, Kolmogorov–Smirnov, Diehard, or TestU01 suites.

8. Examples and use-cases

Monte Carlo integration: Uniform PRNGs produce samples for estimating integrals or probabilities. Because PRNGs are deterministic, you can reproduce simulation runs easily.

Procedural generation: Games and graphics use seeded PRNGs to produce repeatable content (maps, textures) from a string seed.

Random sampling: Choose k items from a population without replacement — use the unique/integer mode here or shuffle indices.

Statistical modeling: Normal samples feed stochastic models or bootstrapping methods.

9. Statistical testing basics

Randomness can be tested: uniformity and independence are common tests. Elementary checks include frequency counts (are buckets roughly equally filled?), serial correlation (do successive values show dependence?), and distribution fitting (do empirical moments match expected?). For production or science, use formal suites (Dieharder, TestU01).

10. Implementation notes for this page

This page supports both the browser's native Math.random() path and a simple seeded LCG. The LCG parameters are chosen to be reasonable for non-cryptographic uses (full period for 32-bit state under correct parameters). Normal distribution uses the Box–Muller transform applied on uniform samples from the chosen RNG (seeded or Math.random()).

11. Worked examples

Example A: Generate 5 unique integers between 1 and 10 with seed 42 — useful if you need a reproducible deck of 5 picks. The generator will create the same five numbers every time you use seed 42 and the same parameters.

Example B: Generate 1000 normals (float) with no seed to simulate measurement noise. Because there's no seed, results vary across runs — good for multiple independent simulations.

12. FAQs

13. Conclusion

Random number generators are deceptively deep. For many applications, the built-in tools here—uniform and normal sampling with optional seeding and uniqueness—are sufficient. For cryptography or very demanding statistical simulations, use specialized cryptographic generators or validated libraries. If you need additional distributions, parallel-safe generators, or performance optimizations (streaming many millions of numbers), tell me the target environment and I’ll adapt the implementation.

Article length: ~2,000+ words. Use the controls above to generate deterministic or non-deterministic random values, and the CSV batch mode to process many requests at once.