Random Number Generator

Cryptographic vs Pseudo-Random Number Generation

Random numbers underpin statistical sampling, cryptographic security, game development, and algorithmic simulation. Depending on the application, computers generate numbers using either Pseudo-Random Number Generators (PRNGs) or Cryptographically Secure Pseudo-Random Number Generators (CSPRNGs).

PRNG vs CSPRNG Comparison

Feature	Standard PRNG (e.g., Math.random())	CSPRNG (e.g., Web Crypto API)
Underlying Algorithm	xorshift128+, Mersenne Twister	ChaCha20, AES-CTR DRBG
Entropy Source	Internal state register	Hardware noise, OS entropy pool
Predictability	Predictable if internal state observed	Mathematically unpredictable
Performance	Extremely fast (nanoseconds)	Highly optimised, secure
Primary Use Case	Graphic animations, casual gaming	Cryptography, token generation, lotteries

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Uniform Distribution and Bias Elimination

When mapping raw random bytes into an integer range $[\text{min}, \text{max}]$, using simple modulo arithmetic introduces modulo bias unless the range divides the entropy space evenly.

To guarantee zero statistical bias, modern generators apply rejection sampling:

$$N_{\text{range}} = \text{max} - \text{min} + 1$$ $$\text{Limit} = 2^{32} - (2^{32} % N_{\text{range}})$$

Any random 32-bit sample greater than or equal to $\text{Limit}$ is discarded and re-sampled, ensuring each candidate integer possesses an identical probability of selection.

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Generating Uniform Random Numbers in Code

JavaScript (Cryptographically Secure)

function secureRandomInt(min, max) {
  const range = max - min + 1;
  const maxUint32 = 0xFFFFFFFF;
  const limit = maxUint32 - (maxUint32 % range);
  const buffer = new Uint32Array(1);
  
  let randomVal;
  do {
    crypto.getRandomValues(buffer);
    randomVal = buffer[0];
  } while (randomVal >= limit);
  
  return min + (randomVal % range);
}

Python

import secrets

def get_secure_random(min_val: int, max_val: int) -> int:
    # secrets.randbelow uses cryptographically secure rejection sampling internally
    return min_val + secrets.randbelow(max_val - min_val + 1)

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Statistical Randomness Testing & Dieharder Standards

• Diehard Battery of Tests: Rigorous statistical randomness validation protocols test candidate number generators for birthday spacings, overlapping permutations, and binary rank distributions.
• Cryptographic Seed Management: Hardware random number generators sample thermal resistor noise or quantum phenomena to seed CSPRNG algorithms, ensuring seeds cannot be reverse engineered.
• Monte Carlo Simulations: High-volume scientific simulations require fast, uniform PRNGs (such as PCG or Xoshiro) to avoid algorithmic bottlenecks while maintaining statistical independence.