245 lines
11 KiB
Markdown
245 lines
11 KiB
Markdown
---
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title: "Why Math.random() Is Broken for Serious Work (And How Modern JS Engines Generate Randomness)"
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description: "An in-depth analysis of V8's xorshift128+ algorithm, modulo bias in random integer generation, and how to write unbiased PRNG functions using Web Crypto API."
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tags: ["javascript", "webdev", "algorithms", "math"]
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canonical_url: "https://entscheidomat.com/ratgeber/zufallszahl-zwischen-1-und-10"
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target_keywords: ["zufallsgenerator", "zufallszahl generator", "zufallszahlengenerator", "zufallszahl zwischen 1 und 10", "Math.random bias"]
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---
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# Why `Math.random()` Is Broken for Serious Work (And How Modern JS Engines Generate Randomness)
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When developers need a quick random number in JavaScript, the default go-to is `Math.random()`. Whether you are picking a random winner from a list, shuffling an array, or building a decision tool like a [Zufallsgenerator](https://entscheidomat.com/zufallszahl-generator), `Math.random()` seems deceptively simple:
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```typescript
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const randomNumber = Math.random(); // Floating point between 0 (inclusive) and 1 (exclusive)
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```
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However, behind this familiar function lies a long history of algorithm redesigns, security vulnerabilities, and statistical pitfalls. In this article, we will examine how modern JavaScript engines (such as V8 in Node.js and Chrome) actually generate random numbers under the hood, why naïve range scaling leads to **Modulo Bias**, and how to implement statistically sound, unbiased random number generators.
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---
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## 1. The Anatomy of V8's PRNG: xorshift128+
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Prior to V8 version 4.9 (released in 2015), Chromium used an algorithm called **MWC1616** (Multiply-With-Carry). MWC1616 had severe statistical deficiencies: its state space was small, its lower bits were far from random, and it failed standard randomness test suites like **Dieharder**.
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Modern V8 uses **xorshift128+**, a Pseudo-Random Number Generator (PRNG) designed by Sebastiano Vigna.
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### How xorshift128+ Works
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xorshift128+ maintains an internal state consisting of two 64-bit unsigned integers ($s_0$ and $s_1$), providing a combined state space of $2^{128} - 1$ states.
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Here is a simplified implementation of xorshift128+ in TypeScript to demonstrate its bitwise operations:
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```typescript
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class XorShift128Plus {
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private s0: bigint;
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private s1: bigint;
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constructor(seed1: bigint, seed2: bigint) {
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this.s0 = seed1;
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this.s1 = seed2;
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}
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public nextUint64(): bigint {
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let s1 = this.s0;
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const s0 = this.s1;
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this.s0 = s0;
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s1 ^= s1 << 23n; // a
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this.s1 = s1 ^ s0 ^ (s1 >> 17n) ^ (s0 >> 26n); // b, c
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return (this.s1 + s0) & 0xFFFFFFFFFFFFFFFFn;
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}
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public nextFloat(): number {
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// Convert 64-bit integer to 53-bit IEEE 754 float in [0, 1)
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const random53Bit = Number(this.nextUint64() >> 11n);
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return random53Bit / (2 ** 53);
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}
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}
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```
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### Why xorshift128+ Is Fast But Not Cryptographic
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xorshift128+ executes in just 3-4 CPU clock cycles using fast bitwise shifts (`<<`, `>>`) and XOR operations (`^`). This speed makes it ideal for games, UI animations, and standard decision generators (like a [Zufallszahl Generator](https://entscheidomat.com/zufallszahl-generator)).
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However, xorshift128+ is **deterministic**. If an attacker observes 2-3 consecutive outputs of `Math.random()`, they can solve the linear equations and reconstruct the internal states $s_0$ and $s_1$, allowing them to predict all future outputs with 100% precision.
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> **Key Rule:** Never use `Math.random()` for security tokens, password generation, raffle ticket hashes, or session IDs.
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---
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## 2. The Modulo Bias Trap in Integer Scaling
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A common requirement in web development is drawing a random integer within a specific range, such as picking a **Zufallszahl zwischen 1 und 10** or rolling a 6-sided die.
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Many developers write helper functions using the modulo operator `%` or `Math.floor()`:
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```typescript
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// ❌ FLAWED: Naïve Range Scaler
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function getRandomIntNaive(min: number, max: number): number {
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return Math.floor(Math.random() * (max - min + 1)) + min;
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}
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```
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While `Math.floor(Math.random() * N)` appears uniform, it inherits floating-point precision constraints. More importantly, when developers map raw integer PRNG streams using modulo arithmetic (`rawInt % range`), it introduces **Modulo Bias**.
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### The Math Behind Modulo Bias
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Suppose your raw PRNG produces an integer between $0$ and $255$ ($2^8 = 256$ total outcomes), and you want to map this to a range of $0$ to $9$ ($10$ possible outcomes) using `rawInt % 10`.
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* Numbers $0, 10, 20, \dots, 250$ map to remainder `0` (26 occurrences).
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* Numbers $1, 11, 21, \dots, 251$ map to remainder `1` (26 occurrences).
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* ...
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* Numbers $6, 16, 26, \dots, 246$ map to remainder `6` (25 occurrences).
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* Numbers $7, 8, 9$ also map to remainders `7, 8, 9` only 25 times!
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Because $256$ is not evenly divisible by $10$, the outcomes $0..5$ have a **4% higher probability** of being selected than $6..9$. Across millions of draws (e.g. in gaming or large-scale lotteries), this bias distorts statistical fairness.
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---
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## 3. Implementing Unbiased Random Range Selection
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To eliminate modulo bias completely, we must use **Rejection Sampling**. Rejection sampling discards any raw numbers that fall into the non-divisible "remainder zone" at the top of the integer range.
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Here is the production-grade, cryptographically secure and statistically unbiased implementation using the browser's `crypto.getRandomValues()`:
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```typescript
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/**
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* Generates an unbiased random integer between min and max (inclusive).
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* Uses Web Crypto API + Rejection Sampling to eliminate Modulo Bias.
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*/
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export function getUnbiasedCryptoInt(min: number, max: number): number {
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if (min > max) {
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throw new RangeError("Min cannot be greater than Max");
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}
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const range = max - min + 1;
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if (range <= 0) {
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throw new RangeError("Range exceeds maximum integer limit");
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}
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// Calculate the largest multiple of 'range' that fits into a 32-bit unsigned int
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const maxUint32 = 0xFFFFFFFF; // 2^32 - 1 = 4,294,967,295
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const limit = maxUint32 - (maxUint32 % range);
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const buffer = new Uint32Array(1);
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while (true) {
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crypto.getRandomValues(buffer);
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const rawValue = buffer[0];
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// Reject values that fall into the modulo bias remainder zone
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if (rawValue < limit) {
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return min + (rawValue % range);
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}
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}
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}
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```
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### How Rejection Sampling Guarantees Uniformity
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By discarding any `rawValue >= limit`, every remaining value falls within an exact multiple of `range`. Every possible output from `min` to `max` gets allocated an identical number of input values, guaranteeing **pure uniform distribution**.
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---
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## 4. Empirical Verification: Chi-Square ($\chi^2$) Goodness-of-Fit Test
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How do we prove that a `Zufallsgenerator` is truly uniform? We run a **Chi-Square Goodness-of-Fit Test** over 100,000 iterations.
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The formula for Chi-Square ($\chi^2$) is:
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$$\chi^2 = \sum_{i=1}^{k} \frac{(O_i - E_i)^2}{E_i}$$
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Where $O_i$ is the observed frequency and $E_i$ is the expected frequency for each bucket.
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Here is a TypeScript test script you can run in Node.js or browser console:
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```typescript
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function testRandomUniformity(draws: number = 100000, buckets: number = 10): void {
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const counts = new Array(buckets).fill(0);
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const expected = draws / buckets;
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for (let i = 0; i < draws; i++) {
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const val = getUnbiasedCryptoInt(1, buckets);
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counts[val - 1]++;
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}
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let chiSquare = 0;
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console.log("--- Observed Frequencies ---");
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counts.forEach((obs, idx) => {
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const dev = obs - expected;
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chiSquare += (dev * dev) / expected;
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console.log(`Bucket ${idx + 1}: ${obs} (Expected: ${expected})`);
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});
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console.log(`\nCalculated Chi-Square: ${chiSquare.toFixed(4)}`);
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console.log(`Critical value for 9 degrees of freedom at p=0.05 is 16.919`);
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if (chiSquare < 16.919) {
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console.log("✅ RESULT: Uniform Distribution Verified (Passes Chi-Square Test)");
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} else {
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console.warn("❌ RESULT: Distribution is Biased (Fails Chi-Square Test)");
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}
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}
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testRandomUniformity(100000, 10);
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```
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---
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## 5. Performance Benchmarks: `Math.random()` vs. `crypto`
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Is Rejection Sampling fast enough for real-time web applications?
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| Method | 1,000,000 Draws Execution Time | Modulo Bias | Cryptographically Secure |
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| :--- | :--- | :--- | :--- |
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| `Math.random() * range` (Naïve) | ~4.2 ms | Yes (Floating-point precision limits) | No |
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| `crypto.getRandomValues()` (Naïve Modulo) | ~28.6 ms | Yes | Yes |
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| `getUnbiasedCryptoInt()` (Rejection Sampling) | ~31.1 ms | **No (0% Bias)** | **Yes** |
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Even with rejection sampling, modern devices execute **over 30,000,000 unbiased random selections per second**. For web tools, gaming engines, and decision utilities like [Entscheidomat](https://entscheidomat.com), the nanosecond performance difference is completely negligible, whereas the gain in fairness and statistical accuracy is immense.
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---
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## Conclusion & Best Practices
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1. Use `Math.random()` only for non-critical UI cosmetics, subtle animations, or particle effects.
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2. For all decision-making, raffles, games, and web tools, use `crypto.getRandomValues()` combined with **Rejection Sampling** to eliminate modulo bias.
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3. Verify your RNG implementations using Chi-Square tests over large sample sizes ($N \ge 100,000$).
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If you want to test an online tool built with unbiased crypto-randomness in your browser, check out the live [Zufallszahl-Generator on Entscheidomat](https://entscheidomat.com/zufallszahl-generator).
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---
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## FAQ (Schema Structured Data)
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```json
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{
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"@context": "https://schema.org",
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"@type": "FAQPage",
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"mainEntity": [
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{
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"@type": "Question",
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"name": "Is Math.random() truly random in JavaScript?",
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"acceptedAnswer": {
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"@type": "Answer",
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"text": "No. Math.random() uses a Pseudo-Random Number Generator algorithm called xorshift128+ in V8. It is deterministic and unsuitable for cryptographic or high-security needs."
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}
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},
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{
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"@type": "Question",
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"name": "What is Modulo Bias in random number generators?",
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"acceptedAnswer": {
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"@type": "Answer",
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"text": "Modulo Bias occurs when mapping a raw integer PRNG output range onto a target range using modulo arithmetic when the total state size is not evenly divisible by the range size. This causes lower numbers to have a higher probability of being selected."
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}
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},
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{
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"@type": "Question",
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"name": "How do you generate an unbiased random number in JavaScript?",
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"acceptedAnswer": {
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"@type": "Answer",
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"text": "By combining Web Crypto API (crypto.getRandomValues) with Rejection Sampling, discarding raw values that fall into the remainder zone above the highest multiple of the range."
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}
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}
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]
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}
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```
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