MathematicsInteractive SimulatorZero Install100% Free

NumberTheory&ModernCryptography

Explore the 'Queen of Mathematics' and the bedrock of modern internet cybersecurity. Sieve for primes, compute greatest common divisors via the Euclidean algorithm, evaluate modular multiplicative inverses, and encrypt/decrypt messages using the RSA asymmetric key cipher.

Launch Interactive LabRead the theory
Number Theory & Modern Cryptography interactive Mathematics simulation illustration
Sieve of Eratosthenes & RSA Cipher EngineInteractive Sieve Grid to 1000 • Euclidean Rectangle Tiling • Live RSA Key Generation (p, q, e, d)

Interactive Experiment Guide

Use this studio like a real-time mathematics workbench

Start with fundamental scientific principles, launch the simulation, and verify mathematical predictions against real-time outcomes.

DisciplineMathematics
Simulation ModeInteractive Numeric Engine
Governing ModelSieve of Eratosthenes & RSA Cipher Engine
DeploymentIn-Browser WebAssembly / GPU
01

Scientific Foundation

What is number theory & modern cryptography?

Number theory studies the algebraic properties of integers (ℤ). By the Fundamental Theorem of Arithmetic, every integer n > 1 has a unique prime factorization. The Euclidean Algorithm computes gcd(a, b) in logarithmic time, with Bézout's Identity guaranteeing integers x, y such that ax + by = gcd(a, b). Modern RSA cryptography exploits the computational asymmetry between prime multiplication (easy) and prime factorization of large semiprimes n = p·q (intractable), relying on Euler's Totient Theorem: a^φ(n) ≡ 1 (mod n).

02

Interactive Simulation Flow

Experiment Execution & Governing Equations

Launch the simulation workspace, adjust parameters in real time, and observe the immediate response in the telemetry and graphical indicator loops.

Bézout's Identity, Euler's Totient Theorem & RSA Encryption\gcd(a,b) = ax + by \quad \text{and} \quad a^{\phi(n)} \equiv 1 \pmod n \quad \text{and} \quad \text{RSA: } c \equiv m^e \pmod n, \; m \equiv c^d \pmod n

Frequently Asked Questions

Number Theory & Modern Cryptography FAQ

2 Answers

Multiplying two large prime numbers p and q (each 1024 or 2048 bits long) takes fractions of a millisecond. However, finding the prime factors p and q given only their product n = p·q has no known polynomial-time classical algorithm and would require millions of years of brute-force computation with current supercomputers.

Knowledge Graph & Related Concepts

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