Concept
The mathematics of secrets — from Caesar ciphers to quantum key distribution. Information security is fundamentally a question of which math is hard.
Understand it in one breath
Design confidentiality, integrity, and authentication so that only holders of the right keys can perform the required operations. RSA is one public-key scheme relying on the classical difficulty of factoring large integers. HTTPS, finance, and messengers combine public-key methods, symmetric encryption, hashes, certificates, and careful implementation. A sufficiently large fault-tolerant quantum computer could use Shor's algorithm against RSA and classical ECC, motivating migration to post-quantum standards.
At a glance
Year | Cipher | Underlying hard mathematical problem | Broken? |
|---|---|---|---|
BCE 50 | Caesar cipher | None (simple shift) | ✗ Frequency analysis |
1553/1586 | Bellaso/Vigenère-style polyalphabetic cipher | Repeated keys and obscured frequencies | ✗ Kasiski, 1863, and other attacks |
1918–1940s | Enigma | Rotors, wiring, and plugboard settings | ✗ Polish Cipher Bureau and Bletchley Park teams |
1976 | Diffie-Hellman | Discrete logarithms are hard | ✓ against classical computers / ✗ against quantum computers (Shor) |
1977 | RSA | Factoring a product of large primes is hard | ✓ against classical computers / ✗ against quantum computers (Shor) |
1985 | ECC (elliptic curves) | Elliptic-curve discrete logarithm | ✓ against classical computers / ✗ against quantum computers |
2024+ | Post-quantum ML-KEM and ML-DSA | Lattice problems | Believed secure against known quantum attacks; standardized by NIST |
Cryptographic security depends not only on hardness assumptions but also on parameters, protocols, implementation, and key management. Quantum risk is driving a staged transition from RSA/ECC toward hybrid and post-quantum systems.
Key formula
Key moments
The Caesar cipher — shifting the alphabet
Julius Caesar used a fixed shift of the alphabet for military communication — one of the simplest substitution ciphers.
Shannon — the birth of information theory
Claude Shannon’s work made information measurable in bits, laying the mathematical foundation for digital compression and communication.
Diffie–Hellman — public-key cryptography
Two parties could now agree on a secret over a channel that everyone could observe — a starting point for modern internet security.
Shor’s algorithm — quantum computing casts a shadow
Peter Shor found a quantum algorithm for factoring large integers efficiently, revealing a future threat to RSA and accelerating research into post-quantum cryptography.
Modern applications
HTTPS, SSH, encrypted messaging, blockchain signatures, password hashing, and zero-knowledge proofs — the trust infrastructure of digital society.
Beyond MathVoyage
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