# Cryptography Cryptography is the mathematics of communicating in the presence of adversaries: transforming messages so that only holders of a secret key can read or forge them. Once a craft of substitution tables, it became a science twice — in 1883, when Kerckhoffs demanded that security rest in the key alone, and in 1949, when [[Claude_Shannon]] of [[Bell_Labs]] reframed secrecy as a branch of [[Information_theory|information theory]], measuring an eavesdropper's [[Uncertainty|uncertainty]] in [[Entropy_(information_theory)|entropy]]. Modern practice splits into symmetric [[Algorithm|algorithms]] that move bulk data, public-key systems whose safety leans on unsolved problems in [[Group_theory|algebra]] and the [[Theory_of_computation|theory of computation]], and protocols that compose them into the trust infrastructure of [[Telecommunications|telecommunications]] and commerce. It is the rare engineering field whose failure modes are intelligent: the adversary is a player, which makes design closer to [[Game_theory|game theory]] than to [[Physics|physics]]. ## Kerckhoffs's principle: assume the enemy has the blueprints Kerckhoffs's *La cryptographie militaire* (1883) laid down the axiom that still sorts serious systems from theater: a cryptosystem must remain secure when everything about it except the key is public. [[Claude_Shannon|Shannon]] restated it bluntly — design as if the enemy knows the system. The reasoning is systemic: algorithms leak, defect, and outlive deployments, while keys are small, replaceable, and auditable; concentrating secrecy in the key turns security into key management, a tractable [[Decision_theory|decision problem]] rather than a hope. Security-through-obscurity fails not because secrecy is worthless but because it cannot be renewed after compromise. The modern corollary is public review: [[Code|codes]] and protocols earn trust by surviving open [[Statistics|statistical]] and mathematical attack, the institutional [[Feedback|feedback loop]] that standards bodies formalize. ## The one perfect cipher and its price The one-time pad — XOR each message bit with a fresh, uniformly random key bit, devised in working form around 1917–19 (Vernam, Mauborgne) — is provably unbreakable in the [[Information_theory|information-theoretic]] sense. [[Claude_Shannon|Shannon's]] 1949 theorem states the exact terms: perfect secrecy requires the key to carry at least as much [[Entropy_(information_theory)|entropy]] as the message and never be reused; then the ciphertext is [[Probability_theory|statistically]] independent of the plaintext, and the attacker's posterior [[Probability_distribution|distribution]] over messages equals the prior. The price — couriered key material as long as all future traffic — confines it to niches. Everything else in the field is a negotiated retreat from perfection: ciphers that are merely *computationally* secure, betting that no adversary can perform ≈2¹²⁸ operations. [[Claude_Shannon|Shannon's]] design vocabulary for that retreat, confusion and diffusion, still names what block ciphers do to smear [[Information|information]] across [[Binary_number|binary]] states. ## Symmetric workhorses: from Enigma to AES Classical ciphers fell to [[Statistics|statistics]]: al-Kindi described frequency analysis in the 9th century, and Kasiski's 1863 test broke the Vigenère. Enigma fell to structure — Rejewski's [[Group_theory|group-theoretic]] reconstruction (1932) and the Turing–Welchman bombes (1940) exploited the machine's fixed-point-free permutations, an early triumph of [[Algorithm|mechanized search]] over a [[Finite-state_machine|finite-state]] device. The modern lineage is standardized: DES (1977), whose 56-bit key fell to a $250,000 brute-force machine in 56 hours in 1998, then AES (Rijndael, standardized 2001), a substitution–permutation network over the finite [[Field_(mathematics)|field]] GF(2⁸), running 10–14 rounds on 128-bit blocks with 128/192/256-bit keys. AES-128's keyspace, ≈3.4 × 10³⁸, is beyond any classical search; real breaks come instead from [[Software_engineering|implementation]] — timing, power, and cache side channels — and from protocol misuse, which is why authenticated modes and [[Error_detection_and_correction|integrity checks]] travel with encryption. Stream ciphers, formally keyed [[Finite-state_machine|finite-state machines]], serve where latency or [[Computer_hardware|hardware]] budgets are tight; their whole security is the unpredictability of the keystream, a strictly harder requirement than the mere uniformity demanded of [[Simulation|simulation]] generators. ## Public keys: mathematics as a trust machine Diffie and Hellman (1976) split the key: publish an encryption key, keep the inverse operation private, and let strangers establish secrets over open channels. RSA (published 1978) rests on the difficulty of factoring; Diffie–Hellman and its elliptic-curve descendants (Koblitz and Miller, 1985) on discrete logarithms in well-chosen [[Group_theory|groups]] — problems from [[Discrete_mathematics|discrete mathematics]] and [[Abstract_algebra|algebra]] with centuries of failed attacks behind them, though no proofs; their hardness is conjecture, calibrated by the best public [[Algorithm|algorithms]]. The honest ledger of key sizes reflects that calibration: the largest publicly factored RSA modulus is 829 bits (RSA-250, 2020, thousands of core-years), so deployments use 2048–3072 bits, while a 256-bit elliptic-curve key delivers security comparable to a 128-bit symmetric key at a fraction of the cost — the asymmetry that put [[Combinatorics|combinatorial]] number theory inside every [[Telecommunications|handset]]. Britain's GCHQ, it emerged in 1997, had found the same constructions secretly in 1970–74. Above the primitives sit the protocols — signatures, key exchange, certificate graphs — where most real-world failures occur, and where [[Graph_theory|graph-theoretic]] trust models do the quiet load-bearing. ## Randomness, automata, and proving without revealing Every key begins as randomness, so generators are attack surface: hardware sources harvest [[Noise_(electronics)|electronic noise]], debias it (a trick going back to [[John_von_Neumann|von Neumann]], 1951), and feed cryptographic generators held to [[Probability_theory|unpredictability]], not just good [[Statistics|statistics]]. Unexpected engines have served: Wolfram proposed the chaotic center column of the Rule 30 [[Elementary_cellular_automaton|elementary cellular automaton]] as a generator in 1986, and [[Cellular_automaton|cellular automata]] remain a live design space for lightweight hardware. At the protocol frontier, zero-knowledge proofs — introduced by Goldwasser, Micali, and Rackoff (1985), made general by the Goldreich–Micali–Wigderson protocol for [[Graph_theory|graph]] three-coloring (1986) — let a prover demonstrate possession of a secret while leaking zero additional [[Information|information]], a construction now underwriting anonymous credentials and [[Theory_of_computation|verifiable computation]]. Here cryptography converges with [[Complexity|complexity]] theory: what can be hidden is exactly shaped by what cannot be computed. ## Quantum futures Shor's algorithm (1994) factors integers and takes discrete logarithms in [[Algorithmic_efficiency|polynomial time]] on a large [[Error_detection_and_correction|fault-tolerant]] [[Quantum_computing|quantum computer]], which would break RSA and elliptic curves outright; Grover's search (1996) merely halves effective symmetric key length, so AES-256 stays comfortable. The response has two tracks. Quantum key distribution (BB84, 1984) moves secrecy onto [[Physics|physics]]: eavesdropping on single [[Photon|photons]] disturbs them detectably, per [[Quantum_mechanics|quantum mechanics]] itself. Post-quantum cryptography stays [[Mathematics|mathematical]]: in 2024 NIST standardized lattice-based key encapsulation and signatures (ML-KEM, ML-DSA) plus a hash-based fallback, and migration has begun now because recorded traffic can be decrypted retroactively — harvest today, break tomorrow. The through-line from Kerckhoffs holds: assume the adversary gets everything, including, eventually, a quantum computer, and let the key — and the [[Theory_of_computation|computational theory]] behind it — carry the whole weight. **On the spine:** [[Information_theory]] · [[Claude_Shannon]] · [[Quantum_computing]] · [[Group_theory]] · [[Cellular_automaton]]. ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Cryptography) : [Wikitube](https://en.wikitube.io/wiki/Cryptography) ## Previous hub tags Hubs: `Systems`. Portals: [[PORTAL_Systems]], [[PORTAL_Cellular_automaton]], [[PORTAL_Graph_theory]], [[PORTAL_Decision_theory]], [[PORTAL_Information_theory]]. --- *Repopulated 2026-08-12 · redlink fill · 0 deletions.*