Cryptography

Part: Computer Science · Chapter: Cryptography · Edition: 2026-07-12
Keywords: confidentiality, integrity, authenticity, public-key, hash
Glossary: cipher, hash, public key, nonce — see GLOSSARY

At a glance

Cryptography provides confidentiality, integrity, and authenticity under adversarial models.
Symmetric and asymmetric primitives compose into protocols.
Security proofs depend on precise threat models.

Online references (live)

Offline snapshots (link-rot shield)

Offline snapshot omitted in this web edition. Offline snapshot · Cryptography · https://en.wikipedia.org/wiki/Cryptography · retrieved 2026-07-12

Cryptography

Offline snapshot omitted in this web edition. Offline snapshot · Public-key cryptography · https://en.wikipedia.org/wiki/Public-key_cryptography · retrieved 2026-07-12

Public-key cryptography

Offline snapshot omitted in this web edition. Offline snapshot · Cryptographic hash function · https://en.wikipedia.org/wiki/Cryptographic_hash_function · retrieved 2026-07-12

Cryptographic hash function

This comprehensive index represents the full breadth of cryptography as an academic, mathematical, and engineering discipline in 2026. It integrates classical foundations with cutting-edge research in quantum resistance and privacy-preserving computation.

1. Mathematical Foundations

The “Hard Problems” that provide computational security.

Number Theory:
Modular Arithmetic & Residue Classes
Primes: Mersenne Primes, Primality Testing (AKS, Miller-Rabin)
Factorization Algorithms: Pollard’s Rho, Quadratic Sieve, GNFS
Euler’s Totient Theorem & Fermat’s Little Theorem
Chinese Remainder Theorem (CRT) applications
Abstract Algebra:
Group Theory: Cyclic groups, Order of elements, Lagrange’s Theorem
Field Theory: Finite Fields (Galois Fields $GF(p^n)$ ), Polynomial Arithmetic
Elliptic Curves: Weierstrass equations, Point addition, Scalar multiplication
Complexity Theory:
Reduction Proofs (Proving B is as hard as A)
Complexity Classes: P, NP, Co-NP, BQP (Bounded-error Quantum Polynomial time)
One-Way Functions (OWF) & Trapdoor Permutations
Information Theory:
Shannon’s Entropy & Diffusion/Confusion principles
Perfect Secrecy & Unconditional Security
The Birthday Paradox & Collision Resistance

2. Symmetric Cryptography (Private Key)

Stream Ciphers:
LFSR (Linear Feedback Shift Registers) & NLFSR
Synchronous vs. Asynchronous stream ciphers
Modern standards: ChaCha20, Salsa20, Trivium
Block Ciphers:
Substitution-Permutation Networks (SPN) vs. Feistel Networks
AES (Advanced Encryption Standard): S-Box design, Key Schedule, MixColumns
Legacy: DES, 3DES, Blowfish, CAST-128
Modes of Operation:
Confidentiality: ECB, CBC, CTR
Authenticated Encryption (AEAD): GCM (Galois/Counter Mode), Poly1305, CCM
Hash Functions:
Merkle-Damgård Construction (MD5, SHA-1, SHA-2)
Sponge Construction (SHA-3/Keccak)
Password Hashing: Argon2, bcrypt, scrypt

3. Asymmetric Cryptography (Public Key)

Factoring-Based: RSA (OAEP padding, PSS signatures)
Discrete Logarithm Problem (DLP):
Diffie-Hellman Key Exchange (DHKE)
ElGamal Encryption & Digital Signature Algorithm (DSA)
Elliptic Curve Cryptography (ECC):
ECDH (Key Exchange) and ECDSA (Signatures)
Edwards-curve Digital Signature Algorithm (EdDSA/Ed25519)
Pairing-based: BLS Signatures (Boneh-Lynn-Shacham)

4. Post-Quantum Cryptography (PQC)

Algorithms designed to survive attacks by a cryptographically relevant quantum computer.

Lattice-Based:
Learning With Errors (LWE) & Ring-LWE
Shortest Vector Problem (SVP) & Closest Vector Problem (CVP)
NIST Standards: ML-KEM (Kyber) and ML-DSA (Dilithium)
Code-Based: McEliece Cryptosystem, Goppa Codes
Multivariate Cryptography: Unbalanced Oil and Vinegar (UOV)
Hash-Based Signatures: LMS, XMSS, SPHINCS+
Isogeny-Based: CSIDH (Commutative Supersingular Isogeny Diffie-Hellman)

5. Advanced Protocols & Zero-Knowledge

Zero-Knowledge Proofs (ZKP):
Interactive Proof Systems (Sigma protocols)
Non-Interactive ZK (NIZK)
zk-SNARKs: Trusted setup, QAPs (Quadratic Arithmetic Programs)
zk-STARKs: Scalability and Transparency (no trusted setup)
Bulletproofs (Range proofs)
Multi-Party Computation (MPC):
Shamir’s Secret Sharing
Garbled Circuits (Yao’s Protocol)
Oblivious Transfer (OT)
Threshold Cryptography (k-of-n signatures)
Homomorphic Encryption:
Partially Homomorphic (RSA, Paillier)
Somewhat Homomorphic (SWHE)
Fully Homomorphic Encryption (FHE): Bootstrapping, BGV/BFV schemes

6. Cryptanalysis & Attack Vectors

Mathematical Attacks: Linear and Differential Cryptanalysis, Meet-in-the-Middle
Side-Channel Attacks (SCA):
Timing Analysis, Power Analysis (SPA/DPA)
Electromagnetic Emissivity (TEMPEST)
Acoustic Cryptanalysis
Quantum Attacks:
Shor’s Algorithm: Breaking RSA/ECC in polynomial time
Grover’s Algorithm: Quadratic speedup for searching (brute-forcing symmetric keys)
Implementation Flaws: Padding Oracles, Bleichenbacher’s Attack, ROCA attack

7. Cryptographic Engineering & Hardware

Random Number Generation:
TRNG (True Random Number Generators - Physical entropy)
PRNG (Pseudo-Random) and CSPRNG (Cryptographically Secure)
Hardware Security:
HSMs (Hardware Security Modules)
TPMs (Trusted Platform Modules)
PUFs (Physical Unclonable Functions)
Side-channel countermeasures (Masking, Blinding, Constant-time code)
Formal Verification: Coq, F*, and JASMIN for provably secure implementations.

8. Applied Infrastructure & Regulation

Public Key Infrastructure (PKI):
Certificate Authorities (CA), X.509 standards
Revocation: CRLs and OCSP Stapling
Certificate Transparency (CT) Logs
Security Protocols:
TLS 1.3, SSHv2, IPsec, QUIC
Signal Protocol (Double Ratchet Algorithm)
Regulatory & Ethics:
Export Controls (ITAR)
Data Protection Laws (GDPR, CCPA)
The “Going Dark” debate: Lawful access vs. End-to-End Encryption (E2EE)

To reach an exhaustive level of detail, we must move beyond standard textbooks and into the granular “specialty” domains found in academic journals, RFCs (Request for Comments), and advanced doctoral research.

Here is the expanded, high-resolution index of cryptography, structured to cover the microscopic details of the field.

9. Specialized Mathematical Structures

Academic research often focuses on these specific, high-level objects.

Non-Abelian Cryptography: Braid groups, Thompson groups, and Matrix groups.
Expander Graphs: Using Ramanujan graphs and Cayley graphs for hash function construction.
Vector Spaces & Bilinear Forms: Orthogonal subspaces and their role in functional encryption.
Ideal Lattices: Ideal-SVP (Shortest Vector Problem) and the Ring-LWE (Learning with Errors) over cyclotomic rings.
Hyperelliptic Curve Cryptography (HECC): * Mumford representation of divisors.
Cantor’s algorithm for group law.
Genus-2 curve optimizations.

10. Privacy-Enhancing Technologies (PETs) & Data Privacy

The intersection of cryptography, statistics, and legal compliance.

Differential Privacy (DP):
Local vs. Centralized DP.
The Laplace and Gaussian mechanisms for noise injection.
Composition theorems and Privacy Budgets ( $\epsilon, \delta$ ).
Private Set Intersection (PSI): Protocols for two parties to find common elements without revealing their entire sets.
Searchable Symmetric Encryption (SSE): Enabling keyword searches over encrypted cloud databases.
Order-Preserving & Order-Revealing Encryption (OPE/ORE): Allowing range queries on encrypted data.
Traffic Analysis Resistance:
Mix Networks (Chaum Mixes).
Onion Routing (Tor) and Garlic Routing (I2P).
Cover traffic and dummy packet injection.

11. Advanced Digital Signature Taxonomy

Beyond simple RSA/DSA, these provide specific properties for complex systems.

Blind Signatures: Signer signs a message without seeing its content (e.g., electronic cash).
Group Signatures: Allowing a member to sign on behalf of a group while remaining anonymous to outsiders (but traceable by a group manager).
Ring Signatures: Signatures where the actual signer is hidden among a set of possible signers (used in Monero).
Aggregate Signatures: Combining $n$ signatures on $n$ different messages into a single short signature.
Proxy Signatures: A primary signer delegates their signing authority to a proxy.
Multi-signatures (MuSig): Specifically for blockchain efficiency, where multiple parties sign a single transaction.

12. Secure Multi-Party Computation (MPC) Niche Topics

Beaver Triples: Pre-computation techniques to speed up online MPC phases.
BGW Protocol: (Ben-Or, Goldwasser, and Wigderson) for information-theoretic MPC.
Cut-and-Choose Protocols: Ensuring fairness and correctness in Garbled Circuits.
Oblivious Transfer Extension: Efficiently scaling thousands of OTs using only a few base OTs.
Malicious vs. Semi-Honest Adversary Models: Designing protocols that survive active interference versus passive “curious” servers.

13. Blockchain & Distributed Ledger Cryptography (DLT)

The technical “guts” of decentralized systems.

Consensus Mechanisms: * BFT (Byzantine Fault Tolerance) and Paxos/Raft.
Proof of Stake (PoS) variants: Ouroboros, Casper, Slashing conditions.
Verifiable Random Functions (VRF): Used for secret leader election in blockchains.
Verifiable Delay Functions (VDF): Preventing “front-running” by forcing a sequential time delay.
Merkle Trees & Their Variants:
Sparse Merkle Trees (SMT).
Merkle Mountain Ranges (MMR).
Patricia Tries (Ethereum state storage).
Layer 2 Cryptography: Optimistic Rollups vs. ZK-Rollups.

14. Hardware-Level Attacks & Countermeasures

Fault Injection Attacks:
Voltage Glitching and Clock Glitching.
Laser Fault Injection (LFI).
Cache-Timing Attacks: Exploiting CPU speculative execution (Spectre, Meltdown, L1TF).
Hardware Trojans: Malicious modifications to circuitry during the manufacturing process.
Micro-architectural Side Channels: Branch predictor leaks and port contention.
Rowhammer: Using memory disturbance to flip bits in adjacent rows, potentially breaking RSA keys in RAM.

15. The Standardization & Lifecycle Process

How academic theory becomes industrial reality.

NIST PQC Standardization: The multi-year vetting process for Kyber, Dilithium, Falcon, and SPHINCS+.
CFRG (Crypto Forum Research Group): The IETF’s bridge between academia and engineering.
FIPS 140-3: Security requirements for cryptographic modules.
Cryptographic Agility Models: How to architect systems for “Algorithm Retirement.”
The “Crypto Wars”: Export controls, the Dual_EC_DRBG backdoor controversy, and the Clipper Chip history.

16. Steganography & Covert Channels

The science of hiding the message itself.

LSB (Least Significant Bit) Encoding: Hiding data in image/audio pixels.
Linguistic Steganography: Hiding data within natural language text patterns.
Network Steganography: Using TCP/IP header fields (TTL, Sequence numbers) to leak data.
Spread Spectrum Steganography: Robust watermarking that survives compression.

17. Quantum Cryptography (The Physics Layer)

Quantum Bit Commitment: Protocols for committing to a value using quantum states.
Quantum Coin Flipping: Achieving fairness between two remote parties using quantum mechanics.
Entanglement-Based Protocols: Using Bell states for secure communication (E91 protocol).
Measurement-Device-Independent QKD (MDI-QKD): Protecting against detector-side-channel attacks.

18. Functional & Identity-Based Schemes

Fuzzy Identity-Based Encryption: Using biometric data (which may have slight variations) as a public key.
Hierarchical IBE (HIBE): Allowing a root authority to delegate key-generation powers to sub-authorities.
Broadcast Encryption: Encrypting a message such that only a specific subset of users can decrypt it (e.g., Pay-Per-View TV).
Traitor Tracing: Identifying which user leaked their decryption key to a pirate site.

19. Historical & Obsolete Cryptography (Academic Context)

Manual Ciphers: Playfair, ADFGVX, Solitaire.
Rotor Machines: Enigma (German), Typex (British), SIGABA (US), Purple (Japanese).
Cylindrical/Disk Ciphers: Jefferson Disk, M-94.
Early Digital: DES (Data Encryption Standard) and the history of S-box design criteria.

Part index

The scientific map, the equations, and the bibliographic spine for this part of the Knowledge Base are collected in the part indexes:

General · Engineering · Computer science · Playbooks
Indexes · Glossary · SUMMARY

Wikipedia snapshots in the chapter card offer offline orientation; the part indexes carry the full citation trails.