Cryptographic hash functions are foundational to the construction of practical cryptographic systems. A hash function transforms input data of arbitrary length into a fixed-length message digest that is easy to compute but impossible to invert. Once a digest is produced, recovering the original message is computationally infeasible. Hash functions are widely used in cryptographic applications, including verifying data integrity, authenticating messages, generating deterministic randomness, and deriving keys for secure communication protocols.
This article provides a detailed design of the FIPS 202 standard. I'll begin by explaining the sponge construction and its security properties, then examine how SHA-3 achieves diffusion and irreversibility through its internal permutation. Along the way, I'll clarify which components of the design are reversible in isolation and why the overall construction remains a one-way function. Finally, the SHA-3 and SHAKE design will highlight that they are particularly well-suited for modern cryptographic systems, including post-quantum cryptography.
A structured visualization of the Elliptic Curve Method (ECM), relating j-invariant classes and curve selection to Palm Jumeirah’s frond layout.
Imagine the Elliptic Curve Method as exploring Palm Jumeirah, Dubai’s iconic palm-shaped island. The island represents an elliptic curve y² = x³ + ax + b mod M, where M is the number to factor (a product of unknown primes). Fronds are j-invariants classifying curve shapes, points (x, y) are coordinates to probe, and the group order (number of points modulo a hidden prime p) is like the frond’s “explorable paths” bounded by Hasse’s theorem: |#E(Fₚ) — (p+1)| ≤ 2√p.
Imagine the Elliptic Curve Method as exploring Palm Jumeirah, Dubai’s iconic palm-shaped island. The island is the elliptic curve (y² = x³ + ax + b (mod M)), a structured algebraic space defined over a composite modulus M. Each frond of the island corresponds to a j-invariant, classifying the curve’s isomorphism class and defining a unique section of the island. Points on the curve (x, y) are like coordinates along a frond, while the group order corresponds to the frond’s area.
This article provides a detailed design of the FIPS 202 standard. I'll begin by explaining the sponge construction and its security properties, then examine how SHA-3 achieves diffusion and irreversibility through its internal permutation. Along the way, I'll clarify which components of the design are reversible in isolation and why the overall construction remains a one-way function. Finally, the SHA-3 and SHAKE design will highlight that they are particularly well-suited for modern cryptographic systems, including post-quantum cryptography.