A permutation rearranges a set so that every element maps to exactly one element, with no collisions. Because the mapping is one-to-one, it can always be reversed. section 2.5.2
In cryptography, permutations provide reversible transformations. SHA-3 uses the fixed Keccak-f permutation inside a sponge construction, while AES uses a key-selected permutation from plaintext blocks to ciphertext blocks. section 2.5.2 section 4.2.2
Track a permutation through these properties:
- Each input has one output. section 2.5.2
- Each output corresponds to one input. section 2.5.2
- A fixed-size input produces a fixed-size output. section 2.5.2 section 4.2.2
Shapes, bits, and encrypted blocks
Imagine arrows connecting four shapes on the left to the same four shapes on the right. Following an arrow transforms a shape; tracing the mapping backward recovers it. The same idea maps every possible 8-bit input to one 8-bit output, or every 128-bit AES plaintext to one 128-bit ciphertext.
