AES Encryption

The Advanced Encryption Standard uses a substitution permutation network. This performs two basic operations on a string: substitution of one block into another, or a permutation of bits. The key is used to define a sequence of such operations. Note that the length of the full string is preserved at each key step. This is one of the simplest ways to satisfy Shannon's 1949 measure of cryptographic strength, and is widely used today.

In category theory the definition of operad is based on substitution, which is usually depicted by tree graftings. This has generalisations that allow any number of inputs and outputs, like in the diagrams of modern M Theory. In this language, a combined AES operation of one key step and one substitution-permutation form an object for the 'operad'. Unlike the usual categorical definition, however, this gadget is terminated by the length of the key string.