16.1 Preliminaries
According to the American security researchers David McGrew and John Viega, ”the Galois/Counter Mode is a block cipher mode of operation that uses universal hashing over a binary Galois field to provide authenticated encryption [114].”
Before studying the internals of the algorithm, we need to quickly cover two mathematical aspects: the finite field used by GCM and the way multiplication is done in that finite field.
16.1.1 The Galois field 𝔽2128
GCM uses multiplication over a finite field. In mathematics, finite fields are also referred to as Galois fields in honor of the 19th-century French mathematician Evariste Galois, hence the name Galois counter mode.
We have already encountered finite fields in section 7.6 Finite Fields in Chapter 7, Public-Key Cryptography. You may go back to that section to refresh your memory, but we are repeating the basic facts here for your convenience.
The field used in GCM has 2128 elements, which...
