Fq×

Fq× refers to the multiplicative group of units of the finite field Fq. A finite field Fq is a set of elements with addition and multiplication operations that satisfy certain algebraic properties, and it contains exactly q elements, where q must be a power of a prime number. The multiplicative group of units of a field consists of all the non-zero elements of that field under the operation of multiplication. Therefore, Fq× is the set of all non-zero elements of Fq.

The number of elements in Fq× is q-1. This is because Fq has q elements in total, and the only element excluded from the multiplicative group of units is the additive identity, which is 0. The operation in Fq× is multiplication modulo the defining polynomials of the field. Fq× is a cyclic group, meaning that there exists at least one element (a generator) such that all other elements in the group can be obtained by repeatedly multiplying this element by itself. The order of the group Fq× is q-1.

The structure and properties of Fq× are fundamental in various areas of mathematics and computer science, including cryptography, coding theory, and number theory. For example, in cryptography, algorithms like Diffie-Hellman key exchange and the ElGamal cryptosystem rely on the properties of these multiplicative groups. The study of discrete logarithms in Fq× is a computationally hard problem, which forms the basis of the security for many cryptographic systems.