GFq×

GFq× is a mathematical structure representing the multiplicative group of the finite field Fq. A finite field Fq, where q is a prime power, consists of q elements and is equipped with addition and multiplication operations that behave similarly to those of integers. The notation "×" signifies the set of non-zero elements of Fq under the operation of multiplication.

The structure GFq× forms a cyclic group of order q-1. This means that all elements in GFq× can be generated by a single element, called a primitive root. The operation within this group is the standard multiplication inherited from the finite field. Important properties include closure, associativity, the existence of an identity element (which is 1), and the existence of an inverse for every element.

The study of GFq× is fundamental in various areas of mathematics and computer science, including abstract algebra, number theory, cryptography, and coding theory. For instance, the properties of these groups are exploited in the design of public-key cryptosystems and error-correcting codes. Understanding the structure of GFq× allows for the analysis of algorithms that rely on finite field arithmetic. The order of the group, q-1, plays a crucial role in determining the security and efficiency of cryptographic protocols.