Evaluation of quadratic residue with the help of Legendre symbol

Cryptography is the study of "Mathematical Systems," which includes two types of security protocols: privacy and authentication. Quadratic residue, a mathematical notion from the discipline of number theory known as Modular arithmetic, is extremely valuable in cryptography. Cryptography is deals with huge numbers, such as integers system with millions of digits or more. In this case, the Legendre symbols may be used to determine if an integer "x" has quadratic residue modulo "p" when p is Prime. This research article explains the mathematical ideas of quadratic residue, Fermat's little theorem, Euler's criteria, and the Legendre symbols. The main goal of this article is to explore the calculation problem of the number of solutions for one kind congruence equation modulo p (an odd prime) using simple methods and character sum properties, and to provide some interesting identities and asymptotic formulas for it.