Cryptography is the study of “Mathematical Systems” involving two kinds of security protocols: Privacy and Authentication. The mathematical concepts from the branch of number theory known as Modular arithmetic, Quadratic residue are significantly useful in cryptography. Cryptography Deals with large number integers i.e. integers as big as hundred digits and more. In such situation identifying whether an integer “a” is quadratic residue modulo “p” where p is Prime can be achieved using Legendre and Jacobi Symbol. This paper introduces to the mathematical concepts of Quadratic Residue, Fermat's little theorem, Euler’s criterion and Legendre and Jacobi symbol. However it has been observed that Jacobi symbol in case of some example fails to give correct result and therefore the Limitation for predicting the quadratic residue.
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