## Cryptology ePrint Archive: Report 2019/638

On the Distribution of Quadratic Residues and Non-residues Modulo Composite Integers and Applications to Cryptography

F.L. Tiplea and S. Iftene and G. Teseleanu and A.-M. Nica

Abstract: We develop exact formulas for the distribution of quadratic residues and non-residues in sets of the form $a+X=\{(a+x)\bmod n\mid x\in X\}$, where $n$ is a prime or the product of two primes and $X$ is a subset of integers with given Jacobi symbols modulo prime factors of $n$. We then present applications of these formulas to Cocks' identity-based encryption scheme and statistical indistinguishability.

Category / Keywords: public-key cryptography / Jacobi symbol, probability distribution, statistical distance, identity-based encryption