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

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\u27 identity-based encryption scheme and statistical indistinguishability