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

Abstract

Abstract We develop exact formulas for the distribution of quadratic residues and non-residues in sets of the form a + X = { ( a + x ) mod n ∣ x ∈ 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.