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We provide unconditional results and conditional ones under the assumption of GRH (Generalized Riemann Hypothesis) on the distribution of quadratic residues and quadratic non-residues inℤ  /  N  ℤ    $\mathbb {Z}/N\mathbb {Z}$   , whereN  =  p  q    $N= pq$    is an RSA modulus used in the Goldwasser–Micali cryptosystem. The paper also discusses cryptographic implications of the results obtained.

The distribution of quadratic residues and non-residues in the Goldwasser–Micali type of cryptosystem