Quadratic residuosity problem

The quadratic residuosity problem in computational number theory is to decide, given integers ${\displaystyle a}$ and ${\displaystyle N}$, whether ${\displaystyle a}$ is a quadratic residue modulo ${\displaystyle N}$ or not. Here ${\displaystyle N=p_{1}p_{2}}$ for two unknown primes ${\displaystyle p_{1}}$ and ${\displaystyle p_{2}}$, and ${\displaystyle a}$ is among the numbers which are not obviously quadratic non-residues (see below).

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