O11a — Quadratic residuosity modulo a composite

Definitions and mathematical statement

Let $n\ge2$ be a composite integer and let $a$ be an integer relatively prime to $n$. Regard $a$ as an element of the unit group $(\mathbb Z/n\mathbb Z)^\times$. It is a quadratic residue modulo $n$ if there exists an integer $x$ such that \[ x^2\equiv a\pmod n. \] Any such $x$ is automatically coprime to $n$. The input consists of the binary encodings of $n$ and $a$; one may reduce $a$ to its representative between $0$ and $n-1$.

Can a deterministic classical algorithm decide this property for every valid pair $(n,a)$ in a number of bit operations polynomial in the total input length? The output is one bit indicating whether a square root exists. Neither a factorization of $n$ nor a square root of $a$ is furnished as auxiliary information or required as output.

The domain includes even composites and prime powers as well as products of distinct odd primes. Restricting attention to odd $n$ makes the Jacobi symbol available, but the condition that this symbol equals one does not characterize squares for all composite moduli. Accordingly, efficiently evaluating that symbol alone does not decide the present problem. The requested bound is uniform and unconditional, and measures time against binary length rather than against the magnitude of the modulus.

Short English statement

Can quadratic residuosity modulo an arbitrary composite integer be decided deterministically in polynomial time?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 8400012 (AMR-083-0012), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Adleman & McCurley - Open Problems in Number Theory Complexity II (1994), O11a, p10 / LNCS p300. Original source indexed by AMR-083-0012. https://mccurley.org/papers/open.ps.gz.