Existence of a hard uniform sequence of 3-RSA moduli

Ascertain whether there exists a deterministic, uniform sequence B(n) of 3-RSA integers—one modulus per input length n—such that no polynomial-time algorithm F(n), given only n, outputs the prime factors of B(n). Additionally, determine whether such a sequence can itself be generated by a uniform algorithm.

Background

To obtain learning separations under a single fixed distribution per input size, the authors consider positing a sequence B(n) of 3-RSA moduli for which no polynomial-time factoring algorithm exists. This would allow fixing the distribution while retaining hardness.

They note this assumption is non-standard and question both the existence of such a hard sequence and whether it can be generated uniformly, expressing doubt about the justification of these requirements. Resolving this would clarify whether fixed-distribution separations in DCR-based constructions are attainable under plausible assumptions.

References

This could be possible if one posits the existence of a sequence $B(n)$ of 3-RSA integers (one per size), for which there exists no poly time algorithm $F(n)$ which returns the factors of $B(n)$. This is not a standard assumption, and it is not clear why such a sequence should exist. Furthermore, to have a reasonable setting for comparison against classical uniform learning algorithms, it would be natural to demand that $B(n)$ itself be a uniform algorithm. This adds to the already demanding assumptions, and it is not clear to the authors whether they could be justified.

— Machine learning with minimal use of quantum computers: Provable advantages in Learning Under Quantum Privileged Information (LUQPI)  (2601.22006 - Bokov et al., 29 Jan 2026) in Section 5.1, Semi-supervised LUQPI