Reductions between LPN and symplectic LPN in the low-noise regime

Determine whether there exists a polynomial-time reduction from Learning Parity with Noise (LPN) with low noise rate p = Θ(1/√n) to symplectic LPN (sympLPN) in the same low-noise regime, or conversely a reduction from sympLPN with p = Θ(1/√n) to LPN. Establishing either direction would clarify whether sympLPN-based schemes inherit security from LPN or constitute a strictly new assumption in this parameter range.

Background

The paper builds cryptographic primitives from symplectic LPN (sympLPN) derived from quantum stabilizer decoding and argues that, unlike standard LPN, sympLPN carries symplectic structure. Prior work reduces LPN to sympLPN only for higher noise; at p = Θ(1/√n) those reductions become vacuous because low-noise LPN is already easy for small k.

Showing a reduction in either direction in the low-noise regime would settle whether sympLPN is merely a rephrasing of LPN or a genuinely distinct assumption for post-quantum cryptography.