Equivalence of search and decision variants of symplectic LPN

Determine whether the search and decision variants of symplectic LPN (sympLPN) are equivalent under polynomial-time reductions; that is, establish whether distinguishing sympLPN instances from random is polynomial-time equivalent to recovering the hidden vector in sympLPN.

Background

In adapting LPN-style constructions to sympLPN, the paper emphasizes decision variants for cryptography, while search is natural for learning-from-noise formulations. For LPN, search and decision are essentially equivalent, but the authors note this is not known for sympLPN.

Clarifying this relationship would impact the tightness of security reductions for sympLPN-based schemes and the design of future primitives.