Polynomial-time reductions between MSI and SIS/LWE

Determine whether any polynomial-time reductions exist between the Modular Symbol Inversion (MSI) problem and the Short Integer Solution (SIS) or Learning With Errors (LWE) problems, given that existing worst-case/average-case reductions for SIS or LWE do not apply to MSI.

Background

The authors analyze how MSI relates to lattice problems such as SIS and LWE. Although MSI can be written as a linear system modulo ℓ^m when path constraints are ignored, the structured sparsity and combinatorial constraints inherent to MSI distinguish it from SIS/LWE.

They explicitly note that known worst-case/average-case reductions for SIS or LWE do not transfer to MSI and that no polynomial-time reductions are currently known between MSI and these lattice problems, highlighting a gap in understanding formal relationships between these problem families.