Reductions between MSI and supersingular isogeny path problems

Determine whether reductions exist between the Modular Symbol Inversion (MSI) problem—recovering a short relative homology class in H_1(X_0(N), C; Z) from its truncated ℓ-adic period vector—and supersingular isogeny path problems on supersingular isogeny graphs, in order to clarify their potential computational relationship.

Background

The paper introduces the Modular Symbol Inversion (MSI) problem, which asks for recovery of a short path-encoded relative homology class from its truncated ℓ-adic period vector. The authors compare MSI to other hard problems, including supersingular isogeny path problems that underlie isogeny-based cryptography such as SQISign.

While both MSI and isogeny path problems involve searching for short paths in exponentially large graphs, the authors emphasize structural differences in the underlying spaces: MSI operates on modular-symbol homology (e.g., quotients of Bruhat–Tits trees or modular-symbol graphs) rather than on supersingular isogeny graphs of elliptic curves.

In discussing this comparison, the authors explicitly state that there are currently no known reductions between MSI and isogeny path problems, leaving open whether any such reductions exist.