On the Quantum Equivalence between $S|LWE\rangle$ and $ISIS$
For researchers in post-quantum cryptography, this work clarifies the relationship between quantum analogues of LWE and ISIS, showing strong connections but also remaining barriers to full equivalence.
The paper establishes the first fully generic reduction from the Inhomogeneous Short Integer Solution (ISIS) problem to the quantum S|LWE⟩ problem, and explores the reverse direction under certain conditions, clarifying the equivalence landscape between these problems.
Chen, Liu, and Zhandry [CLZ22] introduced the problems $S|LWE\rangle$ and $C|LWE\rangle$ as quantum analogues of the Learning with Errors problem, designed to construct quantum algorithms for the Inhomogeneous Short Integer Solution ($ISIS$) problem. Several later works have used this framework for constructing new quantum algorithms in specific cases. However, the general relation between all these problems is still unknown. In this paper, we investigate the equivalence between $S|LWE\rangle$ and $ISIS$. We present the first fully generic reduction from $ISIS$ to $S|LWE\rangle$, valid even in the presence of errors in the underlying algorithms. We then explore the reverse direction, introducing an inhomogeneous variant of $C|LWE\rangle$, denoted $IC|LWE\rangle$, and show that $IC|LWE\rangle$ reduces to $S|LWE\rangle$. Finally, we prove that, under certain recoverability conditions, an algorithm for $ISIS$ can be transformed into one for $S|LWE\rangle$. We instantiate this reverse reduction by tweaking a known algorithm for $(I)SIS_\infty$ in order to construct quantum algorithm for $S|LWE\rangle$ when the alphabet size q is a small power of 2, recovering some results of Bai et al. [BJK+ 25]. Our results thus clarify the landscape of reductions between $S|LWE\rangle$ and $ISIS$, and we show both their strong connection as well as the remaining barriers for showing full equivalence.