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On the Quantum Equivalence between $S|LWE\rangle$ and $ISIS$

André Chailloux, Paul Hermouet

TL;DR

This work analyzes the quantum relationship between ISIS and S|LWE>, introducing a fully generic forward reduction ISIS → S|LWE> and a new intermediate IC|LWE> to bridge the reverse direction. It proves a conditional reverse reduction S|LWE> → IC|LWE> and further shows that, under randomness-recoverable assumptions, IC|LWE> → ISIS, thereby connecting the two problems from both directions. The authors instantiate the reverse path for alphabets with q = 2^l by modifying a Regev-style SIS algorithm, recovering and extending results in the small-power-of-two regime. Overall, the paper clarifies the reduction landscape between S|LWE> and ISIS, identifies key barriers to full equivalence, and opens avenues for new quantum algorithms via these reductions and their intermediate constructions.

Abstract

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.

On the Quantum Equivalence between $S|LWE\rangle$ and $ISIS$

TL;DR

This work analyzes the quantum relationship between ISIS and S|LWE>, introducing a fully generic forward reduction ISIS → S|LWE> and a new intermediate IC|LWE> to bridge the reverse direction. It proves a conditional reverse reduction S|LWE> → IC|LWE> and further shows that, under randomness-recoverable assumptions, IC|LWE> → ISIS, thereby connecting the two problems from both directions. The authors instantiate the reverse path for alphabets with q = 2^l by modifying a Regev-style SIS algorithm, recovering and extending results in the small-power-of-two regime. Overall, the paper clarifies the reduction landscape between S|LWE> and ISIS, identifies key barriers to full equivalence, and opens avenues for new quantum algorithms via these reductions and their intermediate constructions.

Abstract

Chen, Liu, and Zhandry [CLZ22] introduced the problems and as quantum analogues of the Learning with Errors problem, designed to construct quantum algorithms for the Inhomogeneous Short Integer Solution () 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 and . We present the first fully generic reduction from to , valid even in the presence of errors in the underlying algorithms. We then explore the reverse direction, introducing an inhomogeneous variant of , denoted , and show that reduces to . Finally, we prove that, under certain recoverability conditions, an algorithm for can be transformed into one for . We instantiate this reverse reduction by tweaking a known algorithm for in order to construct quantum algorithm for 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 and , and we show both their strong connection as well as the remaining barriers for showing full equivalence.

Paper Structure

This paper contains 23 sections, 25 theorems, 67 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Context
  3. Problems studied
  4. Contributions
  5. Takeaways and Future Work
  6. Preliminaries
  7. Notations
  8. Fourier Transform and Quantum Fourier Transform
  9. Definition
  10. Preliminary calculations around $S|\text{LWE}\rangle$ and $IC|\text{LWE}\rangle$ and tractability results
  11. Preliminary calculations
  12. Tractability bound for $S|\text{LWE}\rangle(\mathbf{A},f)$
  13. The forward direction $\text{ISIS} \rightarrow S|\text{LWE}\rangle$
  14. Characterization of quantum algorithms for $S|\text{LWE}\rangle$
  15. The main algorithm
  16. ...and 8 more sections

Key Result

theorem 1

If we have an efficient algorithm solving $S|\text{LWE}\rangle(\mathbf{A},f)$ with $Im(\widehat{f}) \subseteq T$, then we can construct an efficient algorithm for $\text{ISIS}(\mathbf{A},T)$.

Figures (1)

  • Figure 1: Table of reductions between the different studied problems. An arrow $A \rightarrow B$ means that $A$ reduces to $B$, or in other terms that an efficient algorithm for problem $B$ can be used to construct an efficient algorithm for problem $A$.

Theorems & Definitions (58)

  • definition 1: Inhomogeneous Short Integer Solution $\text{ISIS}(\mathbf{A},T)$
  • definition 2
  • definition 3
  • definition 4
  • theorem 1: Informal
  • theorem 2: informal
  • theorem 3: Informal
  • proposition 1: BJK+25
  • theorem 4: Informal
  • definition 5
  • ...and 48 more