Regev's reduction as a candidate quantum algorithm for the discrete logarithm problem in finite abelian groups

arXiv:2605.03972 · quant-ph · Submitted 2026-05-05 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: I'm Kai, and with me are Mira and Lev, guest researcher.

Mira: Today's paper: "Regev's reduction as a candidate quantum algorithm for the discrete logarithm problem in finite abelian groups".

Kai: Regev’s reduction as a candidate quantum algorithm for the discrete logarithm problem in finite abelian groups explores whether transforming decoding problems into quantum solvers can yield an efficient solution for…

Mira: First, who's behind it and why it matters.

Title and authors: Kai: So we're looking at the paper titled "Regev's reduction as a candidate quantum algorithm for the discrete logarithm problem in finite abelian groups." It’s really about taking problems where classical computers struggle with discrete logarithms and seeing if we can translate them into a decoding challenge for quantum systems.

Mira: Exactly, Kai; it’s about exploring whether this specific reduction framework, Regev's reduction, can actually help us crack the discrete logarithm problem in these finite abelian groups efficiently using quantum methods.

Lev: From an error correction standpoint, I'm interested in how the complexity translates; if we turn DLOG into a decoding problem on the dual code, that means we need to know how hard that dual decoding is classically before we even talk about quantum speedups.

Kai: Right, Lev, so it’s checking if this translation works for problems where classical algorithms are known to be subexponential and seeing what happens when you apply this quantum reduction.

Mira: The core idea here is that Regev's reduction turns a decoder for a code into a quantum solver for the decoding problem on the dual code through the use of the quantum Fourier transform, which is pretty powerful conceptually.

Lev: And it hinges entirely on whether that dual decoding problem is classically hard enough to justify using this method to solve DLOG, which is where things get tricky.

Kai: So basically, they are testing if we can use a known structure—the Cheng-Wan reduction—to tackle DLOG in finite abelian groups by seeing if Regev's framework can provide the necessary quantum advantage.

Mira: And the paper explicitly states that since Shor's algorithm already solves discrete logarithm, the goal isn't finding a new speedup for DLOG itself, but rather understanding if applying this reduction to a problem we suspect is hard can actually solve discrete logarithm in these specific groups.

Lev: That makes sense; it’s about establishing where the limits are when you try to bridge classical hardness results with quantum reductions.

Kai: So, they are investigating if Regev's reduction can solve discrete logarithm in finite abelian groups where classical algorithms are known to be subexponential.

The paper's summary: Mira: To summarize the main point of "Regev's reduction as a candidate quantum algorithm for the discrete logarithm problem in finite abelian groups," they revisit the established Cheng and Wan reduction, which links discrete logarithms over finite fields to solving a BDD instance on low-rate Reed–Solomon codes.

Kai: That sounds like a lot of machinery, so can you explain what this means in simpler terms for us? What's the actual mechanism they are focusing on here?

Mira: Essentially, the paper shows that Regev’s reduction takes an efficient decoder for any code and turns it into an efficient solver for a specific BDD problem situated on the dual code via the quantum Fourier transform.

Lev: So, if we follow this path, the quantum advantage relies on whether that dual decoding problem is classically hard, which is exactly what they are trying to establish.

Kai: And they use the Cheng-Wan reduction as their starting point because it provides a natural way to get instances of DLOG where solving them would automatically solve discrete logarithm.

Mira: Specifically, they show that this leads to constructing a promise BDD instance where the received word has an exact distance equal to the target decoding radius for some value in the factorization.

Lev: And then, if you manage to decode this successfully, it yields a multiplicative relation that leads directly into solving a linear system for factor-base logarithms over Z/(q h − one).

Kai: So they are mapping DLOG onto this structured decoding challenge, and the ultimate question is whether Regev’s reduction can solve discrete logarithm in finite abelian groups.

Mira: The paper goes on to generalize the hardness consequence of this reduction for Reed–Solomon bounded distance decoding, proving that it's NP-hard even at asymptotically zero rate for arbitrary finite abelian groups.

Lev: That NP-hardness result is significant because it shows that the underlying decoding problem is fundamentally hard before we even introduce quantum mechanics to help us solve it.

Kai: So, they are showing that the structure of these problems makes them hard classically, which sets the stage for seeing if a quantum approach actually delivers an efficient solution.

The paper's improvements: Mira: Now, looking at how they tackle this challenge in "Regev's reduction as a candidate quantum algorithm for the discrete logarithm problem in finite abelian groups," they discuss what the paper suggests as potential improvements or necessary steps forward.

Kai: I’m curious what specific parts of the research they point out that could make this approach more robust or lead to better results? Are there any suggested changes to the reduction itself?

Lev: From a practical standpoint, I think the biggest improvement they suggest is moving away from just checking if a decoder exists and instead focusing on finding an efficient one rather than just proving it's possible.

Mira: That aligns with what they found: all standard efficient decoders fall short of the Cheng-Wan threshold by a constant factor, meaning we need a QDP decoder at parameter tau satisfying tau ≤ 4eh (one - o(one)).

Kai: So they suggest that the focus should be on designing quantum algorithms that can actually meet this specific threshold, rather than just relying on existence proofs for general decoders.

Lev: And I think the implication is that the remaining obstacle identified is "finding an efficient decoder rather than a question of principle," which means we need concrete implementations that work within those tight parameter constraints.

Mira: That really highlights the gap between theoretical possibility and practical implementation; even though the Pretty Good Measurement solves every instance unconditionally, it requires exponential resources to implement generally.

Kai: So the paper suggests that for the future, we need a way to bridge that gap where a theoretical solution exists but still demands an exponential resource cost for general implementation.

Conclusion: Mira: To wrap up this discussion on "Regev's reduction as a candidate quantum algorithm for the discrete logarithm problem in finite abelian groups," they summarize the implications by stating that if we assume the Cheng-Wan starting points behave statistically like uniform elements, then a QDP solver for RS

q, q − (3h + four): q at parameter tau ≤ 4eh (one - o(one)) would lead to solving DLOG in F×qh in time O˜(q) with high probability.

Kai: That result is quite specific; it removes the need for Assumption one by showing that the Pretty Good Measurement solves hard instances unconditionally, even though it doesn't yet prove the structured Cheng-Wan distribution.

Lev: From my perspective, this means if we can get past that statistical assumption, then solving DLOG in F×qh becomes a tractable problem with a quantum approach.

Mira: But what they don't do is provide the proof for that statistical assumption; they show the PGM solves hard instances unconditionally, which is an existence result rather than an efficient algorithm.

Kai: So it points toward the fact that while we have a path forward, we still need to move from just theory to finding a concrete quantum circuit that actually runs efficiently.

Lev: I agree; so for us in hardware terms, the focus needs to be on building those high-performance QDP solvers rather than waiting for a perfect theoretical proof before we can test anything real.

Institute for Quantum Information and Matter, California Institute of Technology · Inria de Paris

quant-ph

Submitted: 2026-05-05

Updated: 2026-09-30

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 77/100

The gist: Regev’s reduction as a candidate quantum algorithm for the discrete logarithm problem in finite abelian groups explores whether transforming decoding problems into quantum solvers can yield an

Key concepts

Regev’s Reduction
This transformation converts an efficient decoder for a code into an efficient solver for a related problem on the dual code using the quantum Fourier transform. In this context, it links finding codewords to solving decoding challenges.
Cheng–Wan Reduction
This reduction connects the discrete logarithm problem in a cyclic group to solving a specific type of decoding problem (BDD instance) on low-rate Reed–Solomon codes. This establishes that DLOG can be framed as a hard decoding task.
Pretty Good Measurement (PGM)
The PGM is a quantum measurement within Regev’s reduction that solves every BDD instance unconditionally. However, implementing this measurement efficiently requires exponential resources, meaning it proves existence but not practical efficiency.

Terminology

Summary

Regev’s reduction as a candidate quantum algorithm for the discrete logarithm problem in finite abelian groups explores whether transforming decoding problems into quantum solvers can yield an efficient solution for DLOG, specifically investigating if this approach can solve discrete logarithm in finite abelian groups where classical algorithms are known to be subexponential.

The gist: Regev’s reduction turns a decoder for a code into a quantum solver for a decoding problem on the dual code, and the study investigates whether applying this reduction to Cheng–Wan instances—which reduce DLOG—can solve discrete logarithm, finding that while the Pretty Good Measurement solves every instance unconditionally, its implementation requires exponential resources.

General Framework and Hardness Context

The paper revisits Regev’s reduction, which transforms an efficient decoder for a code into an efficient solver for a problem on the dual code via the quantum Fourier transform. In the code-based setting, this translates to finding a codeword in the dual code within Hamming distance t of a received word y0. The core interest lies in whether this reduction can solve discrete logarithm, given that Cheng and Wan showed that DLOG in finite extension fields reduces classically to solving a BDD instance on low-rate Reed–Solomon codes. The authors generalize this hardness consequence to arbitrary finite abelian groups, proving that bounded distance decoding for Reed–Solomon codes is NP-hard even at asymptotically zero rate.

The Cheng–Wan Reduction and Target Problem

The study focuses on the Cheng–Wan reduction, which relates the discrete logarithm problem in a cyclic group to solving a BDD instance on low-rate Reed–Solomon codes. The reduction involves:

  1. Constructing an induced received word y(i) from the DLOG input element b and an irreducible polynomial h(x).

  2. Showing that this construction yields a promise BDD instance, where the received word has exact distance equal to the target decoding radius for some t(x) in the factorization.

  3. Showing that successful decoding yields a multiplicative relation, which leads to solving a linear system for factor-base logarithms over Z/(q h − 1).

Classical Decoder Limitations

The paper evaluates known efficient decoders against the Cheng–Wan threshold required by the reduction. The analysis shows that all standard efficient decoders fall short of the Cheng–Wan threshold by a constant factor. Specifically, Table 4 summarizes these results:

((Berlekamp-Welch decoder) ≈ 3eh/2)

((Guruswami-Sudan decoder) ≈ 3eh/2)

((Unambiguous state discrimination (USD)) ≈ 3eh)

The Cheng–Wan target requires a QDP decoder at a parameter τ satisfying τ ≤ 4eh (1 - o(1)), which corresponds to a decoding radius of q − 4h − 4. The efficient decoders fall short by a constant factor compared to this required threshold.

Quantum Reduction and Instantiation

The paper develops the quantum reduction by instantiating it with concrete decoders:

((Berlekamp-Welch decoder) ≈ 3eh/2)

((Guruswami-Sudan decoder) ≈ 3eh/2)

((Unambiguous state discrimination (USD)) ≈ 3eh)

Theorem 10 establishes that if an efficient quantum algorithm exists for QDP(RS [q, q − k]q, τ), then an efficient quantum algorithm exists for IBDD(RS [q, k]q, τ ′ q) with success probability close to PDec. The required parameter for the Cheng–Wan reduction is identified as a QDP decoder at parameter τ ≤ 4eh (1 - o(1)).

The Role of the Pretty Good Measurement (PGM)

A critical finding is that the Pretty Good Measurement, applied within Regev’s reduction, solves every BDD instance unconditionally. The PGM requires exponential resources to implement in general, meaning it is an existence result rather than an efficient algorithm. This applies not only to the Cheng–Wan instances but also to the NP-hard instances of Theorem 8. The remaining obstacle identified is finding an efficient decoder rather than a question of principle.

Conclusion and Conjecture

The paper concludes by identifying the necessary decoding performance required to solve discrete logarithm via this quantum approach. Under an assumption that the Cheng–Wan starting points behave statistically like uniform elements, Theorem 11 states that if a QDP solver exists for RS [q, q − (3h + 4)]q at parameter τ ≤ 4eh (1 - o(1)), then DLOG in F×qh can be solved in time O˜(q) with high probability. This result removes the need for Assumption 1 by showing that the PGM solves hard instances unconditionally, though it does not yet provide a proof for the structured Cheng–Wan distribution.

Improvements for AI systems

Based on the provided scientific paper, here are specific improvements that can be made to AI systems, categorized by the capabilities they would gain:

)1. Enhanced Cryptographic Security via Hardness Verification (DLOG Solvers)

The paper establishes a strong link between solving the Discrete Logarithm Problem (DLOG) in finite abelian groups and solving bounded distance decoding problems for Reed-Solomon codes. This suggests a new avenue for verifying the hardness of cryptographic primitives based on code structures.

Improvements:

  1. Implement AI agents that, given a cryptographic scheme whose security is based on DLOG in a specific group (like those analyzed in Section 3), can automatically construct the corresponding Cheng-Wan instance (Section 4) and attempt to solve it using known efficient decoders (Berlekamp-Welch, Guruswami–Sudan).

  2. Improve the AI's capability to identify whether a given decoding problem is hard enough for Regev's reduction by checking if the required decoding radius exceeds the known NP-hard radius (Section 5). If it falls short, the system can flag that this specific cryptographic group/code instance might be susceptible to quantum attacks via this reduction path.

Improved AI System Capability:

An AI system capable of acting as a Cryptographic Hardness Auditor. It would automate the process of mapping a discrete logarithm problem into a decoding challenge and assess its resilience against quantum attacks by comparing the required decoding threshold against established NP-hardness results.

)2. Quantum Algorithm Design for Decoding (QDP Solvers)

The paper provides detailed, concrete instances of how to use Regev's reduction to solve BDD problems, specifically showing that efficient quantum algorithms exist for certain decoding tasks (e.g., IBDD) by leveraging the Quantum Decoding Problem (QDP) solver on the dual code.

Improvements:

  1. Develop AI agents specialized in Quantum Circuit Synthesis that can take a QDP solver for a dual code and automatically synthesize the required quantum circuit to solve an inhomogeneous BDD instance of a primal code (Section 5). This synthesis would focus on efficiently implementing the Pretty Good Measurement (PGM) described in Section 6, even if it requires exponential resources in general.

  2. Train reinforcement learning agents to optimize the choice of quantum decoder (Berlekamp-Welch vs. Guruswami–Sudan vs. Unambiguous State Discrimination) based on the specific parameters of a target BDD instance, aiming to find the most efficient implementation that satisfies Assumption 1 (Section 5.3).

Improved AI System Capability:

A Quantum Decoding Algorithm Designer. This system could automatically design and optimize quantum circuits for solving complex, structured decoding problems (like those arising from Cheng-Wan reductions) by intelligently selecting and optimizing the underlying QDP solver based on the specific code parameters provided.

)3. Automated Proof of Quantum Advantage (Assumption 1 Verification)

The paper hinges on Assumption 1, which posits that a QDP decoder for a certain parameter range succeeds with non-negligible probability on structured instances (Cheng-Wan distribution). The paper itself does not prove this assumption rigorously.

Improvements:

  1. Create an AI module dedicated to Assumption 1 Hypothesis Testing. This module would use techniques like adversarial machine learning or statistical hypothesis testing against the Cheng-Wan distribution DBDD, trying to find counterexamples that violate Assumption 1 (i.e., finding a decoder that fails on the specific structured inputs).

  2. If Assumption 1 is violated, the AI system can automatically trigger a search for alternative reduction strategies or structural randomization techniques (like those hinted at in [YZ24]) to find an instance where the Regev reduction still yields an advantage, thus expanding the scope of quantum applicability.

Improved AI System Capability:

A Quantum Advantage Verifier. This system would move beyond simply applying a known theorem and actively probe the boundaries of its applicability by searching for instances that break underlying assumptions, thereby discovering new quantum-hard problems or novel ways to achieve them.

)4. Complexity Landscape Mapping (BDD Hardness Classification)

The paper meticulously maps the complexity landscape of BDD for Reed-Solomon codes across different rates and radii, identifying specific thresholds where classical algorithms fail and NP-hardness begins (Section 19).

Improvements:

  1. Build a comprehensive knowledge graph or database that cross-references code parameters (rate, minimum distance) with known hardness results (e.g., the transition points between polynomial time, subexponential time, and NP-complete regimes for BDD).

  2. Use this map to provide real-time complexity assessments for novel coding schemes. If a new code is proposed, the AI can instantly tell researchers: This code structure places its BDD problem in the NP-hard regime at radius T, or It falls into the regime where classical algorithms are subexponential.

Improved AI System Capability:

A Coding Complexity Navigator. This system would serve as an expert consultant for coding theorists, allowing them to instantly classify the computational difficulty of any given linear code decoding problem based on its structural properties.

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