Regev's Algorithm: Quantum Lattice Reductions

• Regev's algorithm is a collection of quantum methods that employ high-dimensional lattice sampling and Fourier transforms to reduce complex problems like factorization and discrete logarithms to lattice and subset-sum challenges.
• It integrates quantum subroutines—such as discrete Gaussian state preparation and quantum Fourier transforms—with classical lattice reduction techniques (e.g., LLL, BKZ) to extract secure cryptographic parameters.
• The approach reshapes security assumptions in lattice-based cryptography, driving advances in circuit optimization, noise robustness, and hybrid quantum-classical processing for cryptanalytic tasks.

Regev's algorithm refers to families of quantum algorithms developed by Oded Regev and collaborators that leverage high-dimensional lattice structures and quantum sampling techniques to solve cryptanalytic problems previously intractable on classical computers, including integer factorization, discrete logarithm, the dihedral hidden subgroup problem, and coset/state distinguishing problems in coding theory. Notable instantiations include the high-dimensional quantum factoring/LWE reductions, the dihedral hidden shift approach, and quantum reductions for code-based and lattice-based cryptography. These algorithms have reshaped complexity assumptions for lattice-based cryptography, provided novel quantum resource tradeoffs, and catalyzed extensive research into both algorithmic theory and quantum circuit optimization.

1. Algorithmic Frameworks: Quantum Lattice Sampling and Collimation

Regev's algorithmic schemes share a high-level blueprint: a quantum subroutine samples from a high-dimensional structure (lattice dual, coset, or phase space), reducing the original problem to a lattice or subset-sum instance that is subject to classical post-processing.

For the factoring/discrete-logarithm setting, let $N$ be an $n$-bit composite. The algorithm selects $d$ small coprime bases $a_1,\dots,a_d$ and defines the lattice

$$L = \{ z \in \mathbb{Z}^d : \prod_{i=1}^d a_i^{z_i} \equiv 1 \mod N \},$$

with short nonzero $z\in L$ enabling nontrivial factor recovery via $\gcd(\prod_{i}a_i^{z_i} \pm 1, N)$. A quantum superposition over a discrete Gaussian distribution

$$| \psi_0\rangle = \frac{1}{\sqrt{S}} \sum_{x\in[-2^{k-1},2^{k-1})^d} e^{-\pi \|x\|^2 / s^2} |x\rangle,$$

is prepared; then the unitary

$$|x\rangle |0\rangle \mapsto |x\rangle \left| \prod_i a_i^{x_i} \bmod N \right\rangle,$$

is applied, after which a $d$-fold QFT and measurement yields samples approximating the dual lattice $n$0. Post-quantum sampling, classical lattice-reduction techniques (e.g., LLL, BKZ) extract relations among the $n$1 for factor extraction (Falcó et al., 16 Jun 2026, Pawlitko et al., 13 Feb 2025, Ekerå et al., 2023).

In the dihedral hidden subgroup or hidden shift setting, the quantum circuit repeatedly produces “phase states” from coset oracles, performs QFTs to obtain random phase vectors, and subsequently reduces the search for the hidden shift to a subset-sum instance in classical post-processing (Remaud et al., 2022, Kuperberg, 2011).

2. Complexity and Resource Analysis

Regev's quantum factoring algorithm achieves asymptotic improvements in quantum gate and depth complexity at the cost of increased circuit width and classical post-processing overhead. For $n$2 an $n$3-bit integer:

• Gate Count per Quantum Run: $n$4 for high-dimensional Regev (Ragavan et al., 2023, Yang et al., 22 Nov 2025). By allowing superpolynomial time for classical post-processing (e.g., via harder lattice reduction), the gate count can be further reduced at the expense of increased classical time.
• Width and Qubit Count: The original qubit space scales as $n$5, in contrast to Shor’s $n$6. Optimizations based on space-efficient modular exponentiation (Fibonacci, pebbling) reduce space to $n$7 or even $n$8 at the cost of moderate increases in gate depth (Ragavan et al., 2023, Kahanamoku-Meyer et al., 9 Oct 2025, Yang et al., 22 Nov 2025).
• Number of Quantum Runs: $n$9 parallel quantum runs are needed to extract sufficient lattice samples for post-processing.
• Classical Post-Processing: An $d$0 LLL/BKZ lattice reduction is applied to $d$1 samples; for generic code/coset sampling problems, classical complexity may depend critically on the efficiency of underlying decoders (Chailloux et al., 2024).

For the dihedral hidden shift, query and time tradeoffs are precisely quantified: polynomial quantum space, $d$2 quantum queries, and $d$3 classical time per bit for subset-sum solving, with $d$4 the best-known exponent (Remaud et al., 2022). Subexponential quantum time is available with more aggressive sieve-based approaches (Kuperberg, 2011).

3. Algorithmic Optimizations and Space–Depth Trade-Offs

Substantial research has focused on circuit resource optimization for Regev’s algorithms:

• Intermediate-Uncomputation ("Pebbling") and Space-Efficient Exponentiation: Uncomputation strategies allow modular exponentiation with $d$5 ancilla registers, reducing space from $d$6 to $d$7 (Ragavan et al., 2023, Yang et al., 22 Nov 2025). Binary recursion, $d$8-ary recursion, or parallel/spooky pebbling (where mid-circuit measurements "ghost" registers) brings further improvements. At $d$9, parallel spooky pebbling reduces the depth of modular multiplications from 680 (Fibonacci-based) or 444 (Shor) to 193 using 12 pebbles (space $a_1,\dots,a_d$0 qubits) (Kahanamoku-Meyer et al., 9 Oct 2025).
• Fibonacci-Number Exponentiation: Reversible exponentiation using Zeckendorf (Fibonacci) representations achieves $a_1,\dots,a_d$1 qubit count with $a_1,\dots,a_d$2 gates, effectively matching the lowest asymptotic resource bounds among known schemes (Ragavan et al., 2023).
• Noise Robustness: Lattice-based filtering and robust error models for the classical post-processing allow Regev's approach to tolerate a constant fraction of corrupted quantum runs without compromising overall factor recovery, enhancing feasibility on noisy intermediate-scale devices (Ragavan et al., 2023, Ekerå et al., 2023).

4. Extensions: Discrete Logarithms, Order Finding, and Code-Based Problems

Regev’s lattice-sampling strategy generalizes naturally to discrete logarithm and order-finding problems in various group settings:

• Discrete Logarithms: The lattice is recentered to encode relations among group elements and the unknown exponent, and quantum sampling plus lattice reduction can extract the secret exponent in $a_1,\dots,a_d$3 quantum runs, $a_1,\dots,a_d$4 qubits, and $a_1,\dots,a_d$5 gates per run, under comparable assumptions to factoring (Ekerå et al., 2023).
• Order Finding and Euler Totient: The classical post-processing is adapted to produce the order or Euler’s totient value, enabling full factorization via classical number-theoretical reductions once the order is known (Ekerå et al., 2023).
• Code and Lattice Problems: A general quantum reduction framework leverages periodic quantum states and coset sampling for syndrome decoding, with soft decoders (notably Koetter–Vardy for Reed-Solomon codes) enabling quantum speedups on decoding problems such as $a_1,\dots,a_d$6 and optimal polynomial interpolation (Chailloux et al., 2024). The approach is generic and extends to coset sampling for arbitrary linear codes over finite fields, with complexity dominated by the available classical (soft) decoder oracle.

5. Connections to Lattice-Based Cryptography and Worst-Case/Avg-Case Reductions

Regev’s quantum algorithm for Learning with Errors (LWE) and its associated worst-case/average-case reductions became foundational for post-quantum cryptography:

• LWE and Hardness Reduction: The (quantum) reduction from $a_1,\dots,a_d$7-SVP to LWE establishes the security foundation for LWE-based cryptosystems (Aggarwal et al., 2022). Allowing superpolynomial classical reductions (e.g., via stronger lattice reduction or sampling) tightens the reduction further: polynomially-hard LWE is equivalent to SVP with approximation factor $a_1,\dots,a_d$8 hard for $a_1,\dots,a_d$9-time algorithms.
• Relevance for Cryptography: These reductions underpin the hardness assumptions for PKE and SKC schemes in the post-quantum setting, with quantum attacks relying on the efficiency and parameters of Regev-type reductions and sampling strategies.

6. Dihedral Hidden Subgroup/Sift and Subset-Sum Reductions

Regev’s approach to the dihedral hidden subgroup (DHS) problem introduced a polynomial-space, subexponential-time quantum algorithm:

• Phase Collimation and Subset-Sum Reduction: Sequences of “coset” phase qubits arising from group-oracle queries are processed via a classical subset-sum instance, with each extraction round reducing the bitwise uncertainty of the hidden shift $$L = \{ z \in \mathbb{Z}^d : \prod_{i=1}^d a_i^{z_i} \equiv 1 \mod N \},$$0 (Kuperberg, 2011, Remaud et al., 2022).
• Time/Query Trade-Offs: The Regev approach uses $$L = \{ z \in \mathbb{Z}^d : \prod_{i=1}^d a_i^{z_i} \equiv 1 \mod N \},$$1 coset-state queries and $$L = \{ z \in \mathbb{Z}^d : \prod_{i=1}^d a_i^{z_i} \equiv 1 \mod N \},$$2 classical time (via the current fastest subset-sum algorithms); Kuperberg sieve-based methods interpolate between these and space-rich regimes, achieving lower quantum time with greater space (Remaud et al., 2022).

Algorithm	Quantum Time	Quantum Queries	Classical Space
Ettinger–Høyer	$$L = \{ z \in \mathbb{Z}^d : \prod_{i=1}^d a_i^{z_i} \equiv 1 \mod N \},$$3	$$L = \{ z \in \mathbb{Z}^d : \prod_{i=1}^d a_i^{z_i} \equiv 1 \mod N \},$$4	$$L = \{ z \in \mathbb{Z}^d : \prod_{i=1}^d a_i^{z_i} \equiv 1 \mod N \},$$5
Regev (subset sum)	$$L = \{ z \in \mathbb{Z}^d : \prod_{i=1}^d a_i^{z_i} \equiv 1 \mod N \},$$6	$$L = \{ z \in \mathbb{Z}^d : \prod_{i=1}^d a_i^{z_i} \equiv 1 \mod N \},$$7	$$L = \{ z \in \mathbb{Z}^d : \prod_{i=1}^d a_i^{z_i} \equiv 1 \mod N \},$$8
Kuperberg (sieve)	$$L = \{ z \in \mathbb{Z}^d : \prod_{i=1}^d a_i^{z_i} \equiv 1 \mod N \},$$9	$z\in L$0	subexp/exp in $z\in L$1

7. Experimental Realizations, Practical Performance, and Open Challenges

Experimental benchmarks of Regev’s algorithm, including actual hardware implementations (e.g., IBM’s superconducting devices and QMIO platforms), demonstrate that:

• For small $z\in L$2:
  • Quantum advantage over Shor has not been established; Regev’s circuits are shallower but more fragile to hardware noise due to higher-dimensional Fourier structure (Falcó et al., 16 Jun 2026, Pawlitko et al., 13 Feb 2025).
  • Parameter tuning (e.g., lattice scaling width $z\in L$3) and post-processing can recover or exceed Shor’s success rates on certain small instances (Pawlitko et al., 13 Feb 2025).
• Space-Optimized Circuits: Intermediate-uncomputation schemes have enabled factoring $z\in L$4 on real hardware with total qubit count as low as 9, and favorable output distributions under noise, indicating practical feasibility for larger-scale demonstrations as device reliability and circuit synthesis improve (Yang et al., 22 Nov 2025).
• Open Challenges: Key technical hurdles include scalable, noise-robust quantum state preparation (especially for discrete Gaussian states), optimized lattice reduction for large $z\in L$5 and $z\in L$6, deeper understanding of quantum-classical trade-offs in post-processing, and generalization of the methodology to structures beyond finite fields (e.g., elliptic curves). Further, parameter selection, circuit resource optimization, and hybrid quantum-classical workflows remain active areas of research (Yang et al., 22 Nov 2025, Kahanamoku-Meyer et al., 9 Oct 2025, Ekerå et al., 2024).

• (Falcó et al., 16 Jun 2026): "From Period Finding to Lattice Sampling: Experimental Insights into Shor's and Regev's Factoring Algorithms"
• (Pawlitko et al., 13 Feb 2025): "Implementation and Analysis of Regev's Quantum Factorization Algorithm"
• (Ekerå et al., 2023): "Extending Regev's factoring algorithm to compute discrete logarithms"
• (Ragavan et al., 2023): "Space-Efficient and Noise-Robust Quantum Factoring"
• (Yang et al., 22 Nov 2025): "Space-Optimized and Experimental Implementations of Regev's Quantum Factoring Algorithm"
• (Kahanamoku-Meyer et al., 9 Oct 2025): "Parallel Spooky Pebbling Makes Regev Factoring More Practical"
• (Chailloux et al., 2024): "Quantum advantage from soft decoders"
• (Remaud et al., 2022): "Time and Query Complexity Tradeoffs for the Dihedral Coset Problem"
• (Aggarwal et al., 2022): "Lattice Problems Beyond Polynomial Time"
• (Kuperberg, 2011): "Another subexponential-time quantum algorithm for the dihedral hidden subgroup problem"
• (Ekerå et al., 2024): "A high-level comparison of state-of-the-art quantum algorithms for breaking asymmetric cryptography"