Regev's Factoring Algorithm: Quantum Factorization

• Regev's Factoring Algorithm is a quantum method that generalizes Shor’s approach by leveraging a multidimensional exponent space and tailored small-prime arithmetic.
• The algorithm employs lattice reduction and modular exponentiation optimizations, achieving significant reductions in circuit depth and qubit count over earlier methods.
• Advanced techniques like parallel spooky pebbling and Fibonacci-based optimizations mitigate hardware limitations, enhancing the algorithm’s practical relevance for cryptanalysis.

Regev’s Factoring Algorithm is a family of quantum algorithms for integer factorization that generalizes and extends Shor’s period-finding approach by operating in a higher-dimensional exponent space and utilizing arithmetic on small primes. Its design enables significant quantum circuit depth and gate count reductions over earlier methods. The algorithm’s variants leverage lattice structures, tailored modular arithmetic, and advanced resource optimization techniques, with practical relevance for attacking cryptographically sized integers as quantum hardware progresses.

1. Multidimensional Quantum Factoring: Algorithmic Structure

At its core, Regev’s algorithm employs a $d$-dimensional generalization of Shor’s method. Given an $n$-bit composite $N$, instead of using a single register and period-finding function $z \mapsto a^z \bmod N$, the algorithm constructs a product function over small group elements: $$(z_1, z_2, \ldots, z_d) \mapsto \prod_{i=1}^d b_i^{z_i} \bmod N$$ with $b_i$ chosen as small primes (or their squares), and each $z_i$ bounded to a short interval (typically $|z_i| \leq D/2$ for $D$ set polynomially in $n$). The quantum state is a superposition weighted by a Gaussian function $P_R(z) = \exp(-\pi \|z\|^2/R^2)$: $$\sum_{z \in \{-D/2, \ldots, D/2-1\}^d} P_R(z) \ket{z_1, \ldots, z_d, X(z) \bmod N}$$ The algorithm applies $d$ independent QFTs to the control registers and measures to retrieve vectors $w$ that encode information about the algebraic relations among the $b_i$ modulo $N$. Classical postprocessing extracts a short vector $r = (r_1, \ldots, r_d)$ satisfying $\prod b_i^{r_i} \equiv \pm 1 \pmod{N}$, from which a nontrivial factor is derived by $\gcd(\prod b_i^{r_i}-1, N)$ (Ekerå et al., 2023, Pawlitko et al., 13 Feb 2025).

2. Lattice Structure, Postprocessing, and Robustness

Measurement outcomes correspond to cosets in the dual lattice $C^* / \mathbb{Z}^d$ where $$C = \{ z \in \mathbb{Z}^d : \prod b_i^{z_i} = 1 \bmod N \}$$. A sufficient number of shots (typically $m \geq d+4$) are sampled, and their measurement vectors collected. Lattice reduction (LLL or BKZ) on these vectors yields a short relation vector outside the “trivial” sublattice associated with $\pm 1$, with high probability.

Recent work introduces noise robustness in postprocessing. Even under corruption of a constant fraction of circuit runs (due to hardware errors or sampling noise), filtering based on the “well-spread” condition on sample distributions and careful basis construction ensures recovery of a correct short relation (Ragavan et al., 2023, Ekerå et al., 2023). The modified postprocessing algorithm iterates over vector subsets, performing basis reduction and short vector tests to filter out erroneous samples, guaranteeing success under mild distributional assumptions.

3. Modular Exponentiation and Space Optimization

A distinctive feature of Regev’s quantum arithmetic is its modular exponentiation routine. The original algorithm leverages repeated squaring and modular multiplication, incurring $O(n^{3/2})$ qubit space per run (with $n$ the bit-length of $N$). Space-efficient optimizations—such as implementing the exponentiation via Fibonacci numbers in the Zeckendorf representation—reduce space complexity to $O(n \log n)$ qubits while maintaining $O(n^{3/2} \log n)$ gate depth (Ragavan et al., 2023). This is achieved by expressing exponents $z_i$ as $\sum_j z_{i,j} F_j$, accumulating products in-place using paired accumulator registers, and employing reversible modular multiplication circuits using dirty ancillas and modular inverses.

Table: Quantum Resource Comparison

Method	Qubits	Gate Count	Circuit Depth
Shor's algorithm	$O(n \log n)$	$O(n^2 \log n)$	$O(n^2)$
Regev (original)	$O(n^{3/2})$	$O(n^{3/2} \log n)$	$O(n^{3/2})$
Regev (Fibonacci)	$O(n \log n)$	$O(n^{3/2} \log n)$	$O(n^{3/2})$
Parallel spooky pebbling	$O(\log \ell)$	$2\ell$ depth	Optimal

Here, $\ell$ is the pebbling line graph length, $\log D$ the effective exponent size, and $S_\times(n)$ the multiplication ancilla qubits.

4. Parallel Spooky Pebbling and Circuit Depth Reduction

The recent introduction of parallel spooky pebble games (Kahanamoku-Meyer et al., 9 Oct 2025) enables further reduction of modular multiplication depth in Regev’s arithmetic. By combining mid-circuit measurements (“ghosting”) and parallel scheduling of pebble moves, the modular exponentiation computation on the line graph of intermediate squarings achieves an optimal multiplication depth of $2\ell = 4\log D$ using no more than $2.47\log \ell$ ancillary registers (pebbles).

For 4096-bit modulus $N$, the scheme achieves modular multiplication depth $\approx 193$ per run—surpassing previous Fibonacci-based approaches ($680$ depth) and optimized Shor circuits ($444$ depth). Space usage is strictly logarithmic in the exponent size, dramatically reducing memory requirements and making Regev’s algorithm more competitive in contexts where hardware coherence time is limited.

5. Theoretical Foundation: Number-Theoretic Conjectures and Proofs

Regev’s dimensional exponent space relies on a foundational conjecture: every element in the subgroup generated by small primes $b_1, \ldots, b_d$ modulo $N$ can be written as a short product $\prod b_i^{e_i}$ with $|e_i| \leq e^{O(d)}$, allowing efficient search and modular multiplication.

An unconditional proof of correctness follows from analytic number theory tools, notably zero-density estimates for Dirichlet $L$-functions (Pilatte, 25 Apr 2024). For $b_i$ chosen from primes up to $X = d^{10^3 d}$, every subgroup element is representable in short form with overwhelming probability. These results guarantee that the lattice of multiplicative relations among $b_i$ has a short basis, securing the reliability of the quantum search and the classical postprocessing phase.

6. Extensions: Discrete Logarithms, Order Finding, and Generic Model Limits

Ekerå and Gärtner’s extension (Ekerå et al., 2023) modifies the construction to include arbitrary group elements (not necessarily small), facilitating discrete logarithm and group order finding attacks. The algorithm encodes the DLP instance by mixing in group elements whose exponents encode the unknown logarithm, generating equations of the form $$e_1z_1 + \cdots + e_{d-1}z_{d-1} + ez_d \equiv 0 \pmod{r}$$ and recovering $e$ via modular inversion.

A modified version of Regev’s algorithm is analyzed in the quantum generic ring model (Hhan, 17 Feb 2024). Here, the algorithm outputs a relatively small integer $Z = \prod b_i^{z_i} - 1$ without access to $N$ for in-circuit modular reduction, with factorization achieved via $\gcd(Z, N)$. The paper establishes a lower bound: $Q = \Omega\left( \frac{\log N}{\log(2tw)} \right)$ on the number of quantum ring operations required, using the compression lemma and linear algebra, showing that any “small-output” generic algorithm (including Regev’s) intrinsically requires logarithmic quantum complexity.

7. Practical Implementation, Limitations, and Outlook

Experimental implementations of Regev’s algorithm (Pawlitko et al., 13 Feb 2025) use Qiskit simulators and LLL-based postprocessing on modest-sized $N$. Performance is influenced by parameters $d$ (dimension) and $qd$ (exponent range), with careful tuning needed to balance runtime and success rate. For small integers, Shor’s algorithm remains faster in practice—Regev’s constant factors and circuit overhead dominate asymptotic gains. As $N$ grows, Regev’s approach has theoretical efficiency advantages, but in current practice, further optimizations (e.g., space reduction, improved pebbling strategies) are necessary for cryptographically relevant sizes.

High-level comparisons (Ekerå et al., 23 May 2024) indicate that even space-optimized versions of Regev’s algorithm (utilizing Ragavan–Vaikuntanathan and pebbling improvements) do not yet outperform state-of-the-art Shor variants for large $N$ unless non-computational quantum memory is abundant and cheap. A plausible implication is that further algorithmic and implementation refinements may enable Regev’s algorithm to become a practical candidate for quantum cryptanalysis as hardware matures.

Summary

Regev’s factoring algorithm, through its multidimensional quantum structure, advanced modular arithmetic techniques, and lattice-based postprocessing, establishes a new algorithmic foundation for integer factorization and related cryptanalytic problems. Resource optimizations such as parallel spooky pebbling have delivered significant circuit depth and space savings, propelling Regev’s variants towards greater practicality. Unconditional correctness, robust postprocessing under noise, and extensions to other hard problems further underline its innovative character, yet substantial work remains before it surpasses current optimized quantum methods in large-scale deployments.