Paper 2024/2034
The Jacobi Factoring Circuit: Quantum Factoring with Near-Linear Gates and Sublinear Space and Depth
Abstract
We present a compact quantum circuit for factoring a large class of integers, including some whose classical hardness is expected to be equivalent to RSA (but not including RSA integers themselves). Most notably, we factor $n$-bit integers of the form $P^2 Q$ with $\log Q = \Theta(n^a)$ for $a \in (2/3, 1)$ in space and depth sublinear in n (specifically, $\widetilde{O}(\log Q)$) using $\widetilde{O}(n)$ quantum gates; for these integers, no known classical algorithms exploit the relatively small size of $Q$ to run asymptotically faster than general-purpose factoring algorithms. To our knowledge, this is the first polynomial-time circuit to achieve sublinear qubit count for a classically-hard factoring problem. We thus believe that factoring such numbers has potential to be the most concretely efficient classically-verifiable proof of quantumness currently known. Our circuit builds on the quantum algorithm for squarefree decomposition discovered by Li, Peng, Du, and Suter (Nature Scientific Reports 2012), which relies on computing the Jacobi symbol in quantum superposition. The technical core of our contribution is a new space-efficient quantum algorithm to compute the Jacobi symbol of $A$ mod $B$, in the regime where $B$ is classical and much larger than $A$. Our circuit for computing the Jacobi symbol generalizes to related problems such as computing the greatest common divisor and modular inverses, and thus could be of independent interest.
Note: Minor updates.
Metadata
- Available format(s)
-
PDF
- Category
- Foundations
- Publication info
- Published elsewhere. Major revision. STOC 2025
- Keywords
- factoringquantumShor's algorithmsquarefree decomposition
- Contact author(s)
-
gkm @ mit edu
sragavan @ mit edu
vinodv @ mit edu
kvankirk @ g harvard edu - History
- 2025-06-05: last of 4 revisions
- 2024-12-17: received
- See all versions
- Short URL
- https://ia.cr/2024/2034
- License
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CC BY
BibTeX
@misc{cryptoeprint:2024/2034, author = {Gregory D. Kahanamoku-Meyer and Seyoon Ragavan and Vinod Vaikuntanathan and Katherine Van Kirk}, title = {The Jacobi Factoring Circuit: Quantum Factoring with Near-Linear Gates and Sublinear Space and Depth}, howpublished = {Cryptology {ePrint} Archive, Paper 2024/2034}, year = {2024}, url = {https://eprint.iacr.org/2024/2034} }