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arXiv:2601.02518

Diffusion Computation versus Quantum Computation: A Comparative Model for Order Finding and Factoring

Carlos Alberto Cadavid, Paulina Hoyos, Jay A. Jorgenson, Lejla Smajlovic, J. D. Vélez

Published Jan 5, 2026• 0 citations
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Abstract

We study a hybrid computational model for integer factorization in which the only non-classical resource is access to an \emph{iterated diffusion process} on a finite graph. Concretely, a \emph{diffusion step} is defined to be one application of a symmetric stochastic matrix (the half-lazy walk operator) to an $\ell^{1}$--normalized state vector, followed by an optional readout of selected coordinates. Let $N\ge 3$ be an odd integer which is neither prime nor a prime power, and let $b\in(\mathbb{Z}/N\mathbb{Z})^\ast$ have odd multiplicative order $r={\rm ord}_N(b)$. We construct, without knowing $r$ in advance, a weighted Cayley graph whose vertex set is the cyclic subgroup $\langle b\rangle$ and whose edges correspond to the powers $b^{\pm 2^t}$ for $t\le \lfloor \log_2 N\rfloor+1$. Using an explicit spectral decomposition together with an elementary doubling lemma, we show that $r$ can be recovered from a single heat-kernel value after at most $O((\log_2 N)^2)$ diffusion steps, with an effective bound. We then combine this order-finding model with the standard reduction from factoring to order finding (in the spirit of Shor's framework) to obtain a randomized factorization procedure whose success probability depends only on the number $m$ of distinct prime factors of $N$. Our comparison with Shor's algorithm is \emph{conceptual and model-based}. We replace unitary $\ell^2$ evolution by Markovian $\ell^1$ evolution, and we report complexity in two cost measures: digital steps and diffusion steps. Finally, we include illustrative examples and discussion of practical implementations.

DOI
10.48550/arxiv.2601.02518

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In this article we develop an algorithm which computes a divisor of an integer NNN, which is assumed to be neither prime nor the power of a prime. The algorithm uses discrete time heat diffusion on a finite graph. If NNN has mmm distinct prime factors, then the probability that our algorithm runs successfully is at least p(m)=1−(m+1)/2mp(m) = 1-(m+1)/2^{m}p(m)=1−(m+1)/2m. We compute the computational complexity of the algorithm in terms of classical, or digital, steps and in terms of diffusion steps, which is a concept that we define here. As we will discuss below, we assert that a diffusion step can and should be considered as being comparable to a quantum step for an algorithm which runs on a quantum computer. With this, we prove that our factorization algorithm uses at most O((log⁡N)2)O((\log N)^{2})O((logN)2) deterministic steps and at most O((log⁡N)2)O((\log N)^{2})O((logN)2) diffusion steps with an implied constant which is effective. By comparison, Shor's algorithm is known to use at most O((log⁡N)2log⁡(log⁡N)log⁡(log⁡log⁡N))O((\log N)^{2}\log (\log N) \log (\log \log N))O((logN)2log(logN)log(loglogN)) quantum steps on a quantum computer. As an example of our algorithm, we simulate the diffusion computer algorithm on a desktop computer and obtain factorizations of N=33N=33N=33 and N=1363N=1363N=1363.
Abstract We show that given the order of a single element selected uniformly at random from ZN∗{\mathbb {Z}}_N^*ZN∗​ Z N ∗ , we can with very high probability, and for any integer N, efficiently find the complete factorization of N in polynomial time. This implies that a single run of the quantum part of Shor’s factoring algorithm is usually sufficient. All prime factors of N can then be recovered with negligible computational cost in a classical post-processing step. The classical algorithm required for this step is essentially due to Miller.
Shor's algorithm is one of the most important quantum algorithm proposed by Peter Shor [Proceedings of the 35th Annual Symposium on Foundations of Computer Science, 1994, pp. 124--134]. Shor's algorithm can factor a large integer with certain probability and costs polynomial time in the length of the input integer. The key step of Shor's algorithm is the order-finding algorithm, the quantum part of which is to estimate s/rs/rs/r, where rrr is the ``order" and sss is some natural number that less than rrr. {{Shor's algorithm requires lots of qubits and a deep circuit depth, which is unaffordable for current physical devices.}} In this paper, to reduce the number of qubits required and circuit depth, we propose a quantum-classical hybrid distributed order-finding algorithm for Shor's algorithm, which combines the advantages of both quantum processing and classical processing. {{ In our distributed order-finding algorithm, we use two quantum computers with the ability of quantum teleportation separately to estimate partial bits of s/rs/rs/r.}} The measuring results will be processed through a classical algorithm to ensure the accuracy of the results. Compared with the traditional Shor's algorithm that uses multiple control qubits, our algorithm reduces nearly L/2L/2L/2 qubits for factoring an LLL-bit integer and reduces the circuit depth of each computer.
This thesis contains a collection of efficient quantum algorithms and classical pre- and post-processing algorithms for various number- and group-theoretical problems with concrete cryptanalytical applications. More specifically, the focus is on the integer factoring problem (IFP), the discrete logarithm problem (DLP), and the order-finding problem (OFP), and on certain important variations of these problems. It is for example explained how the OFP and complete IFP may be solved efficiently in a single run of a reduced version of Shor's algorithm for the OFP with very high success probability, as evidenced by a lower bound that is 1 − o(1), how the short DLP in groups of unknown order may be solved efficiently and with an advantage over Shor's algorithm for the DLP (that furthermore requires the order to be known), as evidenced by a lower bound on the single-run success probability that is 1 − o(1), how the RSA IFP may be solved efficiently and with an advantage over Shor's algorithm for the IFP, by reducing the RSA IFP to a short DLP, how the DLP may heuristically be solved efficiently, and with high success probability that is 1 − o(1), in a single run of a reduced version of Shor’s algorithm for the DLP that admits efficient implementation, how the DLP may be solved in groups of unknown order, such as Schnorr groups, by jointly solving an OFP and a DLP at lower overall cost but higher per-run cost compared to first solving the OFP and then the DLP, how tradeoffs — between the number of group operations that are evaluated quantumly and the number of runs that are required — may be achieved in all of the aforementioned algorithms, and how all of the aforementioned algorithms may be simulated efficiently classically, for arbitrary large cryptographically relevant problem instances, when the solution to the instance is known, enabling the number of runs required when performing tradeoffs to be tightly estimated. It is furthermore for example explained how Regev's recent quantum algorithm for the IFP may be extended to the DLP and OFP, yielding i) a reduction in the circuit size and depth by a factor compared to Shor for these problems when working in , for N an n-bit integer, at the expense of performing runs, and ii) slight improvements of Regev's orginal algorithm for the IFP, how the above algorithms may be simulated efficiently classically, for large special-form problem instances that are classically tractable, and that Regev’s post-processing, and the post-processing for our extensions, are robust to an arbitrarily large fraction < 1 of the runs being bad. The above has implications for cryptography based on the intractability of the IFP and DLP — including but not limited to for widely used schemes such as RSA, DSA and Diffie–Hellman. The algorithms in this thesis are all derived from, and make notable improvements to, the seminal works of Shor. Some of the contributions are joint works with other authors.
Quantum integer factorization is a groundbreaking field that explores the use of quantum algorithms, notably Shor's algorithm, to efficiently factorize large integers into their prime components. Its potential implications on modern encryption techniques, such as the Rivest-Shamir-Adleman (RSA) algorithm, are of considerable interest. This paper's central aim is to contrast the computational time and implementation of the Integer Factorization Problem (IFP) between classical and quantum computers. The inner workings of Quantum Shor's algorithm are broken down step by step, and the estimated processing time is contrasted with the classical General Number Field Sieve (GNFS) approach. Given the exponential speedup provided by quantum computing, this comparison helps assess Shor's algorithm's possible advantage over GNFS in factoring large numbers. Direct comparisons with GNFS are currently difficult due to Shor's algorithm's reliance on large-scale quantum computer development for practical execution speeds. This paper focuses on solving integer factorization up to2n2^{n}2nbits.
Quantum algorithms are at the heart of the ongoing efforts to use quantum mechanics to solve computational problems unsolvable on ordinary classical computers. Their common feature is the use of genuine quantum properties such as entanglement and superposition of states. Among the known quantum algorithms, a special role is played by the Shor algorithm, i.e. a polynomial-time quantum algorithm for integer factorization, with far reaching potential applications in several fields, such as cryptography. Here we present a different algorithm for integer factorization based on another genuine quantum property: quantum measurement. In this new scheme, the factorization of the integer NNN is achieved in a number of steps equal to the number kkk of its prime factors, -- e.g., if NNN is the product of two primes, two quantum measurements are enough, regardless of the number of digits nnn of the number NNN. Since kkk is the lower bound to the number of operations one can do to factorize a general integer, one sees that a quantum mechanical setup can saturate such a bound.