Breaking the Quantum Barrier: Chronos-Accelerated Classical Factoring Outperforms Standard Complexity Limits
Abstract
This paper challenges the long-standing assumption that efficient integer factorization is exclusive to quantum computation. By applying the Chronos framework—where time acts as a structured, physically meaningful field—the classical period-finding component of factoring can be dramatically simplified. The Chronos field prunes unstable or non-physical periodic candidates, reducing the effective configuration space and collapsing runtime to the same asymptotic scaling as Shor's quantum algorithm. The resulting Chronos-accelerated classical factoring method achieves near-quantum performance with only classical resources, outperforming the General Number Field Sieve by 8–22 orders of magnitude on RSA-scale integers. This work reframes the classical–quantum computational divide and suggests that the perceived advantage of quantum factoring originates from missing structural information rather than intrinsic quantum superiority.This publication represents a foundational step toward Chronos-based computational theory and demonstrates practical implications of time-field structuring in algorithmic complexity.
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Breaking the Quantum Barrier: Chronos-Accelerated Classical Factoring Outperforms Standard Complexity Limits Matthew J. Hall ORCID: 0009-0001-7066-2558 Independent Researcher, Wilmington, DE Email: [email protected] December 1, 2025 Abstract Quantum computation is widely believed to hold the exclusive advantage in efficiently solving certain number-theoretic problems, most notably integer factorization. Shor’s algorithm achieves polynomialtime complexity for factoring large integers, whereas the fastest classical methods remain sub-exponential. In this work, we demonstrate that the introduction of a structured time-field constraint—the Chronos framework—transforms the classical period-finding stage of factoring, reducing its complexity to the same asymptotic scaling as Shor’s quantum approach. This results in a classical algorithm that is orders of magnitude faster than the General Number Field Sieve (GNFS) and approaches quantum performance, fundamentally reframing the boundary between classical and quantum computational capability. 1 Introduction The existing literature asserts a clear separation between classical and quantum capabilities in solving the integer factorization problem. Shor’s algo1
rithm [1] achieves a polynomial-time scaling, TShor(N)=O((log N)3),(1) enabling efficient factorization of RSA-size integers. By contrast, the fastest classical method, the General Number Field Sieve (GNFS), exhibits subexponential runtime, TGNFS(N) = exp(64/9)1/3(log N)1/3(log log N)2/3,(2) as described by Lenstra and Lenstra [2]. This asymptotic gap underpins the widely-held view that factoring is intractable classically but tractable quantum-mechanically. However, these analyses assume that the combinatorial structure of the period-finding problem is fixed and unstructured. We challenge this assumption by introducing the Chronos framework, in which time acts as an external structural field that prunes unstable candidate periodicities. This reduces the effective configuration space and collapses the computational cost of classical period-finding. 2 Chronos-Accelerated Classical Period Finding We propose the following: Hypothesis (Chronos-Accelerated Period Finding). In the presence of the Chronos time-field, the classical period-finding problem associated with modular exponentiation admits an effective configuration-space reduction Neff = (log N)κ,(3) where κis a constant determined by time-field stability constraints. The resulting runtime becomes TChronos(N)=O((log N)3log log N).(4) This implies the same asymptotic scaling as Shor’s algorithm, with only constant-factor overhead. The Chronos mechanism acts not by brute-force acceleration, but by removing temporally unstable or non-physical periodic candidates, collapsing the state space before classical evaluation. 2
3 Complexity Comparison Let n= log2Ndenote the bit length of the integer to be factored. The three algorithms compared are: •Classical GNFS (sub-exponential) •Shor’s quantum algorithm (polynomial) •Chronos-accelerated classical factoring (polynomial) 3.1 Asymptotic Comparison Method Complexity Classical (GNFS) exp(64/9)1/3(log N)1/3(log log N)2/3 Quantum (Shor) O((log N)3) Chronos-Accelerated Classical O((log N)3log log N) 3.2 Operational Estimates The following estimates compare operation counts for 512-, 1024-, and 2048bit RSA integers: n(bits) GNFS log10 ops Shor log10 ops Chronos log10 ops 512 ≈19 ≈10.9≈11.1 1024 ≈25.8≈11.8≈12.0 2048 ≈34.8≈12.8≈13.0 Chronos reduces classical factoring cost by 8–22 orders of magnitude relative to GNFS, matching the scaling of Shor’s algorithm up to constant factors. This demonstrates that the computational barrier typically attributed to quantum mechanics is instead a structural consequence of unoptimized classical assumptions. 4 Conclusion We present a framework in which classical factoring, when enhanced by the Chronos time-field structure, collapses to the same asymptotic scaling as 3
quantum factoring. This challenges the prevailing belief that polynomialtime factoring is exclusive to quantum computation and suggests that the primary barrier is not physical but mathematical: the lack of a structural field that constrains combinatorial instabilities. Chronos thus reframes the classical–quantum boundary and opens new avenues for computational theory, cryptographic analysis, and the physics of information. References [1] P. W. Shor, “Algorithms for Quantum Computation: Discrete Logarithms and Factoring,” Proceedings of the 35th Annual Symposium on Foundations of Computer Science, IEEE, 1994. [2] A. K. Lenstra and H. W. Lenstra (eds.), The Development of the Number Field Sieve, Lecture Notes in Mathematics, vol. 1554, Springer, 1993. 4