Bridging Number-Theoretic Primality Testing And Algorithmic Optimization: An Adaptive Cryptographic Framework
This paper proposes an adaptive framework for cryptographic primality testing that integrates arbitrary-precision arithmetic, hardware-aware sieving, and a machine-learning-guided Miller-Rabin witness selector anchored on quantum entropy. The core idea is to reduce expensive modular exponentiations by using lightweight structural features of a candidate integer to prioritize bases likely to expose compositeness early. It targets the key-generation bottleneck for RSA and ECC, yet the contribution is more architectural than empirically demonstrated, resting heavily on a withdrawn survey and a toy Python benchmark.
The paper reads as a survey with an unvalidated proof-of-concept rather than a rigorous systems contribution. Its empirical centerpiece—that a Random Forest witness selector costs only $27.58\%$ of a single $1024$-bit modular exponentiation—rests on a Python benchmark on a single CPU core with only $500$ samples, with no comparison to industrial libraries such as GMP or OpenSSL. The authors claim "rigorous mathematical proofs to guarantee security even when the prime distribution is biased," but the argument merely invokes Mihăilescu's existing theorem rather than supplying new proofs. Most damaging, the taxonomic foundations in Section 3 and Table 1, and many complexity bounds, are drawn from AbuDaqa et al. (arXiv:2006.08444), a manuscript withdrawn by its authors because "errors were identified in several sections of the manuscript."
The survey of primality tests is generally accurate standalone material. Section 3.1 correctly notes that AKS "suffers from massive constant overheads" and is "drastically slower than probabilistic methods," and Section 2 accurately describes how Montgomery multiplication eliminates costly trial divisions by mapping values into the Montgomery domain where $\tilde{a} = aR \pmod{N}$. The observation that segmented sieving should be bypassed for isolated point queries to avoid $O(\sqrt{R})$ initialization overhead is also sound engineering, and the analysis of Takhanov et al.'s barren-plateau results correctly explains why continuous gradient descent fails in modular arithmetic spaces.
Four serious flaws undermine the paper. First, its taxonomy and complexity claims in Section 3 and Table 1 rely heavily on AbuDaqa et al. (arXiv:2006.08444), a survey withdrawn by its own authors after identifying multiple errors. Second, the proposed ML witness selector is validated only through a toy Python prototype ($500$ samples, single CPU core) with no adversarial or Carmichael-type inputs, and—critically—no proof that non-uniform, model-dependent base selection preserves the standard Miller-Rabin compositeness bound of $<(1/4)^k$. Third, the repeated claim of "rigorous mathematical proofs to guarantee security even when the prime distribution is biased" is misleading: the paper simply cites Mihăilescu's existing theorem rather than supplying new proofs. Finally, the framework remains an architectural sketch with no reproducible end-to-end implementation, and the quantum-entropy and fastECPP modules are never empirically evaluated.
The empirical evidence is insufficient to support the performance claims. The benchmark compares Python implementations of Miller-Rabin and Solovay-Strassen on $500$ samples rather than benchmarking against optimized arbitrary-precision libraries such as GMP, FLINT, or OpenSSL. The $27.58\%$ overhead figure compares the ML pipeline against a single $1024$-bit modular exponentiation, yet the authors never measure whether the selector actually reduces the mean number of exponentiations across representative candidate distributions, including adversarial pseudoprimes. Comparisons to related work are descriptive rather than competitive: the paper surveys classical algorithms extensively but does not position its end-to-end system against existing cryptographic software stacks.
No source code, trained model weights, or benchmark scripts are provided. Hyperparameters of the Random Forest, the precise feature extraction logic, and the candidate generation distribution are unspecified. The experiments were run in Python 3.10 on a consumer laptop using monotonic timers, but without artifacts, independent reproduction is impossible. The fastECPP and quantum entropy components are described only at the architectural level and are never empirically evaluated, leaving the reader unable to verify whether the full pipeline achieves its claimed security and throughput goals.
The security of modern public-key cryptographic systems, such as RSA, Diffie-Hellman, and ECC, depends entirely on the computational intractability of the integer factorization and discrete logarithm problems. For these asymmetric algorithms to operate, a core requirement is that the system must be capable of generating, testing, and simultaneously proving the primality of extraordinarily large numbers. To meet those requirements, this study proposes a comprehensive framework, bridging clas- sical number theory with modern machine learning models and optimization algorithms. Specifically, the paper delineates an adaptive, arbitrary-precision architecture dedicated exclusively to primality testing. Concurrently, the research deeply analyzes the theoretical foundations of testing methods (both deterministic and probabilistic), hardware-level optimization techniques, and the application of quantum-secured entropy sources, while providing rigorous mathematical proofs to guarantee security even when the prime distribution is biased.
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