Polynomial time factoring algorithm using Bayesian arithmetic

In a previous paper, we have shown that any Boolean formula can be encoded as a linear programming problem in the framework of Bayesian probability theory. When applied to NP-complete algorithms, this leads to the fundamental conclusion that P = NP. Now, we implement this concept in elementary arithmetic and especially in multiplication. This provides a polynomial time deterministic factoring algorithm, while no such algorithm is known to day. This result clearly appeals for a revaluation of the current cryptosystems. The Bayesian arithmetic environment can also be regarded as a toy model for quantum mechanics.

💡 Research Summary

The paper proposes a novel framework that merges Bayesian probability theory with linear programming (LP) to address integer factorisation. Building on the authors’ earlier work, which claimed that any Boolean formula can be encoded as an LP, they argue that this encoding yields a polynomial‑time deterministic algorithm for factoring, and consequently that P = NP. The core idea is to reinterpret elementary arithmetic—specifically multiplication—in a “Bayesian arithmetic” setting. Each binary digit of the multiplicands and product is treated as a discrete random variable that can take the values 0 or 1 with probability 1. Logical operations (AND, OR, NOT) are expressed as conditional probability equations, which are linear and therefore can be incorporated as constraints in an LP model.

In the construction, an n‑bit integer a and an n‑bit integer b are represented by variables (X_{a_i}) and (X_{b_j}). The product c = a·b is expressed bit‑wise: each output bit (c_k) equals the XOR of all pairwise ANDs (a_i ∧ b_j) where i + j = k, together with carry bits. The authors introduce separate variables for the carries and translate every Boolean relation into a linear equality or inequality. The resulting system contains O(n³) variables and constraints, according to the authors, and can be solved by any polynomial‑time LP solver such as an interior‑point method.

To factor a given integer N, the algorithm fixes the bits of N as constants in the LP, leaves the bits of the unknown factors a and b as variables, and solves the LP. A feasible 0‑1 solution corresponds to a factorisation N = a·b. The authors claim that because the LP size grows only polynomially with the bit‑length of N, the whole procedure runs in polynomial time, thereby providing a deterministic factorisation algorithm. They present small‑scale experimental results (up to 64‑bit numbers) showing that the LP can be solved within a few hundred milliseconds, and they argue that the method scales better than classical algorithms such as Pollard‑Rho or the elliptic‑curve method.

The paper then extrapolates to cryptographic implications: if the method truly works in polynomial time for arbitrarily large integers, RSA and other public‑key systems based on the hardness of factoring would be broken, necessitating a complete redesign of modern cryptography. The authors also suggest that the Bayesian arithmetic framework could serve as a toy model for quantum mechanics, hinting at deeper connections between probabilistic computation and quantum phenomena.

From a critical standpoint, several substantial issues arise. First, while the Boolean‑to‑LP translation is mathematically sound, the number of constraints required to capture all pairwise interactions in multiplication grows rapidly. Even if the authors claim O(n³) constraints, a more detailed analysis shows that each output bit depends on O(n) input pairs, and each pair introduces a product term that must be linearised, often leading to O(n⁴) or higher constraint counts. Such growth, though polynomial, may render the approach impractical for the key sizes used in real cryptosystems (2048 bits and above).

Second, standard LP solvers produce real‑valued solutions. The paper assumes that the Bayesian formulation forces every variable to be exactly 0 or 1, but in practice a rounding or additional integer‑programming step is required to enforce binary solutions. Introducing integer constraints transforms the problem into a mixed‑integer linear program (MILP), which is NP‑hard in general and defeats the claimed polynomial‑time guarantee.

Third, the claim that “P = NP” follows from the existence of a polynomial‑size LP for multiplication is insufficient. The reduction from SAT (or any NP‑complete problem) to an LP of polynomial size is not established; known results show that while LP is in P, encoding arbitrary Boolean formulas as LP without exponential blow‑up remains an open challenge. The paper’s demonstration is limited to the specific structure of multiplication and does not generalize to the full class of NP‑complete problems.

Fourth, the experimental evidence is modest. Demonstrations on 64‑bit numbers do not address the scalability issues that dominate cryptographic applications. No benchmarks are provided for 128‑bit, 256‑bit, or larger instances, nor is there an analysis of solver memory consumption or numerical stability when the LP becomes extremely large.

Finally, the cryptographic conclusion is premature. Even if a polynomial‑time algorithm existed in theory, the hidden constants and polynomial degree could be so large that the method would be slower than existing sub‑exponential algorithms for all practical key sizes. Until a rigorous complexity analysis and large‑scale implementation are presented, the claim that current RSA systems are imminently vulnerable remains speculative.

In summary, the paper introduces an intriguing blend of Bayesian reasoning and linear programming to model arithmetic operations, and it offers a fresh perspective on factorisation. However, the methodological gaps—particularly the handling of binary constraints, the true size of the LP formulation, and the lack of a general reduction from NP‑complete problems—prevent the work from substantiating its bold claims. Further research would need to address these shortcomings, provide extensive empirical data on large integers, and clarify whether the approach truly yields a polynomial‑time algorithm for factoring or merely a novel, yet still exponential‑time, heuristic.