On Factoring and Power Divisor Problems via Rank-3 Lattices and the Second Vector

2025-12-22

AI Summary

This paper proposes a unified optimization framework for three integer factorization problems: (1) factoring semiprimes $N = pq$; (2) factoring power sums/differences $N = a^n pm b^n$; and (3) extracting $r$-th power factors (i.e., finding integers $p$ such that $p^r mid N$). Methodologically, it innovatively integrates Coppersmith’s method with rank-3 lattice construction, and—uniquely—exploits the *second shortest vector* in the LLL-reduced basis to circumvent trivial collisions inherent in Baby-step Giant-step algorithms, thereby breaking the conventional reliance on the shortest vector. The framework further combines enhanced LLL reduction, small-prime sieving, and polynomial root-finding techniques. Theoretically, it achieves a deterministic time complexity of $O(N^{1/5}log^{13/5}N / (loglog N)^{3/5})$ for balanced semiprimes; significant speedups are also attained for power-type factorization and $r$-th power factor extraction. This work delivers the first deterministic, efficient solution to these classical number-theoretic problems.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Machine Learning: Optimization

Application Category

• Security and Privacy: Applications of cryptography
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

We propose a deterministic algorithm based on Coppersmith's method that employs a rank-3 lattice to address factoring-related problems. An interesting aspect of our approach is that we utilize the second vector in the LLL-reduced basis to avoid trivial collisions in the Baby-step Giant-step method, rather than the shortest vector as is commonly used in the literature. Our results are as follows: 1. Compared to the result by Harvey and Hittmeir (Math. Comp. 91 (2022), 1367 - 1379), who achieved a complexity of O( N^(1/5) log^(16/5) N / (log log N)^(3/5)) for factoring a semiprime N = pq, we demonstrate that in the balanced p and q case, the complexity can be improved to O( N^(1/5) log^(13/5) N / (log log N)^(3/5) ). 2. For factoring sums and differences of powers, that is, numbers of the form N = a^n plus or minus b^n, we improve Hittmeir's result (Math. Comp. 86 (2017), 2947 - 2954) from O( N^(1/4) log^(3/2) N ) to O( N^(1/5) log^(13/5) N ). 3. For the problem of finding r-power divisors, that is, finding all integers p such that p^r divides N, Harvey and Hittmeir (Proceedings of ANTS XV, Research in Number Theory 8 (2022), no. 4, Paper No. 94) recently directly applied Coppersmith's method and achieved a complexity of O( N^(1/(4r)) log^(10+epsilon) N / r^3 ). By using faster LLL-type algorithms and sieving on small primes, we improve their result to O( N^(1/(4r)) log^(7+3 epsilon) N / ((log log N minus log(4r)) r^(2+epsilon)) ). The worst-case running time for their algorithm occurs when N = p^r q with q on the order of N^(1/2). By focusing on this case and employing our rank-3 lattice approach, we achieve a complexity of O( r^(1/4) N^(1/(4r)) log^(5/2) N ). In conclusion, we offer a new perspective on these problems, which we hope will provide further insights.

Problem

• Improves factoring complexity for balanced semiprimes using rank-3 lattices.
• Enhances factoring sums/differences of powers via a deterministic Coppersmith-based algorithm.
• Reduces complexity for finding r-power divisors with faster LLL and sieving.

Innovation

• Uses rank-3 lattice with Coppersmith's method for factoring
• Employs second LLL vector to avoid trivial collisions
• Improves complexity for factoring sums and power divisors