On Factoring and Power Divisor Problems via Rank-3 Lattices and the Second Vector
Yiming Gao, Yansong Feng, Honggang Hu, Yanbin Pan
TL;DR
The paper introduces a deterministic factoring framework that combines Coppersmith’s method with a Baby-step Giant-step scheme realized on a rank-3 lattice, using the second LLL vector to avoid trivial collisions. This approach yields improved asymptotic complexities for several factoring problems, including balanced semiprimes, sums/differences of powers, and r-power divisors, compared with prior deterministic results. It also provides refined toolkits, loglog-speedups via small-prime sieving, and order-based enhancements that broaden applicability. The results illuminate a new perspective on factoring through lattice-theoretic constructions and raise open questions about extending improvements to the broad Harvey–Hittmeir bounds in the absence of extra information about p or q.
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.