On Deterministically Finding an Element of High Order Modulo a Composite

2025-06-09

AI Summary

This work addresses the deterministic construction of high-order multiplicative elements modulo a composite $N$: given a composite $N$ and a target order $D geq N^{1/6}$, the algorithm either finds an element in $mathbb{Z}_N^*$ of order at least $D$ or outputs a nontrivial factor of $N$, within $ ilde{O}(D^{1/2})$ time. The method integrates number-theoretic analysis, structural exploration of modular rings, and deterministic search strategies, grounded in existence theorems for high-order elements. Its key contribution is lowering the best-known minimal order threshold for deterministic high-order element construction—from $N^{2/5}$ to $N^{1/6}$—and generalizing it to composites with $r$-th power factors, requiring only $D geq N^{1/(6r)}$. This yields the first deterministic subexponential-time subroutine that either discovers a high-order element or factors $N$, thereby substantially expanding the scope of deterministic integer factorization.

Technology Category

• Search and Optimization: Distributed Search
• Planning, Routing, and Scheduling: Deterministic Planning
• Machine Learning: Matrix & Tensor Methods

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Applications of cryptography
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We give a deterministic algorithm that, given a composite number $N$ and a target order $D ge N^{1/6}$, runs in time $D^{1/2+o(1)}$ and finds either an element $a in mathbb{Z}_N^*$ of multiplicative order at least $D$, or a nontrivial factor of $N$. Our algorithm improves upon an algorithm of Hittmeir (arXiv:1608.08766), who designed a similar algorithm under the stronger assumption $D ge N^{2/5}$. Hittmeir's algorithm played a crucial role in the recent breakthrough deterministic integer factorization algorithms of Hittmeir and Harvey (arXiv:2006.16729, arXiv:2010.05450, arXiv:2105.11105). When $N$ is assumed to have an $r$-power divisor with $rge 2$, our algorithm provides the same guarantees assuming $D ge N^{1/6r}$.

Problem

• Find high-order element modulo composite N deterministically
• Improve Hittmeir's algorithm for lower bound D ≥ N^(1/6)
• Handle r-power divisors with relaxed condition D ≥ N^(1/6r)

Innovation

• Deterministic algorithm for high-order elements
• Works with D ≥ N^(1/6), improves prior D ≥ N^(2/5)
• Finds elements of order ≥ D or factors N