Deterministic and Efficient Ideal Arithmetic via Two-Element Representations

2026-06-25

AI Summary

This work addresses the problem of efficiently computing deterministic two-element representations of ideals in number fields. Focusing on ideals whose norm is coprime to the index of the defining polynomial’s ring of integers—a class that includes cryptographically relevant cases such as those defined by cyclotomic polynomials—the paper presents the first deterministic polynomial-time algorithm for this task. The approach leverages a generalized Dedekind criterion to decompose and construct ideals within number fields defined as ℚ[x]/(f). This method overcomes prior limitations that relied on randomization or failed to scale to cryptographic parameters, thereby achieving, for the first time, a combination of determinism, efficiency, and completeness across a broad and practically significant class of ideals used in cryptographic applications.

Abstract

Given an ideal in a number field, it is desirable in many situations to find two elements that generate the ideal over the ring of the integers of the field. Existing algorithms are either randomized, or impractical at cryptographic sizes. In the paper, we present a deterministic polynomial time algorithm to find the two-element representation of an ideal. For a monic irreducible integral polynomial \( f(x) \), let \( K=\Q[x]/(f) \) be the number field, and \( O_K \) be the integral closure. Our algorithm works when the norm of the input ideal is co-prime to the index \( [O_K:\Z[x]/f] \). In particular, it handles all ideals for monogenic \( f(x) \), a class that includes the cyclotomic polynomials widely used in lattice based cryptography. A key technical ingredient in our result is a generalized version of Dedekind criterion.

Problem

• ideal
• two-element representation
• number field
• deterministic algorithm
• monogenic

Innovation

• two-element representation
• deterministic algorithm
• ideal arithmetic
• Dedekind criterion
• monogenic fields