Principal ideal problem and ideal shortest vector over rational primes in power-of-two cyclotomic fields

2026-01-12

AI Summary

This study investigates the length of the shortest vector in the ideal lattice lying above a rational prime \( p \) in cyclotomic fields of conductor a power of two, with a focus on the cases \( p \equiv 7,9 \pmod{16} \). By reformulating the shortest vector problem as that of finding the shortest generator of a principal ideal, and leveraging ideal factorization, lattice embeddings, and algebraic number theory, the authors introduce a novel approach that avoids reliance on traditional lattice basis reduction. This method yields the first exact characterization of the shortest vector length for these congruence classes and establishes a tight upper bound of \( \sqrt[4]{2^{2n+1}p} \), which improves upon the classical Minkowski bound. Consequently, the shortest vector length problem is now fully resolved for all primes \( p \equiv 3,5,7,9 \pmod{16} \).

Technology Category

• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Satisfiability
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Security and Privacy: Applications of cryptography
• Responsible Web: Measurement, analysis, and circumvention of Web censorship
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

The shortest vector problem (SVP) over ideal lattices is closely related to the Ring-LWE problem, which is widely used to build post-quantum cryptosystems. Power-of-two cyclotomic fields are frequently adopted to instantiate Ring-LWE. Pan et al. (EUROCRYPT~2021) explored the SVP over ideal lattices via the decomposition fields and, in particular determined the length of the shortest vector in prime ideals lying over rational primes $p\equiv3,5\pmod{8}$ in power-of-two cyclotomic fields via explicit construction of reduced lattice bases. In this work, we first provide a new method (different from analyzing lattice bases) to analyze the length of the shortest vector in prime ideals in $\mathbb{Z}[\zeta_{2^{n+1}}]$ when $p\equiv3,5\pmod{8}$. Then we precisely characterize the length of the shortest vector in the cases of $p\equiv7,9\pmod{16}$. Furthermore, we derive a new upper bound $\sqrt[4]{2^{2n+1}p}$ for this length, which is tighter than the bound $2^n\sqrt[4]{p}$ obtained from Minkowski's theorem. Our key technique is to investigate whether a generator of a principal ideal can achieve the shortest length after embedding as a vector. If this holds for the ideal, finding the shortest vector in this ideal can be reduced to finding its shortest generator.

Problem

• Shortest Vector Problem
• Ideal Lattices
• Cyclotomic Fields
• Principal Ideal
• Ring-LWE

Innovation

• ideal lattices
• shortest vector problem
• cyclotomic fields
• principal ideal
• post-quantum cryptography