On the Relationship Between Two Embeddings of Ideals into Geometric Space and the Shortest Vector Problem in Principal Ideal Lattices

Abstract

An ideal lattice is the geometric embedding of an ideal in the algebraic integer ring of some number field. Many recent developments in lattice-based cryptography are centered around the use of ideal lattices. The shortest vector problem (SVP) is the most important hard lattice problem. Few algorithms that find a short vector in ideal lattices exploit their additional algebraic structure, and whether or not the SVP can be solved algebraically in ideal lattices remains unknown. We study the relationship between the canonical and coefficient embeddings of ideals in algebraic integer rings of cyclotomic number fields. We examine the algebraic structure of principal ideal lattices under the coefficient embedding by considering them as principal ideals of a cyclotomic quotient ring. Finally, empirical evidence is provided to exhibit a relationship between the algebraic structure of a principal ideal in this quotient ring and the geometric structure of its corresponding ideal lattice. These results demonstrate progress towards solving the SVP in ideal lattices algebraically.