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Many lattice-based crypstosystems employ ideal lattices for high efficiency.However, the additional algebraic structure of ideal lattices usually makes usworry about the security, and it is widely believed that the algebraicstructure will help us solve the hard problems in ideal lattices moreefficiently. In this paper, we study the additional algebraic structure ofideal lattices further and find that a given ideal lattice in some fixedpolynomial ring can be embedded as an ideal in infinitely many differentpolynomial rings. We explicitly present all these polynomial rings for anygiven ideal lattice. The interesting phenomenon tells us that a single ideallattice may have more abundant algebraic structures than we imagine, which willimpact the security of corresponding crypstosystems. For example, it increasesthe difficulties to evaluate the security of crypstosystems based on ideallattices, since it seems that we need consider all the polynomial rings thatthe given ideal lattices can be embedded into if we believe that the algebraicstructure will contribute to solve the corresponding hard problem. It alsoinspires us a new method to solve the ideal lattice problems by embedding thegiven ideal lattice into another well-studied polynomial ring. As a by-product,we also introduce an efficient algorithm to identify if a given lattice is anideal lattice or not.

A Coefficient-Embedding Ideal Lattice can be Embedded into Infinitely Many Polynomial Rings