Cyclic Lattices, Ideal Lattices and Bounds for the Smoothing Parameter

<div>Cyclic lattices and ideal lattices were introduced by Micciancio in \cite{D2}, Lyubashevsky and Micciancio in \cite{L1} respectively, which play an efficient role in Ajtai's construction of a collision resistant Hash function (see \cite{M1} and \cite{M2}) and in Gentry's construction of fully homomorphic encryption (see \cite{G}). Let $R=Z[x]/\langle \phi(x)\rangle$ be a quotient ring of the integer coefficients polynomials ring, Lyubashevsky and Micciancio regarded an ideal lattice as the correspondence of an ideal of $R$, but they neither explain how to extend this definition to whole Euclidean space $\mathbb{R}^n$, nor exhibit the relationship of cyclic lattices and ideal lattices.</div><div>In this paper, we regard the cyclic lattices and ideal lattices as the correspondences of finitely generated $R$-modules, so that we may show that ideal lattices are actually a special subclass of cyclic lattices, namely, cyclic integer lattices. In fact, there is a one to one correspondence between cyclic lattices in $\mathbb{R}^n$ and finitely generated $R$-modules (see Theorem \ref{th4} below). On the other hand, since $R$ is a Noether ring, each ideal of $R$ is a finitely generated $R$-module, so it is natural and reasonable to regard ideal lattices as a special subclass of cyclic lattices (see corollary \ref{co3.4} below). It is worth noting that we use more general rotation matrix here, so our definition and results on cyclic lattices and ideal lattices are more general forms. As application, we provide cyclic lattice with an explicit and countable upper bound for the smoothing parameter (see Theorem \ref{th5} below). It is an open problem that is the shortest vector problem on cyclic lattice NP-hard? (see \cite{D2}). Our results may be viewed as a substantial progress in this direction.</div>

Cyclic Lattices, Ideal Lattices and Bounds for the Smoothing Parameter

<div>Cyclic lattices and ideal lattices were introduced by Micciancio in \cite{D2}, Lyubashevsky and Micciancio in \cite{L1} respectively, which play an efficient role in Ajtai's construction of a collision resistant Hash function (see \cite{M1} and \cite{M2}) and in Gentry's construction of fully homomorphic encryption (see \cite{G}). Let $R=Z[x]/\langle \phi(x)\rangle$ be a quotient ring of the integer coefficients polynomials ring, Lyubashevsky and Micciancio regarded an ideal lattice as the correspondence of an ideal of $R$, but they neither explain how to extend this definition to whole Euclidean space $\mathbb{R}^n$, nor exhibit the relationship of cyclic lattices and ideal lattices.</div><div>In this paper, we regard the cyclic lattices and ideal lattices as the correspondences of finitely generated $R$-modules, so that we may show that ideal lattices are actually a special subclass of cyclic lattices, namely, cyclic integer lattices. In fact, there is a one to one correspondence between cyclic lattices in $\mathbb{R}^n$ and finitely generated $R$-modules (see Theorem \ref{th4} below). On the other hand, since $R$ is a Noether ring, each ideal of $R$ is a finitely generated $R$-module, so it is natural and reasonable to regard ideal lattices as a special subclass of cyclic lattices (see corollary \ref{co3.4} below). It is worth noting that we use more general rotation matrix here, so our definition and results on cyclic lattices and ideal lattices are more general forms. As application, we provide cyclic lattice with an explicit and countable upper bound for the smoothing parameter (see Theorem \ref{th5} below). It is an open problem that is the shortest vector problem on cyclic lattice NP-hard? (see \cite{D2}). Our results may be viewed as a substantial progress in this direction.</div>

Improved Verifier-Based Three-Party Password-Authenticated Key Exchange Protocol from Ideal Lattices

With the advent of large-scale social networks, two communication users need to generate session keys with the help of a remote server to communicate securely. In the existing three-party authenticated key exchange (3PAKE) protocols, users’ passwords need to be stored on the server; it cannot resist the server disclosure attack. To solve this security problem, we propose a more efficient 3PAKE protocol based on the verification element by adopting a public-key cryptosystem and approximate smooth projection hash (ASPH) function on an ideal lattice. Using the structure of separating authentication from the server, the user can negotiate the session key only after two rounds of communication. The analysis results show that it can improve the efficiency of computation and communication and resist the server disclosure attack, quantum algorithm attack, and replay attack; moreover, it has session key privacy to the server. This protocol can meet the performance requirement of the current communication network.

Cevian properties in ideal lattices of Abelian ℓ-groups

We consider the problem of describing the lattices of compact ℓ {\ell} -ideals of Abelian lattice-ordered groups. (Equivalently, describing the spectral spaces of Abelian lattice-ordered groups.) It is known that these lattices have countably based differences and admit a Cevian operation. Our first result says that these two properties are not sufficient: there are lattices having both countably based differences and Cevian operations, which are not representable by compact ℓ {\ell} -ideals of Abelian lattice-ordered groups. As our second result, we prove that every completely normal distributive lattice of cardinality at most ℵ 1 {\aleph_{1}} admits a Cevian operation. This complements the recent result of F. Wehrung, who constructed a completely normal distributive lattice having countably based differences, of cardinality ℵ 2 {\aleph_{2}} , without a Cevian operation.

HOMOMORPHIC OPERATIONS WITHIN IDEAL LATTICE BASED ENCRYPTION SYSTEMS

By 2009 the first system of fully homomorphic encryption had been constructed, and it was thought-provoking for many future works based on it. Instead of legacy encryption systems which depend on sharing a key (public or private) among endpoints involved in exchanging en encrypted message the fully homomorphic encryption can keep service without depending on shared keys and does not necessarily need to access the content. Such property allows any third party to operate on the encrypted data without decrypting it in advance. In this work, the possibility of using the ideal lattices for the construction of homomorphic operations is researched with a detailed level of math.The paper represents the analysis method based on the primitive of a union of ideals in lattice space. A segregated analysis between homomorphic and security properties is the advantage of this method. The work will be based on the analysis of generalized operations over ciphertext using the concept of the base reducing element which shares all about the method above. It will be shown how some non-homomorphic encryption systems can be supplemented by homomorphic operations which invoke different parameters choosing. Thus such systems can be decomposed from ciphertext structure to decryption process which will be affected by separately analyzed base reduction elements. Distinct from the encryption scheme the underlying math can be used to analyze only the homomorphic part, particularly under some simplifications. The building of such ideal-based ciphertext is laying on the assumption that ideals can be extracted further. It will be shown that the “remainder theorem” can be one of the principal ways to do this providing a simple estimate of an upper bound security strength of ciphertext structure.