The LLL algorithm, named after its inventors, Lenstra, Lenstra and Lovász, is one of themost popular lattice reduction algorithms in the literature. In this paper, we propose the first variant of LLL algorithm that is dedicated for ideal lattices, namely, the iLLL algorithm. Our iLLL algorithm takes advantage of the fact that within LLL procedures, previously reduced vectors can be re-used for ...

Lattice-based cryptography is one of the candidates in the area of post-quantum cryptography. Cryptographic schemes with security reductions to hard lattice problems (like the Shortest Vector Problem SVP) offer an alternative to recent number theory-based schemes. In order to guarantee asymptotic efficiency, most lattice-based schemes are instantiated using polynomial rings over integers. These...

Lattice based cryptography is gaining more and more importance in the cryptographic community. It is a common approach to use a special class of lattices, so-called ideal lattices, as the basis of lattice based crypto systems. This speeds up computations and saves storage space for cryptographic keys. The most important underlying hard problem is the shortest vector problem. So far there is no ...

In this paper, we have established bi-approximation semantics for lattice-based logics with the De Morgan negation (unbounded orthologic), and their morphisms. In addition, we have discussed the dual representation between unbounded ortholattices with strict homomorphisms and polarity frames and d-morphisms. Apart from the abstract construction of dual algebras in the series of the present auth...

Partially ordered sets labeled with k labels (k-posets) and their homomorphisms are examined. The homomorphicity order of k-posets is shown to be a distributive lattice. Homomorphicity orders of k-posets and k-lattices are shown to be universal in the sense that every countable poset can be embedded into them. Labeled posets are represented by directed graphs, and a categorical isomorphism betw...

We prove that the theory of EXPTIME degrees with respect to polynomial time Turing and many-one reducibility is undecidable. To do so we use a coding method based on ideal lattices of Boolean algebras which was introduced in [7]. The method can be applied in fact to all hyper-polynomial time classes.

The Dedekind–Birkhoff theorem for finite-height modular lattices has previously been generalized to complete modular lattices, using the theory of regular coverings. In this paper, we investigate regular coverings in lattices of filters and lattices of ideals, and the regularization strategy–embedding the lattice into its lattice of filters or lattice of ideals, thereby possibly converting a co...

Two categories are called Morita equivalent if the categories of functors from these categories to the category of sets are equivalent. We prove that congruence lattices of Morita equivalent small categories are isomorphic. 1. Preliminaries Categories A and B are called Morita equivalent if the functor categories Fun(A, Set) and Fun(B, Set) are equivalent. The basic theory of Morita equivalent ...

We describe plausible lattice-based constructions with properties that approximate the soughtafter multilinear maps in hard-discrete-logarithm groups, and show that some applications of such multi-linear maps can be realized using our approximations. The security of our constructions relies on seemingly hard problems in ideal lattices, which can be viewed as extensions of the assumed hardness o...