Incremental Algorithms for Lattice Problems

We propose a new framework called recursive lattice reduction for finding short non-zero vectors in a lattice or for finding dense sublattices of a lattice. At a high level, the framework works by recursively searching for dense sublattices of dense sublattices (or their duals). Eventually, the procedure encounters a recursive call on a lattice $\mathcal{L}$ with relatively low rank $k$, at which point we simply use a known algorithm to find a short non-zero vector in $\mathc...

Lattice reduction algorithms have numerous applications in number theory, algebra, as well as in cryptanalysis. The most famous algorithm for lattice reduction is the LLL algorithm. In polynomial time it computes a reduced basis with provable output quality. One early improvement of the LLL algorithm was LLL with deep insertions (DeepLLL). The output of this version of LLL has higher quality in practice but the running time seems to explode. Weaker variants of DeepLLL, where ...

Lattice reduction algorithms have numerous applications in number theory, algebra, as well as in cryptanalysis. The most famous algorithm for lattice reduction is the LLL algorithm. In polynomial time it computes a reduced basis with provable output quality. One early improvement of the LLL algorithm was LLL with deep insertions (DeepLLL). The output of this version of LLL has higher quality in practice but the running time seems to explode. Weaker variants of DeepLLL, where ...

We introduce a new canonical form of lattices called the systematic normal form (SNF). We show that for every lattice there is an efficiently computable "nearby" SNF lattice, such that for any lattice one can solve lattice problems on its "nearby" SNF lattice, and translate the solutions back efficiently to the original lattice. The SNF provides direct connections between arbitrary lattices, and various lattice related problems like the Shortest-Integer-Solution, Approximate ...

Given an arbitrary basis for a mathematical lattice, to find a ``good" basis for it is one of the classic and important algorithmic problems. In this note, we give a new and simpler proof of a theorem by Regavim (arXiv:2106.03183): we construct a 18-dimensional lattice that does not have a basis that satisfies the following two properties simultaneously: 1. The basis includes the shortest non-zero lattice vector. 2. The basis is shortest, that is, minimizes the longest basis ...

We present a lattice algorithm specifically designed for some classical applications of lattice reduction. The applications are for lattice bases with a generalized knapsack-type structure, where the target vectors are boundably short. For such applications, the complexity of the algorithm improves traditional lattice reduction by replacing some dependence on the bit-length of the input vectors by some dependence on the bound for the output vectors. If the bit-length of the t...

A lattice is a set of all the integer linear combinations of certain linearly independent vectors. One of the most important concepts on lattice is the successive minima which is of vital importance from both theoretical and practical applications points of view. In this paper, we first study some properties of successive minima and then employ some of them to improve the suboptimal algorithm for solving an optimization problem about maximizing the achievable rate of the inte...

In 2018, the longest vector problem (LVP) and the closest vector problem (CVP) in $p$-adic lattices were introduced. These problems are closely linked to the orthogonalization process. In this paper, we first prove that every $p$-adic lattice has an orthogonal basis and give definition to the successive maxima and the escape distance, as the $p$-adic analogues of the successive minima and the covering radius in Euclidean lattices. Then, we present deterministic polynomial tim...

We show that for those lattices of Voronoi's first kind with known obtuse superbasis, a closest lattice point can be computed in $O(n^4)$ operations where $n$ is the dimension of the lattice. To achieve this a series of relevant lattice vectors that converges to a closest lattice point is found. We show that the series converges after at most $n$ terms. Each vector in the series can be efficiently computed in $O(n^3)$ operations using an algorithm to compute a minimum cut in ...

In this article we perform a computational study of Polyrakis algorithms presented in [12,13]. These algorithms are used for the determination of the vector sublattice and the minimal lattice-subspace generated by a finite set of positive vectors of R^k. The study demonstrates that our findings can be very useful in the field of Economics, especially in completion by options of security markets and portfolio insurance.