New Shortest Lattice Vector Problems of Polynomial Complexity

The Shortest Lattice Vector (SLV) problem is in general hard to solve, except for special cases (such as root lattices and lattices for which an obtuse superbase is known). In this paper, we present a new class of SLV problems that can be solved efficiently. Specifically, if for an $n$-dimensional lattice, a Gram matrix is known that can be written as the difference of a diagonal matrix and a positive semidefinite matrix of rank $k$ (for some constant $k$), we show that the SLV problem can be reduced to a $k$-dimensional optimization problem with countably many candidate points. Moreover, we show that the number of candidate points is bounded by a polynomial function of the ratio of the smallest diagonal element and the smallest eigenvalue of the Gram matrix. Hence, as long as this ratio is upper bounded by a polynomial function of $n$, the corresponding SLV problem can be solved in polynomial complexity. Our investigations are motivated by the emergence of such lattices in the field of Network Information Theory. Further applications may exist in other areas.Comment: 13 page

Faster Deterministic Volume Estimation in the Oracle Model via Thin Lattice Coverings

We give a 2O(n)(1+1/")n time and poly(n)-space deterministic algorithm for computing a (1+")n approximation to the volume of a general convex body K, which comes close to matching the (1+c/")n/2 lower bound for volume estimation in the oracle model by Bárány and Füredi (STOC 1986, Proc. Amer. Math. Soc. 1988). This improves on the previous results of Dadush and Vempala (Proc. Nat’l Acad. Sci. 2013), which gave the above result only for symmetric bodies and achieved a dependence of 2O(n)(1 + log5/2(1/")/"3)n. For our methods, we reduce the problem of volume estimation in K to counting lattice points in K Rn (via enumeration) for a specially constructed lattice L: a so-called thin covering of space with respect to K (more precisely, for which L + K = Rn and voln(K)/ det(L) = 2O(n)). The trade off between time and approximation ratio is achieved by scaling down the lattice. As our main technical contribution, we give the first deterministic 2O(n)-time and poly(n)- space construction of thin covering lattices for general convex bodies. This improves on a recent construction of Alon et al. (STOC 2013) which requires exponential space and only works for symmetric bodies. For our construction, we combine the use of the M-ellipsoid from convex geometry (Milman, C. R. Math. Acad. Sci. Paris 1986) together with lattice sparsification and densification techniques (Dadush and Kun, SODA 2013; Rogers, J. London Math. Soc. 1950)

Near-Optimal Deterministic Algorithms for Volume Computation and Lattice Problems via M-Ellipsoids

We give a deterministic 2^{O(n)} algorithm for computing an M-ellipsoid of a convex body, matching a known lower bound. This has several interesting consequences including improved deterministic algorithms for volume estimation of convex bodies and the shortest and closest lattice vector problems under general norms

Deterministic Construction of an Approximate M-Ellipsoid and its Application to Derandomizing Lattice Algorithms

We give a deterministic O(log n)^n algorithm for the {\em Shortest Vector Problem (SVP)} of a lattice under {\em any} norm, improving on the previous best deterministic bound of n^O(n) for general norms and nearly matching the bound of 2^O(n) for the standard Euclidean norm established by Micciancio and Voulgaris (STOC 2010). Our algorithm can be viewed as a derandomization of the AKS randomized sieve algorithm, which can be used to solve SVP for any norm in 2^O(n) time with high probability. We use the technique of covering a convex body by ellipsoids, as introduced for lattice problems in (Dadush et al., FOCS 2011). Our main contribution is a deterministic approximation of an M-ellipsoid of any convex body. We achieve this via a convex programming formulation of the optimal ellipsoid with the objective function being an n-dimensional integral that we show can be approximated deterministically, a technique that appears to be of independent interest

Compute-and-Forward: Finding the Best Equation

Compute-and-Forward is an emerging technique to deal with interference. It allows the receiver to decode a suitably chosen integer linear combination of the transmitted messages. The integer coefficients should be adapted to the channel fading state. Optimizing these coefficients is a Shortest Lattice Vector (SLV) problem. In general, the SLV problem is known to be prohibitively complex. In this paper, we show that the particular SLV instance resulting from the Compute-and-Forward problem can be solved in low polynomial complexity and give an explicit deterministic algorithm that is guaranteed to find the optimal solution.Comment: Paper presented at 52nd Allerton Conference, October 201