Efficient reductions among lattice problems
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Limits on the Hardness of Lattice Problems in lp Norms
Computational Complexity
Noninteractive Statistical Zero-Knowledge Proofs for Lattice Problems
CRYPTO 2008 Proceedings of the 28th Annual conference on Cryptology: Advances in Cryptology
Deterministic Approximation Algorithms for the Nearest Codeword Problem
APPROX '09 / RANDOM '09 Proceedings of the 12th International Workshop and 13th International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques
Proceedings of the forty-second ACM symposium on Theory of computing
Algorithms for the shortest and closest lattice vector problems
IWCC'11 Proceedings of the Third international conference on Coding and cryptology
The geometry of lattice cryptography
Foundations of security analysis and design VI
On the number of lattice points in a small sphere and a recursive lattice decoding algorithm
Designs, Codes and Cryptography
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We initiate the study of the computational complexity of the covering radius problem for lattices, and approximation versions of the problem for both lattices and linear codes. We also investigate the computational complexity of the shortest linearly independent vectors problem, and its relation to the covering radius problem for lattices. For the covering radius on n-dimensional lattices, we show that the problem can be approximated within any constant factor 驴(n) 1 in random exponential time 2 O(n). We also prove that suitably defined gap versions of the problem lie in AM for 驴(n) = 2, in coAM for $$ \gamma (n) = {\sqrt {n/\log n} }, $$ and in NP 驴 coNP for $$ \gamma (n) = {\sqrt n }. $$ For the covering radius on n-dimensional linear codes, we show that the problem can be solved in deterministic polynomial time for approximation factor $$ \gamma (n) = \log n, $$ but cannot be solved in polynomial time for some $$ \gamma (n) = \Omega (\log \log n) $$ unless NP can be simulated in deterministic $$ n^{{O(\log \log \log n)}} $$ time. Moreover, we prove that the problem is NP-hard for any constant approximation factor, it is 驴2-hard for some constant approximation factor, and that it is unlikely to be 驴2-hard for approximation factors larger than 2 (by giving an AM protocol for the appropriate gap problem). This is a natural hardness of approximation result in the polynomial hierarchy.For the shortest independent vectors problem, we give a coAM protocol achieving approximation factor $$ \gamma (n) = {\sqrt {n/\log n} }, $$ solving an open problem of Blömer and Seifert (STOC'99), and prove that the problem is also in coNP for $$ \gamma (n) = {\sqrt n }. $$ Both results are obtained by giving a gap-preserving nondeterministic polynomial time reduction to the closest vector problem.