On lattices, learning with errors, random linear codes, and cryptography
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
ACM Transactions on Algorithms (TALG)
Hardness of approximating the shortest vector problem in lattices
Journal of the ACM (JACM)
Hardness of Approximating the Closest Vector Problem with Pre-Processing
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
Hardness of approximating the Shortest Vector Problem in high ℓp norms
Journal of Computer and System Sciences - Special issue on FOCS 2003
Progress in computational complexity theory
Journal of Computer Science and Technology
Quantum multiparty communication complexity and circuit lower bounds
Mathematical Structures in Computer Science
Quantum multiparty communication complexity and circuit lower bounds
TAMC'07 Proceedings of the 4th international conference on Theory and applications of models of computation
A note on quantum algorithms and the minimal degree of ε-error polynomials for symmetric functions
Quantum Information & Computation
Lower bounds on matrix rigidity via a quantum argument
ICALP'06 Proceedings of the 33rd international conference on Automata, Languages and Programming - Volume Part I
Theoretical Computer Science
Concurrent zero knowledge without complexity assumptions
TCC'06 Proceedings of the Third conference on Theory of Cryptography
Linear vs. semidefinite extended formulations: exponential separation and strong lower bounds
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
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We show that the problems of approximating the shortest and closest vector in a lattice to within a factor of \sqrt n lie in Np intersect coNP. The result (almost) subsumes the three mutually-incomparable previous results regarding these lattice problems: Banaszczyk [7], Goldreich and Goldwasser [13], and Aharonov and Regev [2]. Our technique is based on a simple fact regarding succinct approximation of functions using their Fourier transform over the lattice. This technique might be useful elsewhere - we demonstrate this by giving a simple and efficient algorithm for one other lattice problem (CVPP) improving on a previous result of Regev [25]. An interesting fact is that our result emerged from a "dequantization" of our previous quantum result in [2]. This route to proving purely classical results might be beneficial elsewhere.