
Faster Lattice Enumeration
A lattice reduction is an algorithm that transforms the given basis of t...
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The nearestcolattice algorithm
In this work, we exhibit a hierarchy of polynomial time algorithms solvi...
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Hardness of Approximate Nearest Neighbor Search under Linfinity
We show conditional hardness of Approximate Nearest Neighbor Search (ANN...
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Effects of Some Lattice Reductions on the Success Probability of the ZeroForcing Decoder
Zeroforcing (ZF) decoder is a commonly used approximation solution of t...
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On Simplifying Dependent Polyhedral Reductions
Reductions combine collections of input values with an associative (and ...
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Simplifying MultipleStatement Reductions with the Polyhedral Model
A Reduction – an accumulation over a set of values, using an associative...
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Scheduling and Tiling Reductions on Realistic Machines
Computations, where the number of results is much smaller than the input...
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DimensionPreserving Reductions Between SVP and CVP in Different pNorms
We show a number of reductions between the Shortest Vector Problem and the Closest Vector Problem over lattices in different ℓ_p norms (_p and _p respectively). Specifically, we present the following 2^ mtime reductions for 1 ≤ p ≤ q ≤∞, which all increase the rank n and dimension m of the input lattice by at most one: ∙ a reduction from O(1/^1/p)γapproximate _q to γapproximate _p; ∙ a reduction from O(1/^1/p) γapproximate _p to γapproximate _q; and ∙ a reduction from O(1/^1+1/p)_q to (1+)unique _p (which in turn trivially reduces to (1+)approximate _p). The last reduction is interesting even in the case p = q. In particular, this special case subsumes much prior work adapting 2^O(m)time _p algorithms to solve O(1)approximate _p. In the (important) special case when p = q, 1 ≤ p ≤ 2, and the _p oracle is exact, we show a stronger reduction, from O(1/^1/p)_p to (exact) _p in 2^ m time. For example, taking = log m/m and p = 2 gives a slight improvement over Kannan's celebrated polynomialtime reduction from √(m)_2 to _2. We also note that the last two reductions can be combined to give a reduction from approximate_p to _q for any p and q, regardless of whether p ≤ q or p > q. Our techniques combine those from the recent breakthrough work of Eisenbrand and Venzin (which showed how to adapt the current fastest known algorithm for these problems in the ℓ_2 norm to all ℓ_p norms) together with sparsificationbased techniques.
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