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Hardness of Approximating the Shortest Vector Problem in Lattices \Lambda



Abstract Let p? 1 be any fixed real. We show that assuming NP 6 ` RP, there is no polynomial time algorithmthat approximates the Shortest Vector Problem (SVP) in `p norm within a constant factor. Under thestronger assumption NP 6 ` RTIME(2poly(log n)), we show that there is no polynomial time algorithm with approximation ratio 2(log n)1=2\Gamma ffl where n is the dimension of the lattice and ffl? 0 is an arbitrarilysmall constant. This greatly improves all previous hardness results in `p norms with 1! p! 1. Thebest results so far gave only a constant factor hardness, namely, 21=p \Gamma ffl by Micciancio [27] and p1\Gamma fflin high `p norms by Khot [20]. We first give a new (randomized) reduction from Closest Vector Problem (CVP) to SVP that achievessome constant factor hardness. The reduction is based on BCH Codes. Its advantage is that the SVP instances produced by the reduction behave well under the augmented tensor product, a new variant oftensor product that we introduce. This enables us to boost the hardness factor to 2(log n)1=2\Gamma ffl

Year: 2004
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