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Nisheeth vishnoi

The Closest Vector is Hard to Approximate and now, for unlimited time only with Pre - Processing !!. Guy Kindler. Microsoft Research. Nisheeth vishnoi. Subhash Khot. Michael Alekhnovich. Joint work with. In this talk: Lattices The closest vector problem: background

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Nisheeth vishnoi

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  1. The Closest Vector is Hard to Approximateand now, for unlimited time onlywith Pre-Processing !! Guy Kindler Microsoft Research Nisheeth vishnoi Subhash Khot Michael Alekhnovich Joint work with

  2. In this talk: • Lattices • The closest vector problem: background • Our results: NP-hardness for CV-PP • Proving hardness with preprocessing • Something about our proof: new property of PCPs

  3. A lattice,L: A discrete additive subgroup of Rn. • A basis for L: b1,…,bn2Rn, s.t. L={iaibi : a1,..,an2Z}.

  4. The Closest Vector Problem (CVP)

  5. The Closest Vector Problem (CVP) • CVP: Given a lattice L and a target vector t, find the point in L closest to t inlpdistance. • [Regev Ronen 05] Hardness results in l2 carry for any lp. • [Ajtai Kumar Sivakumar 01]:2O(nloglog(n)/log n)=2o(n) approx. • [Dinur Kindler Raz Safra 98]:nO(1/loglog n)=no(1) hardness. • [Lagarias Lenstra Schnorr 90, Banaszczyk 93, Goldreich Goldwasser 00, Aharonov Regev 04] NP-hardness of (n/log n)1/2 would collapse the polynomial hierarchy.

  6. Motivation for studying CVP • [Ajtai 96]: Worst case to average case reductions for lattice problems. • [Ajtai Dwork 97] Based cryptosystems on lattice problems. • [Goldreich Goldwasser Halevi 97] Cryptosystem based on CVP. • [Micciancio Vadhan 03] Identification scheme based on (n/log n)1/2 hardness for CVP. t – message. L – coding function: known in advance, and reused.

  7. Is it safe to reuse L as key? • CV-PP: • Preprocess L for unlimited time, • Given t, solve CVP on L,t. • [Kannan 87, Lagarias Lenstra Schnorr 90, Aharonov Regev ] O(n1/2)-approx. for CV-PP. • [Feige Micciancio 02](5/3)1/p approx. hardness for CV-PP. • [Regev 03]31/p approx. hardness for CV-PP.

  8. Our Results • Thm:CV-PP in NP-hard(!) to approximate within any constant. Also applies to NC-PP. • Unless NPµDTIME(2polylog n), • NC-PP is hard to approximate within (log n)1- • CV-PP is hard to approximate within(log n)(1/p)- • 1st Proof : By reduction from E-k-HVC[DGKR 03]. • 2nd proof: Using PCP-PP constructions, plus smoothing technique of [Khot 02].

  9. Reduction I: Instance of ¦2NPC L , t Proving hardness with preprocessing • Hardness of approximation within gap g: I2¦) dist(t,L)· d I¦ ) dist(t,L)¸ d¢g

  10. Size of I PreprocessedL Proving hardness with preprocessing • Hardness of approximation within g, with preprocessing: • Hardness of approximation within gap g: Partial Input Generator Reduction I: Instance of ¦2NPC L , t t I2¦ ) dist(t,L)· d I¦) dist(t,L)¸ d¢g CV-PP

  11. PreprocessedL LEFT t RIGHT CV-PP x2+2xy=7 x2+z2=5 . . PCP with preprocessing (PCP-PP) • PCP: Gap version of Quadratic equations. Partial Input Generator Size of I Reduction I: Instance of ¦2NPC I2¦ ) dist(t,L)· d I¦ ) dist(t,L)¸ d¢g PCP-PP

  12. LEFT RIGHT x2+2xy=7 x2+z2=5 . . PCP with preprocessing (PCP-PP) • PCP: Gap version of Quadratic equations. Partial Input Generator Size of I Reduction I: Instance of ¦2NPC I2¦ ) opt(LEFT,RIGHT)=1 I¦ ) opt(LEFT,RIGHT)·c<1 PCP-PP

  13. LEFT RIGHT PCP with preprocessing (PCP-PP) • PCP: Gap version of Quadratic equations. Partial Input Generator Size of I Reduction I: Instance of ¦2NPC PCP-PP

  14. PreprocessedL LEFT t RIGHT CV-PP PCP with preprocessing (PCP-PP) • PCP: Gap version of Quadratic equations. Size of I I: Instance of ¦2NPC PCP-PP

  15. PCP with preprocessing (PCP-PP) • PCP: Gap version of Quadratic equations. LEFT RIGHT PCP-PP

  16. PCP-PP construction • PCP: Gap version of Quadratic equations. LEFT Just (carefully) apply usualPCP construction! RIGHT PCP-PP

  17. Open problems • Get better hardness parameters for CV-PP (perhaps using methods from [DKRS 98]). • Get improved hardness results for lattice problems, under stronger assumptions than NPP. • Find more uses for PCP-PP constructions.

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