SVP in Hard to Approximate to within some constant

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SVP in Hard to Approximate to within some
constant
Based on Article by Daniele Micciancio -
1998 IEEE Symposium on Foundations of Computer
Science (FOCS 98)
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Overview
?2
11/n?
nO(1)
2n/2
2loglogn
1
O(logn)
O(1)
n1/loglogn
?
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Approximate SVP CVP

• GapSVPg (B,d) B is a Lattice base in Rn, d?R
• Yes Instances ?z?Zn\0 Bz ? d
• No Instances ?z?Zn\0 Bz gt gd

We actually want to prove GapSVP?2 is NP-Hard

• GapCVPg (B,y,d) B?Zkxn , y?Rk, d?R
• Yes Instances ?z?Zn Bz - y ? d
• No Instances ?z?Zn Bz - y gt gd

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Modified CVP

• GapCVPg? (B,y,d) B?Zkxn , y?Rk, d?R
• Yes Instances ?z?0,1n Bz - y ? d
• No Instances ?z?Zn, w?Zn\0 Bz - wy gt gd

GapCVP and GapCVP were proven NP-Hard Babai97
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Sauers Lemma

• There is a probabilistic ploy-time algorithm
  which creates??gt0, for input 1k (input size k)
  the following

Lattice L?R(m1)xm Vector s? R(m1) Matrix C?
Zkxm
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Sauers Lemma (visualization)
Copied from Micc98 article.
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Proving GapSVP NP-hardness
(B,y,d)? and c ?(2/?) ? (V,t) and g?(2/12?)
t?(2/12?) ? ??/d
1. (B,y,d)? is Yes Instance ? ?a s.t. Va2 lt
t2
V
? ??/d
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Proving GapSVP NP-hardness (cont)
(B,y,d)? and c ?(2/?) ? (V,t) and g ?(2/12?)
2. (B,y,d)? is No Instance ?az w z?Zm
Va2 gt g2t2 2
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