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The Complexity of Lattice Problems. Oded Regev, Tel Aviv University. (for more details, see LLL+25 survey). Amsterdam, May 2010. Lattice. For vectors v 1 ,…, v n in R n we define the lattice generated by them as L ={a 1 v 1 +…+ a n v n | a i integers}
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The Complexity of Lattice Problems Oded Regev, Tel Aviv University (for more details, see LLL+25 survey) Amsterdam, May 2010
Lattice • For vectors v1,…,vn in Rn we define the lattice generated by them as • L={a1v1+…+anvn| aiintegers} • We call v1,…,vna basis of L v1+v2 2v2 2v1 2v2-v1 v1 v2 2v2-2v1 0
Lattices from a Computational Complexity Point of View • Lattice problems are among the richest problems in complexity theory, exhibiting a wide range of behaviors: • Some problems are in P (as shown by LLL) • Some problems are NP-hard • Some problems are not known to be in P, but believed not to be NP-hard • As a rule of thumb, ‘algebraic’ problems are easy; ‘geometric’ problems are hard
Shortest Vector Problem (SVP) • GapSVP: Given a lattice, decide if the length of the shortest vector is: • YES: less than 1 • NO: more than v2 v1 0
v Closest Vector Problem (CVP) v2 • GapCVP: Given a lattice and a point v, decide if the distance of v from the lattice is: • YES: less than 1 • NO: more than • GapSVPis not harder than GapCVP[GoldreichMicciancioSafraSeifert99] • Both problems are clearly in NP (for any ) v1 0
n 1 nc/loglogn Cryptography NP-hard [Ajtai96,AjtaiDwork97…] Known Results • Polytime algorithms for gap 2n loglogn/logn[LLL82,Schnorr87,AjtaiKumarSivakumar02] • Hardness is known for: • GapCVP: nc/loglogn[vanEmdeBoas81…,DinurKindlerRazSafra03] • GapSVP: 1 in l1[vanEmdeBoas81] 1 [Ajtai96] 2 [Micciancio98] 2^(log½-εn) [Khot04] nc/loglogn[HavivR07] ? 2n loglogn/logn P
Known ResultsLimits on Inapproximability • GapCVPn2 NP∩coNP[LagariasLenstraSchnorr90, Banaszczyk93] • GapCVPn/logn2 NP∩coAM[GoldreichGoldwasser98] • GapCVPn2NP∩coNP[AharonovRegev04] 1 nc/loglogn n n 2n loglogn/logn NP∩coNP NP∩coAM NP∩coNP P NP-hard
What’s ahead? • GapCVPn/logn2 NP∩coAM[GoldreichGoldwasser98] • GapCVPn2NP∩coNP[AharonovRegev04]
What’s ahead? • GapCVPn/logn2 coAM[GoldreichGoldwasser98] • GapCVPn2coNP[AharonovRegev04]
Our Goal Given: - Lattice L (specified by a basis) - Point v We want to: Be convinced that v is far from L by interacting with an (all powerful) prover (using a constant number of rounds)
Basic High-dimensional Geometry • How big is the intersection of two balls of radius 1 in n dimensions whose centers are at distance apart? • When 2, balls disjoint • When =0, balls exactly overlap • When =0.1, intersection is exponentially small • When =1/n, intersection is constant fraction
The Protocol • Flip a fair coin • If heads, choose a random point in L+B • If tails, choose a random point in L+B+v • Send the resulting point to the prover • The prover is supposed to tell whether the coin was heads of tails (Can be implemented efficiently)
Analysis • If dist(v,L)>2 then prover can always answer correctly • If dist(v,L)<1/n then with some constant probability, the prover has no way to tell what the coin outcome was • Hence we catch the prover cheating with some constant probability • This completes the proof
Our Goal Given: - Lattice L (specified by a basis) - Point v We want: A witness for the fact that v is far from L
Overview Step 1:Define f Its value depends on the distance from L: • Almost zero if distance > n • More than zero if distance < log n Step 2:Encode f Show that the function f has a short description Step 3:Verifier Construct the NP verifier
The function f Consider the Gaussian: Periodize over L: Normalize by g(0):
f distinguishes between far and close vectors (a) d(x,L)≥n f(x)≤2-Ω(n) (b) d(x,L)≤logn f(x)>n-5 Proof:(a) [Banaszczyk93] (b)Not too difficult
The function f (again) Let’s consider its Fourier transform !
Proof: g is a convolution of a Gaussian and δL f̂ is a probability distribution Claim: f̂: L*R+is a probability distribution on L*
f as an expectation In fact, itis an expectation of a real variable between -1 and 1: Chernoff
Encoding f Pick W=(w1,w2,…,wN)with N=poly(n) according to the f̂distribution on L* (Chernoff) This is true even pointwise!
The Approximating Function (with N=1000 dual vectors)
Interlude: CVPP GapCVPP Solve GapCVP on a preprocessed lattice (allowed infinite computational power, but before seeing v) (ideas led to [MicciancioVoulgaris10]’s recent deterministic 2n algorithm for lattice problems) Algorithm for GapCVPP: Prepare the function fW in advance; When given v, calculate fW(v). Algorithm for GapCVPP(n/logn)(best known!)
This concludes Step 2: Encode fThe encoding is a list W of vectors in L*fW(x) ≈ f(x)
0.01 The Verifier (First Attempt) Given input L,v, and witness W, accept iff 1.fW(v) < n-10, and 2.fW(x) > n-5 for all x within distance logn from L • This verifier is correct • But: how to check (2) efficiently? • - First check that fWis periodic over L (true if W in L*) • - Then check that >n-5aroundorigin • We don’t know how to do this for distance logn • Instead, we do this for distance 0.01
The Verifier (Second Attempt) Given input L,v, and witness W, accept iff 1.fW(v) < n-10, and 2. w1,…,wN L*, and 3. 2 implies that fW is periodic on L:
The Verifier (Second Attempt) Given input L,v, and witness W, accept iff 1.fW(v) < n-10, and 2. w1,…,wN L*, and 3. 3 implies that fW is at least 0.8 within distance 0.01 of the origin: fW(x) 0 .01 -.01
The Final Verifier Given input L,v, and witness W, accept iff 1.fW(v) < n-10, and 2. w1,…,wNL*, and 3. ||WWT||<N where 3 checks that in any direction the w’s are not too long:
The Final Verifier Given input L,v, and witness W, accept iff 1.fW(v) < n-10, and 2. w1,…,wNL*, and 3. ||WWT||<N where
Conclusion and Open Questions • Lattice problems with approximation factors >n are unlikely to be NP-hard • These are the problems used for crypto • Can we say anything about their hardness? • Perhaps relate to hardness of other problems, say factoring? • Extremely important question for crypto • Can the containment in NP∩coNP be improved to (n/logn) or even below?
