80 research outputs found
Improved Hardness of BDD and SVP Under Gap-(S)ETH
We show improved fine-grained hardness of two key lattice problems in the
norm: Bounded Distance Decoding to within an factor of the
minimum distance () and the (decisional)
-approximate Shortest Vector Problem (),
assuming variants of the Gap (Strong) Exponential Time Hypothesis (Gap-(S)ETH).
Specifically, we show:
1. For all , there is no -time algorithm for
for any constant ,
where and
is the kissing-number constant, unless non-uniform Gap-ETH is false.
2. For all , there is no -time algorithm for
for any constant , where
is explicit and satisfies for , , and as , unless randomized Gap-ETH is false.
3. For all and all , there
is no -time algorithm for for any constant
, where is explicit and
satisfies as for any fixed , unless non-uniform Gap-SETH is false.
4. For all , , and all , there is no -time algorithm for for
some constant , where is explicit and satisfies as , unless randomized Gap-SETH is false.Comment: ITCS 202
On the Quantitative Hardness of CVP
For odd
integers (and ), we show that the Closest Vector Problem
in the norm (\CVP_p) over rank lattices cannot be solved in
2^{(1-\eps) n} time for any constant \eps > 0 unless the Strong Exponential
Time Hypothesis (SETH) fails. We then extend this result to "almost all" values
of , not including the even integers. This comes tantalizingly close
to settling the quantitative time complexity of the important special case of
\CVP_2 (i.e., \CVP in the Euclidean norm), for which a -time
algorithm is known. In particular, our result applies for any
that approaches as .
We also show a similar SETH-hardness result for \SVP_\infty; hardness of
approximating \CVP_p to within some constant factor under the so-called
Gap-ETH assumption; and other quantitative hardness results for \CVP_p and
\CVPP_p for any under different assumptions
Parameterized Intractability of Even Set and Shortest Vector Problem from Gap-ETH
The k-Even Set problem is a parameterized variant of the Minimum Distance Problem of linear codes over F_2, which can be stated as follows: given a generator matrix A and an integer k, determine whether the code generated by A has distance at most k. Here, k is the parameter of the problem. The question of whether k-Even Set is fixed parameter tractable (FPT) has been repeatedly raised in literature and has earned its place in Downey and Fellows\u27 book (2013) as one of the "most infamous" open problems in the field of Parameterized Complexity.
In this work, we show that k-Even Set does not admit FPT algorithms under the (randomized) Gap Exponential Time Hypothesis (Gap-ETH) [Dinur\u2716, Manurangsi-Raghavendra\u2716]. In fact, our result rules out not only exact FPT algorithms, but also any constant factor FPT approximation algorithms for the problem. Furthermore, our result holds even under the following weaker assumption, which is also known as the Parameterized Inapproximability Hypothesis (PIH) [Lokshtanov et al.\u2717]: no (randomized) FPT algorithm can distinguish a satisfiable 2CSP instance from one which is only 0.99-satisfiable (where the parameter is the number of variables).
We also consider the parameterized k-Shortest Vector Problem (SVP), in which we are given a lattice whose basis vectors are integral and an integer k, and the goal is to determine whether the norm of the shortest vector (in the l_p norm for some fixed p) is at most k. Similar to k-Even Set, this problem is also a long-standing open problem in the field of Parameterized Complexity. We show that, for any p > 1, k-SVP is hard to approximate (in FPT time) to some constant factor, assuming PIH. Furthermore, for the case of p = 2, the inapproximability factor can be amplified to any constant
Improved Lower Bounds for Approximating Parameterized Nearest Codeword and Related Problems under ETH
In this paper we present a new gap-creating randomized self-reduction for
parameterized Maximum Likelihood Decoding problem over
(-MLD). The reduction takes a -MLD instance with
vectors as input, runs in time for some computable function ,
outputs a -Gap--MLD instance for any
, where . Using this reduction, we show that
assuming the randomized Exponential Time Hypothesis (ETH), no algorithms can
approximate -MLD (and therefore its dual problem -NCP) within
factor in time for any
.
We then use reduction by Bhattacharyya, Ghoshal, Karthik and Manurangsi
(ICALP 2018) to amplify the -gap to any constant. As a
result, we show that assuming ETH, no algorithms can approximate -NCP
and -MDP within -factor in
time for some constant . Combining with the
gap-preserving reduction by Bennett, Cheraghchi, Guruswami and Ribeiro (STOC
2023), we also obtain similar lower bounds for -MDP, -CVP and
-SVP.
These results improve upon the previous lower bounds for these problems under ETH using reductions by
Bhattacharyya et al. (J.ACM 2021) and Bennett et al. (STOC 2023).Comment: 32 pages, 3 figure
Parameterized Inapproximability of the Minimum Distance Problem over All Fields and the Shortest Vector Problem in All ℓpNorms
Funding Information: M. Cheraghchi’s research was partially supported by the National Science Foundation under Grants No. CCF-2006455 and CCF-2107345. V. Guruswami’s research was supported in part by NSF grants CCF-2228287 and CCF-2210823 and a Simons Investigator award. J. Ribeiro’s research was supported by NOVA LINCS (UIDB/04516/2020) with the financial support of FCT - Fundação para a Ciência e a Tecnologia and by the NSF grants CCF-1814603 and CCF-2107347 and the following grants of Vipul Goyal: the NSF award 1916939, DARPA SIEVE program, a gift from Ripple, a DoE NETL award, a JP Morgan Faculty Fellowship, a PNC center for financial services innovation award, and a Cylab seed funding award. Publisher Copyright: © 2023 ACM.We prove that the Minimum Distance Problem (MDP) on linear codes over any fixed finite field and parameterized by the input distance bound is W[1]-hard to approximate within any constant factor. We also prove analogous results for the parameterized Shortest Vector Problem (SVP) on integer lattices. Specifically, we prove that SVP in the p norm is W[1]-hard to approximate within any constant factor for any fixed p >1 and W[1]-hard to approximate within a factor approaching 2 for p=1. (We show hardness under randomized reductions in each case.) These results answer the main questions left open (and explicitly posed) by Bhattacharyya, Bonnet, Egri, Ghoshal, Karthik C. S., Lin, Manurangsi, and Marx (Journal of the ACM, 2021) on the complexity of parameterized MDP and SVP. For MDP, they established similar hardness for binary linear codes and left the case of general fields open. For SVP in p norms with p > 1, they showed inapproximability within some constant factor (depending on p) and left open showing such hardness for arbitrary constant factors. They also left open showing W[1]-hardness even of exact SVP in the 1 norm.publishersversionpublishe
Hardness of Bounded Distance Decoding on Lattices in ?_p Norms
Bounded Distance Decoding BDD_{p,?} is the problem of decoding a lattice when the target point is promised to be within an ? factor of the minimum distance of the lattice, in the ?_p norm. We prove that BDD_{p, ?} is NP-hard under randomized reductions where ? ? 1/2 as p ? ? (and for ? = 1/2 when p = ?), thereby showing the hardness of decoding for distances approaching the unique-decoding radius for large p. We also show fine-grained hardness for BDD_{p,?}. For example, we prove that for all p ? [1,?) ? 2? and constants C > 1, ? > 0, there is no 2^((1-?)n/C)-time algorithm for BDD_{p,?} for some constant ? (which approaches 1/2 as p ? ?), assuming the randomized Strong Exponential Time Hypothesis (SETH). Moreover, essentially all of our results also hold (under analogous non-uniform assumptions) for BDD with preprocessing, in which unbounded precomputation can be applied to the lattice before the target is available.
Compared to prior work on the hardness of BDD_{p,?} by Liu, Lyubashevsky, and Micciancio (APPROX-RANDOM 2008), our results improve the values of ? for which the problem is known to be NP-hard for all p > p? ? 4.2773, and give the very first fine-grained hardness for BDD (in any norm). Our reductions rely on a special family of "locally dense" lattices in ?_p norms, which we construct by modifying the integer-lattice sparsification technique of Aggarwal and Stephens-Davidowitz (STOC 2018)
Lattice Problems Beyond Polynomial Time
We study the complexity of lattice problems in a world where algorithms,
reductions, and protocols can run in superpolynomial time, revisiting four
foundational results: two worst-case to average-case reductions and two
protocols. We also show a novel protocol.
1. We prove that secret-key cryptography exists if
-approximate SVP is hard for -time
algorithms. I.e., we extend to our setting (Micciancio and Regev's improved
version of) Ajtai's celebrated polynomial-time worst-case to average-case
reduction from -approximate SVP to SIS.
2. We prove that public-key cryptography exists if
-approximate SVP is hard for -time
algorithms. This extends to our setting Regev's celebrated polynomial-time
worst-case to average-case reduction from -approximate
SVP to LWE. In fact, Regev's reduction is quantum, but ours is classical,
generalizing Peikert's polynomial-time classical reduction from
-approximate SVP.
3. We show a -time coAM protocol for -approximate
CVP, generalizing the celebrated polynomial-time protocol for -CVP due to Goldreich and Goldwasser. These results show
complexity-theoretic barriers to extending the recent line of fine-grained
hardness results for CVP and SVP to larger approximation factors. (This result
also extends to arbitrary norms.)
4. We show a -time co-non-deterministic protocol for
-approximate SVP, generalizing the (also celebrated!)
polynomial-time protocol for -CVP due to Aharonov and Regev.
5. We give a novel coMA protocol for -approximate CVP with a
-time verifier.
All of the results described above are special cases of more general theorems
that achieve time-approximation factor tradeoffs
- …