5 research outputs found
Fourier Growth of Structured ??-Polynomials and Applications
We analyze the Fourier growth, i.e. the L? Fourier weight at level k (denoted L_{1,k}), of various well-studied classes of "structured" m F?-polynomials. This study is motivated by applications in pseudorandomness, in particular recent results and conjectures due to [Chattopadhyay et al., 2019; Chattopadhyay et al., 2019; Eshan Chattopadhyay et al., 2020] which show that upper bounds on Fourier growth (even at level k = 2) give unconditional pseudorandom generators.
Our main structural results on Fourier growth are as follows:
- We show that any symmetric degree-d m F?-polynomial p has L_{1,k}(p) ? Pr [p = 1] ? O(d)^k. This quadratically strengthens an earlier bound that was implicit in [Omer Reingold et al., 2013].
- We show that any read-? degree-d m F?-polynomial p has L_{1,k}(p) ? Pr [p = 1] ? (k ? d)^{O(k)}.
- We establish a composition theorem which gives L_{1,k} bounds on disjoint compositions of functions that are closed under restrictions and admit L_{1,k} bounds.
Finally, we apply the above structural results to obtain new unconditional pseudorandom generators and new correlation bounds for various classes of m F?-polynomials
Counting Simplices in Hypergraph Streams
We consider the problem of space-efficiently estimating the number of
simplices in a hypergraph stream. This is the most natural hypergraph
generalization of the highly-studied problem of estimating the number of
triangles in a graph stream. Our input is a -uniform hypergraph with
vertices and hyperedges. A -simplex in is a subhypergraph on
vertices such that all possible hyperedges among exist in .
The goal is to process a stream of hyperedges of and compute a good
estimate of , the number of -simplices in .
We design a suite of algorithms for this problem. Under a promise that
, our algorithms use at most four passes and together imply a
space bound of for each fixed , in order to
guarantee an estimate within with probability at least
. We also give a simpler -pass algorithm that achieves
space, where (respectively, ) denotes
the maximum number of -simplices that share a hyperedge (respectively, a
vertex). We complement these algorithmic results with space lower bounds of the
form , , and
for multi-pass algorithms and
for -pass algorithms, which show that some of the dependencies on parameters
in our upper bounds are nearly tight. Our techniques extend and generalize
several different ideas previously developed for triangle counting in graphs,
using appropriate innovations to handle the more complicated combinatorics of
hypergraphs
Fine-Grained Completeness for Optimization in P
We initiate the study of fine-grained completeness theorems for exact and
approximate optimization in the polynomial-time regime. Inspired by the first
completeness results for decision problems in P (Gao, Impagliazzo, Kolokolova,
Williams, TALG 2019) as well as the classic class MaxSNP and
MaxSNP-completeness for NP optimization problems (Papadimitriou, Yannakakis,
JCSS 1991), we define polynomial-time analogues MaxSP and MinSP, which contain
a number of natural optimization problems in P, including Maximum Inner
Product, general forms of nearest neighbor search and optimization variants of
the -XOR problem. Specifically, we define MaxSP as the class of problems
definable as , where is a quantifier-free
first-order property over a given relational structure (with MinSP defined
analogously). On -sized structures, we can solve each such problem in time
. Our results are:
- We determine (a sparse variant of) the Maximum/Minimum Inner Product
problem as complete under *deterministic* fine-grained reductions: A strongly
subquadratic algorithm for Maximum/Minimum Inner Product would beat the
baseline running time of for *all* problems in MaxSP/MinSP by
a polynomial factor.
- This completeness transfers to approximation: Maximum/Minimum Inner Product
is also complete in the sense that a strongly subquadratic -approximation
would give a -approximation for all MaxSP/MinSP problems in
time , where can be chosen
arbitrarily small. Combining our completeness with~(Chen, Williams, SODA 2019),
we obtain the perhaps surprising consequence that refuting the OV Hypothesis is
*equivalent* to giving a -approximation for all MinSP problems in
faster-than- time.Comment: Full version of APPROX'21 paper, abstract shortened to fit ArXiv
requirement
Communication Lower Bounds of Key-Agreement Protocols via Density Increment Arguments
Constructing key-agreement protocols in the random oracle model (ROM) is a viable method to assess the feasibility of developing public-key cryptography within Minicrypt. Unfortunately, as shown by Impagliazzo and Rudich (STOC 1989) and Barak and Mahmoody (Crypto 2009), such protocols can only guarantee limited security: any -query protocol can be attacked by an -query adversary. This quadratic gap matches the key-agreement protocol proposed by Merkle (CACM 78), known as Merkle\u27s Puzzles.
Besides query complexity, the communication complexity of key-agreement protocols in the ROM is also an interesting question in the realm of find-grained cryptography, even though only limited security is achievable. Haitner et al. (ITCS 2019) first observed that in Merkle\u27s Puzzles, to obtain secrecy against an eavesdropper with queries, the honest parties must exchange bits. Therefore, they conjectured that high communication complexity is unavoidable, i.e., any -query protocols with bits of communication could be attacked by an -query adversary. This, if true, will suggest that Merkle\u27s Puzzle is also optimal regarding communication complexity. Building upon techniques from communication complexity, Haitner et al. (ITCS 2019) confirmed this conjecture for two types of key agreement protocols with certain natural properties.
This work affirms the above conjecture for all non-adaptive protocols with perfect completeness. Our proof uses a novel idea called density increment argument. This method could be of independent interest as it differs from previous communication lower bounds techniques (and bypasses some technical barriers)
LIPIcs, Volume 244, ESA 2022, Complete Volume
LIPIcs, Volume 244, ESA 2022, Complete Volum