2,393 research outputs found
An Algorithm for Constructing a Smallest Register with Non-Linear Update Generating a Given Binary Sequence
Registers with Non-Linear Update (RNLUs) are a generalization of Non-Linear
Feedback Shift Registers (NLFSRs) in which both, feedback and feedforward,
connections are allowed and no chain connection between the stages is required.
In this paper, a new algorithm for constructing RNLUs generating a given binary
sequence is presented. Expected size of RNLUs constructed by the presented
algorithm is proved to be O(n/log(n/p)), where n is the sequence length and p
is the degree of parallelization. This is asymptotically smaller than the
expected size of RNLUs constructed by previous algorithms and the expected size
of LFSRs and NLFSRs generating the same sequence. The presented algorithm can
potentially be useful for many applications, including testing, wireless
communications, and cryptography
SPRIGHT: A Fast and Robust Framework for Sparse Walsh-Hadamard Transform
We consider the problem of computing the Walsh-Hadamard Transform (WHT) of
some -length input vector in the presence of noise, where the -point
Walsh spectrum is -sparse with scaling sub-linearly in
the input dimension for some . Over the past decade, there has
been a resurgence in research related to the computation of Discrete Fourier
Transform (DFT) for some length- input signal that has a -sparse Fourier
spectrum. In particular, through a sparse-graph code design, our earlier work
on the Fast Fourier Aliasing-based Sparse Transform (FFAST) algorithm computes
the -sparse DFT in time by taking noiseless samples.
Inspired by the coding-theoretic design framework, Scheibler et al. proposed
the Sparse Fast Hadamard Transform (SparseFHT) algorithm that elegantly
computes the -sparse WHT in the absence of noise using
samples in time . However, the SparseFHT algorithm explicitly
exploits the noiseless nature of the problem, and is not equipped to deal with
scenarios where the observations are corrupted by noise. Therefore, a question
of critical interest is whether this coding-theoretic framework can be made
robust to noise. Further, if the answer is yes, what is the extra price that
needs to be paid for being robust to noise? In this paper, we show, quite
interestingly, that there is {\it no extra price} that needs to be paid for
being robust to noise other than a constant factor. In other words, we can
maintain the same sample complexity and the computational
complexity as those of the noiseless case, using our SParse
Robust Iterative Graph-based Hadamard Transform (SPRIGHT) algorithm.Comment: Part of our results was reported in ISIT 2014, titled "The SPRIGHT
algorithm for robust sparse Hadamard Transforms.
Approximate Gradient Coding via Sparse Random Graphs
Distributed algorithms are often beset by the straggler effect, where the
slowest compute nodes in the system dictate the overall running time.
Coding-theoretic techniques have been recently proposed to mitigate stragglers
via algorithmic redundancy. Prior work in coded computation and gradient coding
has mainly focused on exact recovery of the desired output. However, slightly
inexact solutions can be acceptable in applications that are robust to noise,
such as model training via gradient-based algorithms. In this work, we present
computationally simple gradient codes based on sparse graphs that guarantee
fast and approximately accurate distributed computation. We demonstrate that
sacrificing a small amount of accuracy can significantly increase algorithmic
robustness to stragglers
Mathematical foundations of modern cryptography: computational complexity perspective
Theoretical computer science has found fertile ground in many areas of
mathematics. The approach has been to consider classical problems through the
prism of computational complexity, where the number of basic computational
steps taken to solve a problem is the crucial qualitative parameter. This new
approach has led to a sequence of advances, in setting and solving new
mathematical challenges as well as in harnessing discrete mathematics to the
task of solving real-world problems.
In this talk, I will survey the development of modern cryptography -- the
mathematics behind secret communications and protocols -- in this light. I will
describe the complexity theoretic foundations underlying the cryptographic
tasks of encryption, pseudo-randomness number generators and functions, zero
knowledge interactive proofs, and multi-party secure protocols. I will attempt
to highlight the paradigms and proof techniques which unify these foundations,
and which have made their way into the mainstream of complexity theory
On the Efficiency of Solving Boolean Polynomial Systems with the Characteristic Set Method
An improved characteristic set algorithm for solving Boolean polynomial
systems is proposed. This algorithm is based on the idea of converting all the
polynomials into monic ones by zero decomposition, and using additions to
obtain pseudo-remainders. Three important techniques are applied in the
algorithm. The first one is eliminating variables by new generated linear
polynomials. The second one is optimizing the strategy of choosing polynomial
for zero decomposition. The third one is to compute add-remainders to eliminate
the leading variable of new generated monic polynomials. By analyzing the depth
of the zero decomposition tree, we present some complexity bounds of this
algorithm, which are lower than the complexity bounds of previous
characteristic set algorithms. Extensive experimental results show that this
new algorithm is more efficient than previous characteristic set algorithms for
solving Boolean polynomial systems
Pseudorandomness for Multilinear Read-Once Algebraic Branching Programs, in any Order
We give deterministic black-box polynomial identity testing algorithms for
multilinear read-once oblivious algebraic branching programs (ROABPs), in
n^(lg^2 n) time. Further, our algorithm is oblivious to the order of the
variables. This is the first sub-exponential time algorithm for this model.
Furthermore, our result has no known analogue in the model of read-once
oblivious boolean branching programs with unknown order, as despite recent work
there is no known pseudorandom generator for this model with sub-polynomial
seed-length (for unbounded-width branching programs).
This result extends and generalizes the result of Forbes and Shpilka that
obtained a n^(lg n)-time algorithm when given the order. We also extend and
strengthen the work of Agrawal, Saha and Saxena that gave a black-box algorithm
running in time exp((lg n)^d) for set-multilinear formulas of depth d. We note
that the model of multilinear ROABPs contains the model of set-multilinear
algebraic branching programs, which itself contains the model of
set-multilinear formulas of arbitrary depth. We obtain our results by
recasting, and improving upon, the ideas of Agrawal, Saha and Saxena. We phrase
the ideas in terms of rank condensers and Wronskians, and show that our results
improve upon the classical multivariate Wronskian, which may be of independent
interest.
In addition, we give the first n^(lglg n) black-box polynomial identity
testing algorithm for the so called model of diagonal circuits. This model,
introduced by Saxena has recently found applications in the work of Mulmuley,
as well as in the work of Gupta, Kamath, Kayal, Saptharishi. Previously work
had given n^(lg n)-time algorithms for this class. More generally, our result
holds for any model computing polynomials whose partial derivatives (of all
orders) span a low dimensional linear space.Comment: 38 page
Short seed extractors against quantum storage
Some, but not all, extractors resist adversaries with limited quantum
storage. In this paper we show that Trevisan's extractor has this property,
thereby showing an extractor against quantum storage with logarithmic seed
length
Monotone Boolean Functions, Feasibility/Infeasibility, LP-type problems and MaxCon
This paper outlines connections between Monotone Boolean Functions, LP-Type
problems and the Maximum Consensus Problem. The latter refers to a particular
type of robust fitting characterisation, popular in Computer Vision (MaxCon).
Indeed, this is our main motivation but we believe the results of the study of
these connections are more widely applicable to LP-type problems (at least
'thresholded versions', as we describe), and perhaps even more widely. We
illustrate, with examples from Computer Vision, how the resulting perspectives
suggest new algorithms. Indeed, we focus, in the experimental part, on how the
Influence (a property of Boolean Functions that takes on a special form if the
function is Monotone) can guide a search for the MaxCon solution.Comment: Parts under conference review, work in progress. Keywords: Monotone
Boolean Functions, Consensus Maximisation, LP-Type Problem, Computer Vision,
Robust Fitting, Matroid, Simplicial Complex, Independence System
Hyperparameter Optimization in Neural Networks via Structured Sparse Recovery
In this paper, we study two important problems in the automated design of
neural networks -- Hyper-parameter Optimization (HPO), and Neural Architecture
Search (NAS) -- through the lens of sparse recovery methods. In the first part
of this paper, we establish a novel connection between HPO and structured
sparse recovery. In particular, we show that a special encoding of the
hyperparameter space enables a natural group-sparse recovery formulation, which
when coupled with HyperBand (a multi-armed bandit strategy), leads to
improvement over existing hyperparameter optimization methods. Experimental
results on image datasets such as CIFAR-10 confirm the benefits of our
approach. In the second part of this paper, we establish a connection between
NAS and structured sparse recovery. Building upon ``one-shot'' approaches in
NAS, we propose a novel algorithm that we call CoNAS by merging ideas from
one-shot approaches with a techniques for learning low-degree sparse Boolean
polynomials. We provide theoretical analysis on the number of validation error
measurements. Finally, we validate our approach on several datasets and
discover novel architectures hitherto unreported, achieving competitive (or
better) results in both performance and search time compared to the existing
NAS approaches.Comment: arXiv admin note: text overlap with arXiv:1906.0286
sAVSS: Scalable Asynchronous Verifiable Secret Sharing in BFT Protocols
This paper introduces a new way to incorporate verifiable secret sharing
(VSS) schemes into Byzantine Fault Tolerance (BFT) protocols. This technique
extends the threshold guarantee of classical Byzantine Fault Tolerant
algorithms to include privacy as well. This provides applications with a
powerful primitive: a threshold trusted third party, which simplifies many
difficult problems such as a fair exchange. In order to incorporate VSS into
BFT, we introduced sAVSS, a framework that transforms any VSS scheme into an
asynchronous VSS scheme with constant overhead. By incorporating Kate et al.'s
scheme into our framework, we obtain an asynchronous VSS that has constant
overhead on each replica -- the first of its kind. We show that a key-value
store built using BFT replication and sAVSS supports writing secret-shared
values with about a 30% - 50% throughput overhead with less than 35 millisecond
request latencies
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