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Electronic Colloquium on Computational Complexity

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All reports by Author Benny Applebaum:

TR17-189 | 25th December 2017
Benny Applebaum, Barak Arkis

Conditional Disclosure of Secrets and $d$-Uniform Secret Sharing with Constant Information Rate

Consider the following secret-sharing problem. Your goal is to distribute a long file $s$ between $n$ servers such that $(d-1)$-subsets cannot recover the file, $(d+1)$-subsets can recover the file, and $d$-subsets should be able to recover $s$ if and only if they appear in some predefined list $L$. How small ... more >>>

TR17-067 | 21st April 2017
Benny Applebaum

Garbled Circuits as Randomized Encodings of Functions: a Primer

Yao's garbled circuit construction is a central cryptographic tool with numerous applications. In this tutorial, we study garbled circuits from a foundational point of view under the framework of \emph{randomized encoding} (RE) of functions. We review old and new constructions of REs, present some lower bounds, and describe some applications. ... more >>>

TR17-063 | 10th April 2017
Benny Applebaum

Exponentially-Hard gap-CSP and local PRG via Local Hardcore Functions

The gap-ETH assumption (Dinur 2016; Manurangsi and Raghavendra 2016) asserts that it is exponentially-hard to distinguish between a satisfiable 3-CNF formula and a 3-CNF formula which is at most 0.99-satisfiable. We show that this assumption follows from the exponential hardness of finding a satisfying assignment for *smooth* 3-CNFs. Here smoothness ... more >>>

TR17-038 | 23rd February 2017
Benny Applebaum, Barak Arkis, Pavel Raykov, Prashant Nalini Vasudevan

Conditional Disclosure of Secrets: Amplification, Closure, Amortization, Lower-bounds, and Separations

Revisions: 1

In the \emph{conditional disclosure of secrets} problem (Gertner et al., J. Comput. Syst. Sci., 2000) Alice and Bob, who hold inputs $x$ and $y$ respectively, wish to release a common secret $s$ to Carol (who knows both $x$ and $y$) if only if the input $(x,y)$ satisfies some predefined predicate ... more >>>

TR17-008 | 14th January 2017
Benny Applebaum, Naama Haramaty, Yuval Ishai, Eyal Kushilevitz, Vinod Vaikuntanathan

Low-Complexity Cryptographic Hash Functions

Cryptographic hash functions are efficiently computable functions that shrink a long input into a shorter output while achieving some of the useful security properties of a random function. The most common type of such hash functions is {\em collision resistant} hash functions (CRH), which prevent an efficient attacker from finding ... more >>>

TR16-082 | 22nd May 2016
Benny Applebaum, Pavel Raykov

Fast Pseudorandom Functions Based on Expander Graphs

We present direct constructions of pseudorandom function (PRF) families based on Goldreich's one-way function. Roughly speaking, we assume that non-trivial local mappings $f:\{0,1\}^n\rightarrow \{0,1\}^m$ whose input-output dependencies graph form an expander are hard to invert. We show that this one-wayness assumption yields PRFs with relatively low complexity. This includes weak ... more >>>

TR15-206 | 15th December 2015
Benny Applebaum, Pavel Raykov

From Private Simultaneous Messages to Zero-Information Arthur-Merlin Protocols and Back

Goos, Pitassi and Watson (ITCS, 2015) have recently introduced the notion of Zero-Information Arthur-Merlin Protocols (ZAM). In this model, which can be viewed as a private version of the standard Arthur-Merlin communication complexity game, Alice and Bob are holding a pair of inputs $x$ and $y$ respectively, and Merlin, the ... more >>>

TR15-186 | 24th November 2015
Benny Applebaum, Pavel Raykov

On the Relationship between Statistical Zero-Knowledge and Statistical Randomized Encodings

\emph{Statistical Zero-knowledge proofs} (Goldwasser, Micali and Rackoff, SICOMP 1989) allow a computationally-unbounded server to convince a computationally-limited client that an input $x$ is in a language $\Pi$ without revealing any additional information about $x$ that the client cannot compute by herself. \emph{Randomized encoding} (RE) of functions (Ishai and Kushilevitz, FOCS ... more >>>

TR15-172 | 3rd November 2015
Benny Applebaum, Shachar Lovett

Algebraic Attacks against Random Local Functions and Their Countermeasures

Revisions: 1

Suppose that you have $n$ truly random bits $x=(x_1,\ldots,x_n)$ and you wish to use them to generate $m\gg n$ pseudorandom bits $y=(y_1,\ldots, y_m)$ using a local mapping, i.e., each $y_i$ should depend on at most $d=O(1)$ bits of $x$. In the polynomial regime of $m=n^s$, $s>1$, the only known solution, ... more >>>

TR15-061 | 14th April 2015
Benny Applebaum, Jonathan Avron, Christina Brzuska

Arithmetic Cryptography

Revisions: 1

We study the possibility of computing cryptographic primitives in a fully-black-box arithmetic model over a finite field F. In this model, the input to a cryptographic primitive (e.g., encryption scheme) is given as a sequence of field elements, the honest parties are implemented by arithmetic circuits which make only a ... more >>>

TR15-051 | 5th April 2015
Benny Applebaum, Sergei Artemenko, Ronen Shaltiel, Guang Yang

Incompressible Functions, Relative-Error Extractors, and the Power of Nondeterminsitic Reductions

Revisions: 2

A circuit $C$ \emph{compresses} a function $f:\{0,1\}^n\rightarrow \{0,1\}^m$ if given an input $x\in \{0,1\}^n$ the circuit $C$ can shrink $x$ to a shorter $\ell$-bit string $x'$ such that later, a computationally-unbounded solver $D$ will be able to compute $f(x)$ based on $x'$. In this paper we study the existence of ... more >>>

TR15-045 | 1st April 2015
Benny Applebaum, Yuval Ishai, Eyal Kushilevitz

Minimizing Locality of One-Way Functions via Semi-Private Randomized Encodings

Revisions: 1

A one-way function is $d$-local if each of its outputs depends on at most $d$ input bits. In (Applebaum, Ishai, and Kushilevitz, FOCS 2004) it was shown that, under relatively mild assumptions, there exist $4$-local one-way functions (OWFs). This result is not far from optimal as it is not hard ... more >>>

TR15-027 | 25th February 2015
Benny Applebaum

Cryptographic Hardness of Random Local Functions -- Survey

Revisions: 1

Constant parallel-time cryptography allows to perform complex cryptographic tasks at an ultimate level of parallelism, namely, by local functions that each of their output bits depend on a constant number of input bits. A natural way to obtain local cryptographic constructions is to use \emph{random local functions} in which each ... more >>>

TR13-098 | 28th June 2013
Benny Applebaum, Yoni Moses

Locally Computable UOWHF with Linear Shrinkage

We study the problem of constructing locally computable Universal One-Way Hash Functions (UOWHFs) $H:\{0,1\}^n \rightarrow \{0,1\}^m$. A construction with constant \emph{output locality}, where every bit of the output depends only on a constant number of bits of the input, was established by [Applebaum, Ishai, and Kushilevitz, SICOMP 2006]. However, this ... more >>>

TR12-058 | 5th May 2012
Benny Applebaum, Yuval Ishai, Eyal Kushilevitz

How to Garble Arithmetic Circuits

Revisions: 1

Yao's garbled circuit construction transforms a boolean circuit $C:\{0,1\}^n\to\{0,1\}^m$
into a ``garbled circuit'' $\hat{C}$ along with $n$ pairs of $k$-bit keys, one for each
input bit, such that $\hat{C}$ together with the $n$ keys
corresponding to an input $x$ reveal $C(x)$ and no additional information about $x$.
The garbled circuit ... more >>>

TR11-126 | 17th September 2011
Benny Applebaum, Andrej Bogdanov, Alon Rosen

A Dichotomy for Local Small-Bias Generators

We consider pseudorandom generators in which each output bit depends on a constant number of input bits. Such generators have appealingly simple structure: they can be described by a sparse input-output dependency graph and a small predicate that is applied at each output. Following the works of Cryan and Miltersen ... more >>>

TR11-007 | 17th January 2011
Benny Applebaum

Pseudorandom Generators with Long Stretch and Low locality from Random Local One-Way Functions

Revisions: 3

We continue the study of pseudorandom generators (PRG) $G:\{0,1\}^n \rightarrow \{0,1\}^m$ in NC0. While it is known that such generators are likely to exist for the case of small sub-linear stretch $m=n+n^{1-\epsilon}$, it remains unclear whether achieving larger stretch such as $m=2n$ or even $m=n+n^2$ is possible. The existence of ... more >>>

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