4,574 research outputs found
Non-Malleable Codes for Small-Depth Circuits
We construct efficient, unconditional non-malleable codes that are secure
against tampering functions computed by small-depth circuits. For
constant-depth circuits of polynomial size (i.e. tampering
functions), our codes have codeword length for a -bit
message. This is an exponential improvement of the previous best construction
due to Chattopadhyay and Li (STOC 2017), which had codeword length
. Our construction remains efficient for circuit depths as
large as (indeed, our codeword length remains
, and extending our result beyond this would require
separating from .
We obtain our codes via a new efficient non-malleable reduction from
small-depth tampering to split-state tampering. A novel aspect of our work is
the incorporation of techniques from unconditional derandomization into the
framework of non-malleable reductions. In particular, a key ingredient in our
analysis is a recent pseudorandom switching lemma of Trevisan and Xue (CCC
2013), a derandomization of the influential switching lemma from circuit
complexity; the randomness-efficiency of this switching lemma translates into
the rate-efficiency of our codes via our non-malleable reduction.Comment: 26 pages, 4 figure
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On Resilience to Computable Tampering
Non-malleable codes, introduced by Dziembowski, Pietrzak, and Wichs (ICS 2010), provide a means of encoding information such that if the encoding is tampered with, the result encodes something either identical or completely unrelated. Unlike error-correcting codes (for which the result of tampering must always be identical), non-malleable codes give guarantees even when tampering functions are allowed to change every symbol of a codeword.
In this thesis, we will provide constructions of non-malleable codes secure against a variety tampering classes with natural computational semantics:
âą Bounded-Communication: Functions corresponding to 2-party protocols where each party receives half the input (respectively) and then may communicate </4 bits before returning their (respective) half of the tampered output.
âąLocal Functions (Juntas):} each tampered output bit is only a function of nÂč-áș inputs bits, where áș>0 is any constant (the efficiency of our code depends on áș). This class includes NCâ°.
âąDecision Trees: each tampered output bit is a function of nÂč/âŽ-â°(Âč) adaptively chosen bits.
âąSmall-Depth Circuits: each tampered output bit is produced by a log(n)/log log(n)-depth circuit of polynomial size, for some constant . This class includes ACâ°.
âąLow Degree Polynomials: each tampered output field element is produced by a low-degree (relative to the field size) polynomial.
âąPolynomial-Size Circuit Tampering: each tampered codeword is produced by circuit of size ᶠwhere is any constant (the efficiency of our code depends on ). This result assumes that E is hard for exponential size nondeterministic circuits (all other results are unconditional).
We stress that our constructions are efficient (encoding and decoding can be performed in uniform polynomial time) and (with the exception of the last result, which assumes strong circuit lower bounds) enjoy unconditional, statistical security guarantees. We also illuminate some potential barriers to constructing codes for more complex computational classes from simpler assumptions
Limits to Non-Malleability
There have been many successes in constructing explicit non-malleable codes for various classes of tampering functions in recent years, and strong existential results are also known. In this work we ask the following question:
When can we rule out the existence of a non-malleable code for a tampering class ??
First, we start with some classes where positive results are well-known, and show that when these classes are extended in a natural way, non-malleable codes are no longer possible. Specifically, we show that no non-malleable codes exist for any of the following tampering classes:
- Functions that change d/2 symbols, where d is the distance of the code;
- Functions where each input symbol affects only a single output symbol;
- Functions where each of the n output bits is a function of n-log n input bits.
Furthermore, we rule out constructions of non-malleable codes for certain classes ? via reductions to the assumption that a distributional problem is hard for ?, that make black-box use of the tampering functions in the proof. In particular, this yields concrete obstacles for the construction of efficient codes for NC, even assuming average-case variants of P ? NC
Algebra in Computational Complexity
At its core, much of Computational Complexity is concerned with combinatorial objects and structures. But it has often proven true that the best way to prove things about these combinatorial objects is by establishing a connection to a more well-behaved algebraic setting. Indeed, many of the deepest and most powerful results in Computational Complexity rely on algebraic proof techniques. The Razborov-Smolensky polynomial-approximation method for proving constant-depth circuit lower bounds, the PCP characterization of NP, and the Agrawal-Kayal-Saxena polynomial-time primality test are some of the most prominent examples. The algebraic theme continues in some of the most exciting recent progress in computational complexity. There have been significant recent advances in algebraic circuit lower bounds, and the so-called "chasm at depth 4" suggests that the restricted models now being considered are not so far from ones that would lead to a general result. There have been similar successes concerning the related problems of polynomial identity testing and circuit reconstruction in the algebraic model, and these are tied to central questions regarding the power of randomness in computation. Representation theory has emerged as an important tool in three separate lines of work: the "Geometric Complexity Theory" approach to P vs. NP and circuit lower bounds, the effort to resolve the complexity of matrix multiplication, and a framework for constructing locally testable codes. Coding theory has seen several algebraic innovations in recent years, including multiplicity codes, and new lower bounds. This seminar brought together researchers who are using a diverse array of algebraic methods in a variety of settings. It plays an important role in educating a diverse community about the latest new techniques, spurring further progress
Improved Pseudorandom Generators from Pseudorandom Multi-Switching Lemmas
We give the best known pseudorandom generators for two touchstone classes in
unconditional derandomization: an -PRG for the class of size-
depth- circuits with seed length , and an -PRG for the class of -sparse
polynomials with seed length . These results bring the state of the art for
unconditional derandomization of these classes into sharp alignment with the
state of the art for computational hardness for all parameter settings:
improving on the seed lengths of either PRG would require breakthrough progress
on longstanding and notorious circuit lower bounds.
The key enabling ingredient in our approach is a new \emph{pseudorandom
multi-switching lemma}. We derandomize recently-developed
\emph{multi}-switching lemmas, which are powerful generalizations of
H{\aa}stad's switching lemma that deal with \emph{families} of depth-two
circuits. Our pseudorandom multi-switching lemma---a randomness-efficient
algorithm for sampling restrictions that simultaneously simplify all circuits
in a family---achieves the parameters obtained by the (full randomness)
multi-switching lemmas of Impagliazzo, Matthews, and Paturi [IMP12] and
H{\aa}stad [H{\aa}s14]. This optimality of our derandomization translates into
the optimality (given current circuit lower bounds) of our PRGs for
and sparse polynomials
Continuously non-malleable codes with split-state refresh
Non-malleable codes for the split-state model allow to encode a message into two parts, such that arbitrary independent tampering on each part, and subsequent decoding of the corresponding modified codeword, yields either the same as the original message, or a completely unrelated value. Continuously non-malleable codes further allow to tolerate an unbounded (polynomial) number of tampering attempts, until a decoding error happens. The drawback is that, after an error happens, the system must self-destruct and stop working, otherwise generic attacks become possible. In this paper we propose a solution to this limitation, by leveraging a split-state refreshing procedure. Namely, whenever a decoding error happens, the two parts of an encoding can be locally refreshed (i.e., without any interaction), which allows to avoid the self-destruct mechanism. An additional feature of our security model is that it captures directly security against continual leakage attacks. We give an abstract framework for building such codes in the common reference string model, and provide a concrete instantiation based on the external Diffie-Hellman assumption. Finally, we explore applications in which our notion turns out to be essential. The first application is a signature scheme tolerating an arbitrary polynomial number of split-state tampering attempts, without requiring a self-destruct capability, and in a model where refreshing of the memory happens only after an invalid output is produced. This circumvents an impossibility result from a recent work by Fuijisaki and Xagawa (Asiacrypt 2016). The second application is a compiler for tamper-resilient RAM programs. In comparison to other tamper-resilient compilers, ours has several advantages, among which the fact that, for the first time, it does not rely on the self-destruct feature
Non-malleable codes for space-bounded tampering
Non-malleable codesâintroduced by Dziembowski, Pietrzak and Wichs at ICS 2010âare key-less coding schemes in which mauling attempts to an encoding of a given message, w.r.t. some class of tampering adversaries, result in a decoded value that is either identical or unrelated to the original message. Such codes are very useful for protecting arbitrary cryptographic primitives against tampering attacks against the memory. Clearly, non-malleability is hopeless if the class of tampering adversaries includes the decoding and encoding algorithm. To circumvent this obstacle, the majority of past research focused on designing non-malleable codes for various tampering classes, albeit assuming that the adversary is unable to decode. Nonetheless, in many concrete settings, this assumption is not realistic
Two-Source Condensers with Low Error and Small Entropy Gap via Entropy-Resilient Functions
In their seminal work, Chattopadhyay and Zuckerman (STOC\u2716) constructed a two-source extractor with error epsilon for n-bit sources having min-entropy {polylog}(n/epsilon). Unfortunately, the construction\u27s running-time is {poly}(n/epsilon), which means that with polynomial-time constructions, only polynomially-small errors are possible. Our main result is a {poly}(n,log(1/epsilon))-time computable two-source condenser. For any k >= {polylog}(n/epsilon), our condenser transforms two independent (n,k)-sources to a distribution over m = k-O(log(1/epsilon)) bits that is epsilon-close to having min-entropy m - o(log(1/epsilon)). Hence, achieving entropy gap of o(log(1/epsilon)).
The bottleneck for obtaining low error in recent constructions of two-source extractors lies in the use of resilient functions. Informally, this is a function that receives input bits from r players with the property that the function\u27s output has small bias even if a bounded number of corrupted players feed adversarial inputs after seeing the inputs of the other players. The drawback of using resilient functions is that the error cannot be smaller than ln r/r. This, in return, forces the running time of the construction to be polynomial in 1/epsilon.
A key component in our construction is a variant of resilient functions which we call entropy-resilient functions. This variant can be seen as playing the above game for several rounds, each round outputting one bit. The goal of the corrupted players is to reduce, with as high probability as they can, the min-entropy accumulated throughout the rounds. We show that while the bias decreases only polynomially with the number of players in a one-round game, their success probability decreases exponentially in the entropy gap they are attempting to incur in a repeated game
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