24,993 research outputs found
A No-Go Theorem for Derandomized Parallel Repetition: Beyond Feige-Kilian
In this work we show a barrier towards proving a randomness-efficient
parallel repetition, a promising avenue for achieving many tight
inapproximability results. Feige and Kilian (STOC'95) proved an impossibility
result for randomness-efficient parallel repetition for two prover games with
small degree, i.e., when each prover has only few possibilities for the
question of the other prover. In recent years, there have been indications that
randomness-efficient parallel repetition (also called derandomized parallel
repetition) might be possible for games with large degree, circumventing the
impossibility result of Feige and Kilian. In particular, Dinur and Meir
(CCC'11) construct games with large degree whose repetition can be derandomized
using a theorem of Impagliazzo, Kabanets and Wigderson (SICOMP'12). However,
obtaining derandomized parallel repetition theorems that would yield optimal
inapproximability results has remained elusive.
This paper presents an explanation for the current impasse in progress, by
proving a limitation on derandomized parallel repetition. We formalize two
properties which we call "fortification-friendliness" and "yields robust
embeddings." We show that any proof of derandomized parallel repetition
achieving almost-linear blow-up cannot both (a) be fortification-friendly and
(b) yield robust embeddings. Unlike Feige and Kilian, we do not require the
small degree assumption.
Given that virtually all existing proofs of parallel repetition, including
the derandomized parallel repetition result of Dinur and Meir, share these two
properties, our no-go theorem highlights a major barrier to achieving
almost-linear derandomized parallel repetition
Anchoring games for parallel repetition
Two major open problems regarding the parallel repetition of games are whether an analogue of Raz's parallel-repetition theorem holds for (a) games with more than two players, and (b) games with quantum players using entanglement. We make progress on both problems: we introduce a class of games we call anchored, and prove exponential-decay parallel repetition theorems for anchored games in the multiplayer and entangled-player settings. We introduce a simple transformation on games called anchoring and show that this transformation turns any game into an anchored game. Together, our parallel repetition theorem and our anchoring transformation provide a simple and efficient hardness-amplification technique in both the classical multiplayer and quantum settings
An efficient quantum parallel repetition theorem and applications
We prove a tight parallel repetition theorem for -message computationally-secure quantum interactive protocols between an efficient challenger and an efficient adversary. We also prove under plausible assumptions that the security of -message computationally secure protocols does not generally decrease under parallel repetition. These mirror the classical results of Bellare, Impagliazzo, and Naor [BIN97]. Finally, we prove that all quantum argument systems can be generically compiled to an equivalent -message argument system, mirroring the transformation for quantum proof systems [KW00, KKMV07].
As immediate applications, we show how to derive hardness amplification theorems for quantum bit commitment schemes (answering a question of Yan [Yan22]), EFI pairs (answering a question of Brakerski, Canetti, and Qian [BCQ23]), public-key quantum money schemes (answering a question of Aaronson and Christiano [AC13]), and quantum zero-knowledge argument systems. We also derive an XOR lemma [Yao82] for quantum predicates as a corollary
Parallel Repetition of Entangled Games
We consider one-round games between a classical referee and
two players. One of the main questions in this area is the
parallel repetition question: Is there a way to decrease the
maximum winning probability of a game without increasing
the number of rounds or the number of players? Classically,
efforts to resolve this question, open for many years, have
culminated in Raz’s celebrated parallel repetition theorem
on one hand, and in efficient product testers for PCPs on
the other.
In the case where players share entanglement, the only
previously known results are for special cases of games, and
are based on techniques that seem inherently limited. Here
we show for the first time that the maximum success probability
of entangled games can be reduced through parallel
repetition, provided it was not initially 1. Our proof is inspired
by a seminal result of Feige and Kilian in the context
of classical two-prover one-round interactive proofs. One of
the main components in our proof is an orthogonalization
lemma for operators, which might be of independent interest
Derandomized Parallel Repetition via Structured PCPs
A PCP is a proof system for NP in which the proof can be checked by a
probabilistic verifier. The verifier is only allowed to read a very small
portion of the proof, and in return is allowed to err with some bounded
probability. The probability that the verifier accepts a false proof is called
the soundness error, and is an important parameter of a PCP system that one
seeks to minimize. Constructing PCPs with sub-constant soundness error and, at
the same time, a minimal number of queries into the proof (namely two) is
especially important due to applications for inapproximability.
In this work we construct such PCP verifiers, i.e., PCPs that make only two
queries and have sub-constant soundness error. Our construction can be viewed
as a combinatorial alternative to the "manifold vs. point" construction, which
is the only construction in the literature for this parameter range. The
"manifold vs. point" PCP is based on a low degree test, while our construction
is based on a direct product test. We also extend our construction to yield a
decodable PCP (dPCP) with the same parameters. By plugging in this dPCP into
the scheme of Dinur and Harsha (FOCS 2009) one gets an alternative construction
of the result of Moshkovitz and Raz (FOCS 2008), namely: a construction of
two-query PCPs with small soundness error and small alphabet size.
Our construction of a PCP is based on extending the derandomized direct
product test of Impagliazzo, Kabanets and Wigderson (STOC 09) to a derandomized
parallel repetition theorem. More accurately, our PCP construction is obtained
in two steps. We first prove a derandomized parallel repetition theorem for
specially structured PCPs. Then, we show that any PCP can be transformed into
one that has the required structure, by embedding it on a de-Bruijn graph
Gap Amplification for Small-Set Expansion via Random Walks
In this work, we achieve gap amplification for the Small-Set Expansion
problem. Specifically, we show that an instance of the Small-Set Expansion
Problem with completeness and soundness is at least as
difficult as Small-Set Expansion with completeness and soundness
, for any function which grows faster than
. We achieve this amplification via random walks -- our gadget
is the graph with adjacency matrix corresponding to a random walk on the
original graph. An interesting feature of our reduction is that unlike gap
amplification via parallel repetition, the size of the instances (number of
vertices) produced by the reduction remains the same
Predictable arguments of knowledge
We initiate a formal investigation on the power of predictability for argument of knowledge systems for NP. Specifically, we consider private-coin argument systems where the answer of the prover can be predicted, given the private randomness of the verifier; we call such protocols Predictable Arguments of Knowledge (PAoK).
Our study encompasses a full characterization of PAoK, showing that such arguments can be made extremely laconic, with the prover sending a single bit, and assumed to have only one round (i.e., two messages) of communication without loss of generality.
We additionally explore PAoK satisfying additional properties (including zero-knowledge and the possibility of re-using the same challenge across multiple executions with the prover), present several constructions of PAoK relying on different cryptographic tools, and discuss applications to cryptography
- …