On Fortification of Projection Games

A recent result of Moshkovitz \cite{Moshkovitz14} presented an ingenious method to provide a completely elementary proof of the Parallel Repetition Theorem for certain projection games via a construction called fortification. However, the construction used in \cite{Moshkovitz14} to fortify arbitrary label cover instances using an arbitrary extractor is insufficient to prove parallel repetition. In this paper, we provide a fix by using a stronger graph that we call fortifiers. Fortifiers are graphs that have both $\ell_1$ and $\ell_2$ guarantees on induced distributions from large subsets. We then show that an expander with sufficient spectral gap, or a bi-regular extractor with stronger parameters (the latter is also the construction used in an independent update \cite{Moshkovitz15} of \cite{Moshkovitz14} with an alternate argument), is a good fortifier. We also show that using a fortifier (in particular $\ell_2$ guarantees) is necessary for obtaining the robustness required for fortification.Comment: 19 page

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

Gaming security by obscurity

Shannon sought security against the attacker with unlimited computational powers: *if an information source conveys some information, then Shannon's attacker will surely extract that information*. Diffie and Hellman refined Shannon's attacker model by taking into account the fact that the real attackers are computationally limited. This idea became one of the greatest new paradigms in computer science, and led to modern cryptography. Shannon also sought security against the attacker with unlimited logical and observational powers, expressed through the maxim that "the enemy knows the system". This view is still endorsed in cryptography. The popular formulation, going back to Kerckhoffs, is that "there is no security by obscurity", meaning that the algorithms cannot be kept obscured from the attacker, and that security should only rely upon the secret keys. In fact, modern cryptography goes even further than Shannon or Kerckhoffs in tacitly assuming that *if there is an algorithm that can break the system, then the attacker will surely find that algorithm*. The attacker is not viewed as an omnipotent computer any more, but he is still construed as an omnipotent programmer. So the Diffie-Hellman step from unlimited to limited computational powers has not been extended into a step from unlimited to limited logical or programming powers. Is the assumption that all feasible algorithms will eventually be discovered and implemented really different from the assumption that everything that is computable will eventually be computed? The present paper explores some ways to refine the current models of the attacker, and of the defender, by taking into account their limited logical and programming powers. If the adaptive attacker actively queries the system to seek out its vulnerabilities, can the system gain some security by actively learning attacker's methods, and adapting to them?Comment: 15 pages, 9 figures, 2 tables; final version appeared in the Proceedings of New Security Paradigms Workshop 2011 (ACM 2011); typos correcte

A parallel repetition theorem for all entangled games

The behavior of games repeated in parallel, when played with quantumly entangled players, has received much attention in recent years. Quantum analogues of Raz's classical parallel repetition theorem have been proved for many special classes of games. However, for general entangled games no parallel repetition theorem was known. We prove that the entangled value of a two-player game $G$ repeated $n$ times in parallel is at most $c_G n^{-1/4} \log n$ for a constant $c_G$ depending on $G$, provided that the entangled value of $G$ is less than 1. In particular, this gives the first proof that the entangled value of a parallel repeated game must converge to 0 for all games whose entangled value is less than 1. Central to our proof is a combination of both classical and quantum correlated sampling.Comment: To appear in the 43rd International Colloquium on Automata, Languages, and Programming (ICALP

Parallel repetition via fortification: analytic view and the quantum case

In a recent work, Moshkovitz [FOCS'14] presented a transformation n two-player games called "fortification", and gave an elementary proof of an (exponential decay) parallel repetition theorem for fortified two-player projection games. In this paper, we give an analytic reformulation of Moshkovitz's fortification framework, which was originally cast in combinatorial terms. This reformulation allows us to expand the scope of the fortification method to new settings. First, we show any game (not just projection games) can be fortified, and give a simple proof of parallel repetition for general fortified games. Then, we prove parallel repetition and fortification theorems for games with players sharing quantum entanglement, as well as games with more than two players. This gives a new gap amplification method for general games in the quantum and multiplayer settings, which has recently received much interest. An important component of our work is a variant of the fortification transformation, called "ordered fortification", that preserves the entangled value of a game. The original fortification of Moshkovitz does not in general preserve the entangled value of a game, and this was a barrier to extending the fortification framework to the quantum setting

Parallel Repetition From Fortification

The Parallel Repetition Theorem upper-bounds the value of a repeated (tensored) two prover game in terms of the value of the base game and the number of repetitions. In this work we give a simple transformation on games – “fortification” – and show that for fortified games, the value of the repeated game decreases perfectly exponentially with the number of repetitions, up to an arbitrarily small additive error. Our proof is combinatorial and short. As corollaries, we obtain: (1) Starting from a PCP Theorem with soundness error bounded away from 1, we get a PCP with arbitrarily small constant soundness error. In particular, starting with the combinatorial PCP of Dinur, we get a combinatorial PCP with low error. The latter can be used for hardness of approximation as in the work of Hastad. (2) Starting from the work of the author and Raz, we get a projection PCP theorem with the smallest soundness error known today. The theorem yields nearly a quadratic improvement in the size compared to previous work. We then discuss the problem of derandomizing parallel repetition, and the limitations of the fortification idea in this setting. We point out a connection between the problem of derandomizing parallel repetition and the problem of composition. This connection could shed light on the so-called Projection Games Conjecture, which asks for projection PCP with minimal error.National Science Foundation (U.S.) (Grant 1218547

Spartan Daily, June 7, 1943

Volume 31, Issue 149https://scholarworks.sjsu.edu/spartandaily/10809/thumbnail.jp

Spartan Daily, June 7, 1943

Volume 31, Issue 149https://scholarworks.sjsu.edu/spartandaily/10809/thumbnail.jp

Spartan Daily, June 7, 1943

Volume 31, Issue 149https://scholarworks.sjsu.edu/spartandaily/10809/thumbnail.jp