Small Memory Robust Simulation of Client-Server Interactive Protocols over Oblivious Noisy Channels

Abstract

We revisit the problem of low-memory robust simulation of interactive protocols over noisy channels. Haeupler [FOCS 2014] considered robust simulation of two-party interactive protocols over oblivious, as well as adaptive, noisy channels. Since the simulation does not need to have fixed communication pattern, the achieved communication rates can circumvent the lower bound proved by Kol and Raz [STOC 2013]. However, a drawback of this approach is that each party needs to remember the whole history of the simulated transcript. In a subsequent manuscript, Haeupler and Resch considered low-memory simulation. The idea was to view the original protocol as a computational DAG and only the identities of the nodes are saved (as opposed to the whole transcript history) for backtracking to reduce memory usage.In this paper, we consider low-memory robust simulation of more general client-server interactive protocols, in which a leader communicates with other members/servers, who do not communicate among themselves; this setting can be applied to information-theoretic multi-server Private Information Retrieval (PIR) schemes. We propose an information-theoretic technique that converts any correct PIR protocol that assumes reliable channels, into a protocol which is both correct and private in the presence of a noisy channel while keeping the space complexity to a minimum. Despite the huge attention that PIR protocols have received in the literature, the existing works assume that the parties communicate using noiseless channels.Moreover, we observe that the approach of Haeupler and Resch to just save the nodes in the aforementioned DAG without taking the transcript history into account will lead to a correctness issue even for oblivious corruptions. We resolve this issue by saving hashes of prefixes of past transcripts. Departing from the DAG representation also allows us to accommodate scenarios where a party can simulate its part of the protocol without any extra knowledge (such as the DAG representation of the whole protocol). In the the two-party setting, our simulation has the same dependence on the error rate as in the work of Haeupler, and in the client-server setting it also depends on the number of servers. Furthermore, since our approach does not remember the complete transcript history, our current technique can defend only against oblivious corruptions.1. If protocol Π is run where the communication channel between Alice and each Bob has oblivious error rate , then, except with probability δ, Π correctly simulates Π.2. In Π , for each (noisy) communication channel, the number of bits transmitted is at most3. If party X has a memory usage of M X bits and is involved in κ (= 1 for each Bob, or m for Alice) communication channels in Π, then its memory usage (in bits) in Π is at mostPaper Organization. While we are trying to give the precise results of this paper in the above description as soon as possible, readers unfamiliar with the formal definitions and settings can first refer to Section 2. The research backgroun...

Circuits resilient to short-circuit errors

“…In particular, we conjecture that, unlike in the case of formulas [3,18], a polynomial overhead will not suffice. Unfortunately, proving so, even in an existential manner, may be currently out-of-reach as it would imply P/ poly ⊈ NC 1 ; since P/ poly ⊆ NC 1 implies that every circuit of polynomial size has an equivalent formula of polynomial size 4 , and thus, due to [3,18], also has an error-resilient formula of polynomial size.…”

“…Motivated by the problem of constructing resilient circuits, [15] (which is an unpublished manuscript by a subset of the current authors and is an earlier version of this work) initiated the study of interactive codes that incur a small overhead in memory 5 . Building on [15], the work of [4] gives an interactive coding scheme that is resilient to a constant fraction of adversarial errors and only incurs an O (log d ) overhead in the memory, where d is the length of Π. However, unlike our setting, the scheme of [4] assumes an oblivious adversary who makes all its decisions in advance (i.e., independently of the random choices of the interactive coding scheme or the communication history).…”

“…Building on [15], the work of [4] gives an interactive coding scheme that is resilient to a constant fraction of adversarial errors and only incurs an O (log d ) overhead in the memory, where d is the length of Π. However, unlike our setting, the scheme of [4] assumes an oblivious adversary who makes all its decisions in advance (i.e., independently of the random choices of the interactive coding scheme or the communication history). Moreover, as explained above, to get our result, aside from dealing with small memory, we also need to deal with memory corruptions and rectangular correctness (see Section 1.2).…”

“…Remark 3.5. We emphasize the correctness requirement of dagprotocols is significantly stronger than that of memory-limited communication protocols [4]. Namely, memory-limited communication protocols (defined with the same tuple as a dag-protocol) only require that o v ∈ S (x, y), where v is the unique leaf that is reached from the root on input (x, y), whereas, dag-protocols require that o v ∈ S (x, y) for every leaf v such that (x, y) ∈ R v , even if this leaf is not "reached from the root" on inputs (x, y).…”

Interactive Coding with Small Memory

“…The question of interactive coding with bounded space was first considered in the unpublished manuscript [18] 4 , which we consider to be an earlier version of this work. Building on ideas from [18], the space bounded interactive coding question was studied in weaker error models: The work of [7] gives an interactive coding scheme that is space efficient (i.e., incurs only logarithmic overhead to the space complexity, similarly to our work) and also achieves the (conjectured) optimal rate as approaches 0. However, unlike our setting, the scheme of [7] assumes an oblivious adversary who makes all its corruption decisions in advance.…”