Biblio

This paper introduces the notion of one-way communication schemes with partial noisy feedback. To support this communication, the schemes suppose that Alice and Bob wish to communicate: Alice sends a sequence of alphabets over a channel to Bob, while Alice receives feedback bits from Bob for δ fraction of the transmissions. An adversary is allowed to tamper up to a constant fraction of these transmissions for both forward rounds and feedback rounds separately. This paper intends to determine the Maximum Error Rate (MER), as a function of δ (0 ≤ δ ≤ 1), under the MER rate, so that Alice can successfully communicate the messages to Bob via some protocols with δ fraction of noisy feedback. To provide a reasonable solution for the above problem, we need to explore a new kind of coding scheme for the interactive communication. In this paper, we use the notion of “non-malleable codes” (NMC) which relaxes the notions of error-correction and error-detection to some extent in communication. Informally, a code is non-malleable if the message contained in a modified codeword is either the original message or a completely unrelated value. This property largely enforces the way to detect the transmission errors. Based on the above knowledge, we provide an alphabet-based encoding scheme, including a pair of (Enc, Dec). Suppose the message needing to be transmitted is m; if m is corrupted unintentionally, then the encoding scheme Dec(Enc(m)) outputs a symbol `⊥' to denote that some potential corruptions happened during transmission. In this work, based on the previous results, we show that for any δ ∈ (0; 1), there exists a deterministic communication scheme with noiseless full feedback(δ = 1), such that the maximal tolerable error fraction γ (on Alice's transmissions) can be up to 1/2, theoretically. Moreover, we show that for any δ ∈ (0; 1), there exists a communication scheme with noisy feedback, denoting the forward and backward rounds noised with error fractions of γ0and γ1respectively, such that the maximal tolerable error fraction γ0(on forward rounds) can be up to 1/2, as well as the γ1(on feedback rounds) up to 1.

Randomness extractors and error correcting codes are fundamental objects in computer science. Recently, there have been several natural generalizations of these objects, in the context and study of tamper resilient cryptography. These are seeded non-malleable extractors, introduced by Dodis and Wichs; seedless non-malleable extractors, introduced by Cheraghchi and Guruswami; and non-malleable codes, introduced by Dziembowski, Pietrzak and Wichs. Besides being interesting on their own, they also have important applications in cryptography, e.g, privacy amplification with an active adversary, explicit non-malleable codes etc, and often have unexpected connections to their non-tampered analogues. However, the known constructions are far behind their non-tampered counterparts. Indeed, the best known seeded non-malleable extractor requires min-entropy rate at least 0.49; while explicit constructions of non-malleable two-source extractors were not known even if both sources have full min-entropy, and was left as an open problem by Cheraghchi and Guruswami. In this paper we make progress towards solving the above problems and other related generalizations. Our contributions are as follows. (1) We construct an explicit seeded non-malleable extractor for polylogarithmic min-entropy. This dramatically improves all previous results and gives a simpler 2-round privacy amplification protocol with optimal entropy loss, matching the best known result. In fact, we construct more general seeded non-malleable extractors (that can handle multiple adversaries) which were used in the recent construction of explicit two-source extractors for polylogarithmic min-entropy. (2) We construct the first explicit non-malleable two-source extractor for almost full min-entropy thus resolving the open question posed by Cheraghchi and Guruswami. (3) We motivate and initiate the study of two natural generalizations of seedless non-malleable extractors and non-malleable codes, where the sources or the codeword may be tampered many times. By using the connection found by Cheraghchi and Guruswami and providing efficient sampling algorithms, we obtain the first explicit non-malleable codes with tampering degree t, with near optimal rate and error. We call these stronger notions one-many and many-manynon-malleable codes. This provides a stronger information theoretic analogue of a primitive known as continuous non-malleable codes. Our basic technique used in all of our constructions can be seen as inspired, in part, by the techniques previously used to construct cryptographic non-malleable commitments.