Shamir's celebrated secret sharing scheme provides an efficient method for encoding a secret of arbitrary length @math among any @math players such that for a threshold parameter @math , (i) the knowledge of any @math shares does not reveal any information about the secret and, (ii) any choice of @math shares fully reveals the secret. It is known that any such threshold secret sharing scheme necessarily requires shares of length @math , and in this sense Shamir's scheme is optimal. The more general notion of ramp schemes requires the reconstruction of secret from any @math shares, for a positive integer gap parameter @math . Ramp secret sharing scheme necessarily requires shares of length @math . Other than the bound related to secret length @math , the share lengths of ramp schemes can not go below a quantity that depends only on the gap ratio @math . In this work, we study secret sharing in the extremal case of bit-long shares and arbitrarily small gap ratio @math , where standard ramp secret sharing becomes impossible. We show, however, that a slightly relaxed but equally effective notion of semantic security for the secret, and negligible reconstruction error probability, eliminate the impossibility. Moreover, we provide explicit constructions of such schemes. One of the consequences of our relaxation is that, unlike standard ramp schemes with perfect secrecy, adaptive and non-adaptive adversaries need different analysis and construction. For non-adaptive adversaries, we explicitly construct secret sharing schemes that provide secrecy against any @math fraction of observed shares, and reconstruction from any @math fraction of shares, for any choices of @math . Our construction achieves secret length @math , which we show to be optimal. For adaptive adversaries, we construct explicit schemes attaining a secret length @math .
The notion of secrecy in wiretap codes has evolved over years. More recently the notion of semantic security for wiretap model has been introduced @cite_6 , which allows arbitrary message distribution and is shown to be equivalent to negligible leakage with respect to statistical distance. There remains one last distinction between semantically secure wiretap model and secret sharing with fixed share size. That is the nature of the main and wiretapper channels are typically stochastic (e.g., the erasure channel with random i.i.d. erasures), whereas for secret sharing a worst-case guarantee for the erasure patterns is required. Namely, in secret sharing, reconstruction with overwhelming probability is required for every choice of @math or more shares, and privacy of the secret is required for every (adaptive or non-adaptive) choice of the @math shares observed by the adversary.
@cite_6
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In this work, the critical role of noisy feedback in enhancing the secrecy capacity of the wiretap channel is established. Unlike previous works, where a noiseless public discussion channel is used for feedback, the feed-forward and feedback signals share the same noisy channel in the present model. Quite interestingly, this noisy feedback model is shown to be more advantageous in the current setting. More specifically, the discrete memoryless modulo-additive channel with a full-duplex destination node is considered first, and it is shown that the judicious use of feedback increases the secrecy capacity to the capacity of the source-destination channel in the absence of the wiretapper. In the achievability scheme, the feedback signal corresponds to a private key, known only to the destination. In the half-duplex scheme, a novel feedback technique that always achieves a positive perfect secrecy rate (even when the source-wiretapper channel is less noisy than the source-destination channel) is proposed. These results hinge on the modulo-additive property of the channel, which is exploited by the destination to perform encryption over the channel without revealing its key to the source. Finally, this scheme is extended to the continuous real valued modulo-Lambda channel where it is shown that the secrecy capacity with feedback is also equal to the capacity in the absence of the wiretapper. We consider a fading wiretap channel model where the transmitter has only statistical channel state information, and the legitimate receiver and eavesdropper have perfect channel state information. We propose a sequence of non-random lattice codes which achieve strong secrecy and semantic security over ergodic fading channels. The construction is almost universal in the sense that it achieves the same constant gap to secrecy capacity over Gaussian and ergodic fading models. We derive single-letter characterizations of (strong) secrecy capacities for models with an arbitrary number of terminals, each of which observes a distinct component of a discrete memoryless multiple source, with unrestricted and interactive public communication permitted between the terminals. A subset of these terminals can serve as helpers for the remaining terminals in generating secrecy. According to the extent of an eavesdropper's knowledge, three kinds of secrecy capacity are considered: secret key (SK), private key (PK), and wiretap secret key (WSK) capacity. The characterizations of the SK and PK capacities highlight the innate connections between secrecy generation and multiterminal source coding without secrecy requirements. A general upper bound for WSK capacity is derived which is tight in the case when the eavesdropper can wiretap noisy versions of the components of the underlying multiple source, provided randomization is permitted at the terminals. These secrecy capacities are seen to be achievable with noninteractive communication between the terminals. The achievability results are also shown to be universal. Suppose that Alice wishes to send messages to Bob through a communication channel C1, but her transmissions also reach an eavesdropper Eve through another channel C2. This is the wiretap channel model introduced by Wyner in 1975. The goal is to design a coding scheme that makes it possible for Alice to communicate both reliably and securely. Reliability is measured in terms of Bob's probability of error in recovering the message, while security is measured in terms of the mutual information between the message and Eve's observations. Wyner showed that the situation is characterized by a single constant Cs, called the secrecy capacity, which has the following meaning: for all e >; 0, there exist coding schemes of rate R ≥ Cs-e that asymptotically achieve the reliability and security objectives. However, his proof of this result is based upon a random-coding argument. To date, despite consider able research effort, the only case where we know how to construct coding schemes that achieve secrecy capacity is when Eve's channel C2 is an erasure channel, or a combinatorial variation thereof. Polar codes were recently invented by Arikan; they approach the capacity of symmetric binary-input discrete memoryless channels with low encoding and decoding complexity. In this paper, we use polar codes to construct a coding scheme that achieves the secrecy capacity for a wide range of wiretap channels. Our construction works for any instantiation of the wiretap channel model, as long as both C1 and C2 are symmetric and binary-input, and C2 is degraded with respect to C1. Moreover, we show how to modify our construction in order to provide strong security, in the sense defined by Maurer, while still operating at a rate that approaches the secrecy capacity. In this case, we cannot guarantee that the reliability condition will also be satisfied unless the main channel C1 is noiseless, although we believe it can be always satisfied in practice.