Wyner's elegant model of wiretap channel exploits noise in the communication channel to provide perfect secrecy against a computationally unlimited eavesdropper without requiring a shared key. We consider an adversarial model of wiretap channel proposed in [18,19] where the adversary is active: it selects a fraction @math of the transmitted codeword to eavesdrop and a fraction @math of the codeword to corrupt by "adding" adversarial error. It was shown that this model also captures network adversaries in the setting of 1-round Secure Message Transmission [8]. It was proved that secure communication (1-round) is possible if and only if @math . In this paper we show that by allowing communicants to have access to a public discussion channel (authentic communication without secrecy) secure communication becomes possible even if @math . We formalize the model of protocol and for two efficiency measures, information rate and message round complexity derive tight bounds. We also construct a rate optimal protocol family with minimum number of message rounds. We show application of these results to Secure Message Transmission with Public Discussion (SMT-PD), and in particular show a new lower bound on transmission rate of these protocols together with a new construction of an optimal SMT-PD protocol.
Other models of adversarial wiretap, @cite_21 @cite_25 @cite_22 @cite_2 , and their relationship to the model considered here, have been discussed in @cite_0 . The paper also establishes a relation between a special subset of adversarial wiretap codes and 1-round Secure Message Transmission (SMT) @cite_16 protocols for networks. In SMT, Alice and Bob are connected by @math node disjoint communication paths in a network, a subset of which is controlled by an adversary who can see and arbitrarily change the transmissions over the controlled paths. SMT protocols provide security and reliability against this adversary.
@cite_22 @cite_21 @cite_0 @cite_2 @cite_16 @cite_25
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In a secure message transmission (SMT) scenario, a sender wants to send a message in a private and reliable way to a receiver. Sender and receiver are connected by n wires, t of which can be controlled by an adaptive adversary with unlimited computational resources. In Eurocrypt 2008, Garay and Ostrovsky considered an SMT scenario where sender and receiver have access to a public discussion channel and showed that secure and reliable communication is possible when n ≥ t + 1. In this paper, we will show that a secure protocol requires at least three rounds of communication and two rounds invocation of the public channel and hence give a complete answer to the open question raised by Garay and Ostrovsky. We also describe a round optimal protocol that has constant transmission rate over the public channel. 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. We consider the MIMO wiretap channel, where a legitimate transmitter (Alice) wishes to communicate a confidential message to the legitimate receiver (Bob) over a MIMO channel, while its messages are being eavesdropped by Eve through another MIMO channel. Supposing that Alice uses the nested lattice code proposed in [8], we characterize the expected value of amount of information leakage to Eve, and from minimizing this expected value, we derive a code design criterion for MIMO lattice wiretap codes. The special cases of block and fast fading channels are also studied. We also illustrate our lattice code design criterion using the Alamouti Code as the simplest MIMO code available in the literature. We study the question of basing symmetric key cryptography on weak secrets. In this setting, Alice and Bob share an n-bit secret W, which might not be uniformly random, but the adversary has at least k bits of uncertainty about it (formalized using conditional min-entropy). Since standard symmetric-key primitives require uniformly random secret keys, we would like to construct an authenticated key agreement protocol in which Alice and Bob use W to agree on a nearly uniform key R, by communicating over a public channel controlled by an active adversary Eve. We study this question in the information theoretic setting where the attacker is computationally unbounded. We show that single-round (i.e. one message) protocols do not work when k ≤ n 2, and require poor parameters even when n 2 On the other hand, for arbitrary values of k, we design a communication efficient two-round (challenge-response) protocol extracting nearly k random bits. This dramatically improves the previous construction of Renner and Wolf [32], which requires Θ(λ + log(n)) rounds where λ is the security parameter. Our solution takes a new approach by studying and constructing "non-malleable" seeded randomness extractors -- if an attacker sees a random seed X and comes up with an arbitrarily related seed X', then we bound the relationship between R= Ext(W;X) and R' = Ext(W;X'). We also extend our two-round key agreement protocol to the "fuzzy" setting, where Alice and Bob share "close" (but not equal) secrets WA and WB, and to the Bounded Retrieval Model (BRM) where the size of the secret W is huge.