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arXiv:1004.4334 · International Symposium on Information Theory and its Applications

New results on Secret Key Establishment over a pair of broadcast channels

Hadi Ahmadi, Reihaneh Safavi–Naini

Published Oct 1, 2010• 4 citations
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Abstract

Secret Key Establishment (SKE) over a pair of independent Discrete Memoryless Broadcast Channels (DMBCs) was studied in where lower and upper bounds on the secret-key capacity were provided. In this paper, we study the above setup for two cases: (1) the DMBCs have “secrecy potential”, and (2) the DMBCs are stochastically degraded with independent channels. For (1), we propose a simple SKE protocol using the novel Interactive Channel Coding (ICC) method and prove that it achieves the lower bound. For (2), we give a simplified expression for the lower bound and prove that, when one of the legitimate parties sends only i.i.d. variables, the lower bound is tight and the capacity is achieved by a two-round protocol.

DOI
10.1109/isita.2010.5650104

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Secret Key Establishment (SKE) over a pair of independent Discrete Memoryless Broadcast Channels (DMBCs) was studied in where lower and upper bounds on the secret-key capacity were provided. In this paper, we study the above setup for two cases: (1) the DMBCs have “secrecy potential”, and (2) the DMBCs are stochastically degraded with independent channels. For (1), we propose a simple SKE protocol using the novel Interactive Channel Coding (ICC) method and prove that it achieves the lower bound. For (2), we give a simplified expression for the lower bound and prove that, when one of the legitimate parties sends only i.i.d. variables, the lower bound is tight and the capacity is achieved by a two-round protocol.
This is the second part of a two-part paper on information-theoretically secure secret key agreement. This part covers the secret key capacity under the channel model. In this model, multiple terminals wish to create a shared secret key that is secure from an eavesdropper with unlimited computational resources. The terminals are all connected to a noiseless and authenticated but insecure channel, called the “public channel.” Furthermore, the terminals have access to a secure but noisy discrete memoryless broadcast channel (DMBC). The first terminal can choose a sequence of inputs to the DMBC, which has outputs at the other terminals and at the eavesdropper. After each channel use, the terminals can engage in arbitrarily many rounds of interactive authenticated communication over the public channel. At the end, each legitimate terminal should be able to generate the secret key. In this paper, we derive new lower and upper bounds on the secrecy capacity. In each case, an example is provided to show that the new bound represents a strict improvement over the previously best known bound. This part of the paper is not standalone, and is written under the assumption that the reader has access to Part I, which is published in the same issue.
Abstract. We study the problem of unconditionally secure Secret Key Establishment (SKE) when Alice and Bob are connected by two noisy channels in opposite directions, and the channels are eavesdropped by Eve. We consider the case that Alice and Bob do not have any sources of initial randomness at their disposal. We start by discussing special cases of interest where SKE is impossible, and then provide a simple SKE construction over a binary symmetric channel that achieves some rates of secret key. We next focus on the Secret Key (SK) capacity, i.e., the highest rate of secure and reliable key establishment (in bits per channel use) that the parties can achieve. Relying on the existence of capacity-achieving coding schemes, we propose a multi-round SKE protocol, called the main protocol, that proves a lower bound on the SK capacity. The main protocol consists of an initialization round, followed by repeated use of a two-round SKE protocol, called the basic protocol. We also provide an upper bound on the SK capacity and show that the two bounds coincide when channels do not leak information to the adversary. We apply the results to the case that communicants are connected by binary symmetric channels. 1
Abstract. Common Randomness Generation (CRG) and Secret Key Establishment (SKE) are fundamental primitives that are used in information-theoretic coding and cryptography. We study these two problems over the two-way channel model of communication, introduced by Shannon. In this model, the common randomness (CK) capacity is defined as the maximum number of random bits per channel use that the two parties can generate. The secret key (SK) capacity is defined similarly when the random bits are also required to be secure against a passive adversary. We provide lower bounds on the two capacities. These lower bounds are tighter than those one might derive based on the previously known results. We prove our lower bounds by proposing a two-round, two-level coding construction over the two-way channel. We show that the lower bound on the common randomness capacity can also be achieved using a simple interactive channel coding (ICC) method. We furthermore provide upper bounds on these capacities and show that the lower and the upper bounds coincide when the two-way channel consists of two independent (physically degraded) one-way channels. We apply the results to the case where the channels are binary symmetric.
The problem of secure broadcasting with independent secret keys is studied. The particular scenario is analyzed in which a common message has to be broadcast to two legitimate receivers, while keeping an external eavesdropper ignorant of it. The transmitter shares independent secret keys of sufficiently high rates with both legitimate receivers, which can be used in different ways: they can be used as one-time pads to encrypt the common message, as fictitious messages for wiretap coding, or as a hybrid of these. In this paper, capacity results are established when the broadcast channels involving the three receivers are degraded. If both legitimate channels are degraded versions of the eavesdropper's channel, it is shown that the one-time pad approach is optimal for several cases, yielding corresponding capacity expressions. Alternatively, the wiretap coding approach is shown to be optimal if the eavesdropper's channel is degraded with respect to both legitimate channels, establishing capacity in this case as well. If the eavesdropper's channel is neither the strongest nor the weakest, an intricate scheme that carefully combines both concepts of one-time pad and wiretap coding with fictitious messages turns out to be capacity-achieving. Finally we also obtain some results for the general non-degraded broadcast channel.
The broadcast channel with independent secret keys models a communication scenario in which a common message has to be securely broadcast to two legitimate receivers keeping an eavesdropper ignorant of it.The transmitter shares an independent secret key of arbitrary rate with each legitimate receiver.Depending on the channel qualities of the legitimate receivers and the eavesdropper, these secret keys must be used as one-time pads to encrypt the message, as fictitious messages for randomization in wiretap coding, or a combination of both to achieve the secrecy capacity.In this paper, communication takes place over independent parallel subchannels and the secrecy capacity is established for these cases, in which all parallel channels follow the same order of degradation.