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arXiv:1501.06287 · IEEE Information Theory Workshop (ITW)

On the secrecy exponent of the wire-tap channel

Mani Bastani Parizi, Emre Telatar

Published Oct 1, 2015• 6 citations
arXivHTMLPDF

Abstract

We derive an exponentially decaying upper-bound on the unnormalized amount of information leaked to the wire-tapper in Wyner's wire-tap channel setting. We characterize the exponent of the bound as a function of the randomness used by the encoder. This exponent matches that of the recent work of Hayashi [12] which is, to the best of our knowledge, the best exponent that exists in the literature. Our proof (like those of [17], [18]) is exclusively based on an i.i.d. random coding construction while that of [12], in addition, requires the use of random universal hash functions.

DOI
10.1109/itwf.2015.7360781

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We derive an exponentially decaying upper-bound on the unnormalized amount of information leaked to the wire-tapper in Wyner's wire-tap channel setting. We characterize the exponent of the bound as a function of the randomness used by the encoder. This exponent matches that of the recent work of Hayashi [12] which is, to the best of our knowledge, the best exponent that exists in the literature. Our proof (like those of [17], [18]) is exclusively based on an i.i.d. random coding construction while that of [12], in addition, requires the use of random universal hash functions.
Abstract — We derive a new upper bound for Eve’s information in secret key generation from a common random number without communication. This bound improves on Bennett[7]’s bound based on the Rényi entropy of order 2 because the bound obtained here uses the Rényi entropy of order 1+s for s ∈ [0, 1]. This bound is applied to a wire-tap channel. Then, we derive an exponential upper bound for Eve’s information. Our exponent is compared with Hayashi[8]’s exponent. For the additive case, the bound obtained here is better. The result is applied to secret key agreement by public discussion. Index Terms — exponential rate, non-asymptotic setting, secret key agreement, universal hash function, wire-tap channel I.
We derive a new upper bound for Eve's information in secret key generation from a common random number without communication. This bound improves on Bennett 's bound based on the Rényi entropy of order 2 because the bound obtained here uses the Rényi entropy of order 1+sfors∈ [0,1]. This bound is applied to a wire-tap channel. Then, we derive an exponential upper bound for Eve's information. Our exponent is compared with Hayashi 's exponent. For the additive case, the bound obtained here is better. The result is applied to secret key agreement by public discussion.
The compound wire-tap channel is studied, which is based on Wynerpsilas wire-tap model with both the channel from the source to the destination and the channel from the source to the wire-tapper taking a number of states. No matter which states occur for the two channels, the source wishes to guarantee that the destination decodes its message successfully and that the wire-tapper does not obtain the source message. The semideterministic compound wire-tap channel is first studied, in which the channel from the source to the destination is deterministic and has only one state. The secrecy capacity is obtained. An example parallel Gaussian compound wire-tap channel is then studied, in which both channels have two states. Three schemes are studied, and it is shown that introducing randomness either into the source message or into the encoder achieves the maximal secrecy degree of freedom. Both channels studied in this paper demonstrate that creating an auxiliary input, and hence adding a prefix channel from this auxiliary input to the actual channel input, improves the secrecy rate.
With the advent of quantum key distribution (QKD) systems, perfect (i.e. information-theoretic) security can now be achieved for distribution of a cryptographic key. QKD systems and similar protocols use classical error-correcting codes for both error correction (for the honest parties to correct errors) and privacy amplification (to make an eavesdropper fully ignorant). From a coding perspective, a good model for such a setting is the wire tap channel system introduced by Wyner in 1975. In this paper, we study fundamental limits and coding methods for wire tap channel systems. We provide a novel proof for the secrecy capacity theorem for wire tap channels and show how capacity achieving codes can be used to achieve the secrecy capacity for any wiretap channel. We also consider binary erasure channel and binary symmetric channel special cases for the wiretap channel system and propose specific practical codes. In some cases our designs achieve the secrecy capacity and in others the codes provide complete security at rates below secrecy capacity in some specific cases.
The compound wire-tap channel is studied, which is based on the classical wire-tap channel with the channel from the source to the destination and the channel from the source to the wire-tapper taking a number of states, respectively. This channel can also be viewed as the wire-tap channel with mul-tiple destinations and multiple wire-tappers, i.e., multicast with multiple wire-tappers. The source wishes to transmit information to all destinations and wants to keep the information secret from all wire-tappers. For the discrete memoryless compound wire-tap channel, lower and upper bounds on the secrecy capacity are derived and are shown to match for the degraded channel. The parallel Gaussian compound wire-tap channel is further studied, for which the secrecy capacity and the characterization of an optimal power allocation are obtained. The secrecy degree of freedom (s.d.o.f.) is also derived, which connects the secure communication rate in the high SNR regime to secure networking coding for deterministic networks. Finally, the multi-antenna (i.e., MIMO) compound wire-tap channel is studied. The secrecy capacity is established for the degraded MIMO compound wire-tap channel and an achievable s.d.o.f. is given for the general MIMO compound wire-tap channel.