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arXiv:0904.0308 · IEEE Transactions on Information Theory

Exponential Decreasing Rate of Leaked Information in Universal Random Privacy Amplification

Masahito Hayashi

Published May 25, 2011• 269 citations
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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 '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.

DOI
10.1109/tit.2011.2110950

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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.
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 adopt L1 distance as Eve's distinguishability in secret key generation from a common random number without communication. Under this secrecy criterion, using the Renyi entropy of order 1 + s for s in [0, 1], we derive a new upper bound of Eve's distinguishability under the application of the universal2 hash function. It is also shown that this bound gives the tight exponential decreasing rate. The result is applied to wire-tap channel model and secret key distillation (agreement) by public discussion.
Motivated by the desirability of universal composability, we analyze in terms of L1 distinguishability the task of secret key generation from a joint random variable. Under this secrecy criterion, using the Rényi entropy of order 1+s for s ∈ [0,1], we derive a new upper bound of Eve's distinguishability under the application of the universal2 hash functions. It is also shown that this bound gives the tight exponential rate of decrease in the case of independent and identical distributions. The result is applied to the wiretap channel model and to secret key generation (distillation) by public discussion.
In this paper without tailoring to any presumed condition on communication channel or any restriction on adversary's resources we design a secrecy scheme with information leakage that decays at an exponential rate. The only requirement for such an exponentially secure system is existence of a common key source between legitimate users that is partially known by Eve. A key extractor based on a sampler and a Rényi extractor derives secret keys with the required entropy from this source. A general cipher uses this key to ensure the required equivocation for the plaintext and to establish the first layer of secrecy. Using privacy amplification on top of this cipher based on inverse universal2hashing constitutes the second layer of secrecy for highly confidential message transmission with information leakage that is exponentially decreasing. We provide exponent analysis and optimization to minimize information leakage in terms of Eve's distinguishability based on L1norm distance from uniformity. The required key rate is characterized for different source entropies in order to guarantee the secrecy that is demanded in terms of secrecy exponent for the second layer and Eve's error rate for the first layer.
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.
In this paper without tailoring to any physical channel condition or any restriction on adversary's computational resources we design a secrecy scheme with information leakage that decays at an exponential rate. The only advantage for legitimate users over eavesdropper is existence of a key source that is partially known by Eve. A key extractor based on a sampler and a Renyi extractor derives secret keys with the required entropy from this source. Encryption using this key with a general cipher maps Eve's uncertainty about the key to uncertainty about the plaintext that can also be interpreted as degradation in her channel. Using privacy amplification on top of this cipher based on inverse universal2 hashing results in information leakage with decreasing exponent that can be optimized to guarantee a higher level of secrecy.