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Security and Communication Networks

Analysis of the information theoretically secret key agreement by public discussion

Qiuhua Wang, Xiaojun Wang, Qiuyun Lv, Xueyi Ye, Yi Luo, Lin You

Published Jan 23, 2015• 12 citations

Abstract

Abstract Generally, the information theoretically secret key agreement by public discussion consists of three phases, namely, advantage distillation, information reconciliation, and privacy amplification. In the existing literatures, these three phases are studied separately, and no comprehensive consideration has been given. In this paper, we analyze the restrictive relationship among these three phases from an overall point of view. For the satellite scenario, we analyze in detail the Winnow protocol for the first time. We also present the mutual restrictive relationship between the parameters of the advantage distillation phase and the information reconciliation phase. We further address how to set the parameters of the advantage distillation and the information reconciliation to maximize the total secret key agreement efficiency. Moreover, we determine the required length range of the initial random key string to guarantee the validation of the secret key agreement. Copyright © 2015 John Wiley & Sons, Ltd.

DOI: 10.1002/sec.1192

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Abstract Generally, the information theoretically secret key agreement by public discussion consists of three phases, namely, advantage distillation, information reconciliation, and privacy amplification. In the existing literatures, these three phases are studied separately, and no comprehensive consideration has been given. In this paper, we analyze the restrictive relationship among these three phases from an overall point of view. For the satellite scenario, we analyze in detail the Winnow protocol for the first time. We also present the mutual restrictive relationship between the parameters of the advantage distillation phase and the information reconciliation phase. We further address how to set the parameters of the advantage distillation and the information reconciliation to maximize the total secret key agreement efficiency. Moreover, we determine the required length range of the initial random key string to guarantee the validation of the secret key agreement. Copyright © 2015 John Wiley & Sons, Ltd.

Aultmore and Balvenie want to agree on a common secret key. They have at their disposal a noisy channel and an error free public channel. The situation is complicated by an adversary, Ellen, who eavesdrops the noisy channel and the public channel. Fortunately for Aultmore and Balvenie, the eavesdropping channel is noisy as well. This will enable them to generate a secret key. In the literature, this is done in three phases; the advantage distillation phase, the information reconciliation phase, and the privacy amplification phase. We show that the first two phases are based on the same concept and that presently known techniques used in both phases are basically special cases of a more general technique.

Privacy Amplification, introduced by Bennett et. al in [1] and generalized in [2], is the technique of shrinking a partially secret string to a highly secret one by public discussion. More precisely, in the privacy amplification phase, two legitimate parties, Alice and Bob, who share a partially secret string S about an opponent Eve has partial information to extract, and a shorter but highly-secret string S' by communication through an public channel such that Eve’s information about S' is negligible, i.e., ε= ≥ −S' U uH ( / ) log S2 | '| holds with very high probability for some small ε > 0, where the random variable U summarizes Eve’s complete knowledge about S', and u is the particular value known to Eve [2]. Privacy amplification is an important step in both quantum cryptography and key theoretically secret information agreement [3].

Aultmore and Balvenie want to agree on a common secret key. They have to their disposal a noisy channel and an error free public channel. The situation is complicated by an adversary, Ellen, who eavesdrops the noisy channel and the public channel. Fortunately for Aultmore and Balvenie, the eaves- dropping channel is noisy as well. This will enable them to generate a secret key. In the literature, this is done in three phases; the advantage distillation phase, the information reconciliation phase, and the privacy amplification phase. We will show that the first two phases are based on the same concept and that presently known tech- niques used in both phases are basically special cases of a more general technique.

The classical secret-key agreement (SKA) scheme includes three phases: (a) advantage distillation (AD), (b) reconciliation, and (c) privacy amplification. Define the transmission rate as the ratio between the number of raw key bits obtained by the AD phase and the number of transmitted bits in the AD. The unidirectional SKA, whose transmission rate is 0 . 5, can be realized by using the original two-way wiretap channel as the AD phase. In this paper, we establish an efficient bidirectional SKA whose transmission rate is nearly 1 by modifying the two-way wiretap channel and using the modified two-way wiretap channel as the AD phase. The bidirectional SKA can be extended to multiple rounds of SKA with the same performance and transmission rate. For multiple rounds of bidirectional SKA, we have provided the bit error rate performance of the main channel and eavesdropper’s channel and the secret-key capacity. It is shown that the bit error rate (BER) of the main channel was lower than the eavesdropper’s channel and we prove that the transmission rate was nearly 1 when the number of rounds was large. Moreover, the secret-key capacity C s was from 0 . 04 to 0 . 1 as the error probability of channel was from 0 . 01 to 0 . 15 in binary symmetric channel (BSC). The secret-key capacity was close to 0 . 3 as the signal-to-noise ratio increased in the additive white Gaussian noise (AWGN) channel.

This paper, provides a general treatment of privacy amplification by public discussion, a concept introduced by Bennett, Brassard, and Robert for a special scenario. Privacy amplification is a process that allows two parties to distil a secret key from a common random variable about which an eavesdropper has partial information. The two parties generally know nothing about the eavesdropper's information except that it satisfies a certain constraint. The results have applications to unconditionally secure secret-key agreement protocols and quantum cryptography, and they yield results on wiretap and broadcast channels for a considerably strengthened definition of secrecy capacity.