IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY, VOL. 13, NO. 6, JUNE 2018 1351
Polar Coding for the Wiretap
Channel With Shared Key
Haowei Wang , Xiaofeng Tao, Senior Member, IEEE,NaLi,Member, IEEE,andZhuHan,Fellow, IEEE
Abstract—The wiretap channel with shared key (WTC-K)
refers to the classic wiretap channel with a secret key of
arbitrary rate shared between the transmitter and the legitimate
receiver. While the secrecy capacity of WTC-K has been obtained
using random coding arguments, in this paper, we propose a low-
complexity polar coding scheme achieving the secrecy capacity
under the strong secrecy condition. Our construction consists
of choosing the sets of information indices and using the shared
key properly. Specifically, we utilize the structure of superposition
coding in the proposed scheme.
Index Terms— Polar codes, secret key, superposition coding,
wiretap channel.
I. INTRODUCTION
I
NFORMATION-THEORETIC security was introduced by
Shannon in [1]. In this pioneering work, Shannon assumed
that the legitimate users share a secret key and proved that
the message can be perfectly secure from the eavesdropper
by using the shared key as one-time pad. Later on, Wyner
studied information-theoretic security from a new perspective
in his seminal work [2], where he introduced the wiretap
channel model and showed that secure communication can
be realized by exploiting the property of the noisy channel
without using any secret key. Wyner’s results were extended
by Csiszár and Körner [3] to more general broadcast channels.
Since then, information-theoretic security has attracted wide
research interests. Interested readers are referred to [4]–[9]
and references therein for further details.
Secure communication cannot always be guaranteed solely
by exploiting the property of noisy channels. There have
Manuscript received March 25, 2017; revised August 31, 2017 and
October 24, 2017; accepted October 31, 2017. Date of publication
November 16, 2017; date of current version February 7, 2018. This work
was supported in part by the National Natural Science Foundation of
China under Grant 61325006 and Grant 61501057, in part by the Shen-
zhen Science and Technology Innovation Commission Project under Grant
JCYJ20170307172830043, in part by the Fundamental Research Funds for the
Central Universities under Grant 2016RCGD07, in part by the 111 Project of
China under Grant B16006, and in part by the U.S. NSF under Grant CNS-
1717454, Grant CNS-1731424, Grant CNS-1702850, Grant CNS-1646607,
Grant ECCS-1547201, Grant CMMI-1434789, Grant CNS-1443917, and
Grant ECCS-1405121. The associate editor coordinating the review of this
manuscript and approving it for publication was Dr. Tobias Oechtering.
(Corresponding author: Xiaofeng Tao.)
H. Wang, X. Tao, and N. Li are with the National Engineering Laboratory
for Mobile Network Technologies, Beijing University of Posts and Telecom-
munications, Beijing 100876, China (e-mail: wanghaowei@bupt.edu.cn;
taoxf@bupt.edu.cn; Lina_Lena@bupt.edu.cn).
Z. Han is with the University of Houston, Houston, TX 77004 USA, and
also with the Department of Computer Science and Engineering, Kyung Hee
University, Gwangneung 12001, South Korea (e-mail: zhan2@uh.edu).
Digital Object Identifier 10.1109/TIFS.2017.2774499
been efforts to study the wiretap channel with a secret
key available at legitimate users (WTC-K). Yamamoto [10]
first studied the rate-distortion problem of this model. Later,
Merhav [11] further explored this model by taking the side
information into consideration. Kang and Liu [12] general-
ized Yamamoto’s results and obtained the secrecy capacity
of WTC-K. Wiretap channels with secure feedback are also
investigated in [13]–[15]. The secure feedback can be lever-
aged to create a secret key known only to the legitimate
users, and thus enhancing the secure communication. Recently,
Schaefer studied the broadcast channel with multiple shared
keys in [16]. Note that all of these works are based on random
coding arguments.
Developing practical coding schemes is crucial for the
implementation of information-theoretic security. In this paper,
we aim to develop a polar coding scheme for WTC-K. Polar
codes, introduced by Arıkan [17], are the first class of provable
capacity-achieving codes for symmetric binary-input mem-
oryless channels with low coding complexity. Recent years
have seen great advances in exploring polar codes in various
scenarios, such as source coding [18], [19], and channel
coding [20], [21].
Polar coding schemes, presented in [22]–[25], have achieved
the secrecy capacity of degraded wiretap channels with weak
secrecy. In [26], a subtle scheme using chaining construction
was proposed to achieve strong secrecy for degraded wiretap
channels. Later on, a polar coding technique for asymmet-
ric models was introduced in [27]. Using this technique,
along with chaining construction [28], schemes presented
in [29], [30] were proved to achieve the secrecy capacity
of general wiretap channels. Several other schemes have
also been proposed for multi-user wiretap channels, including
broadcast and interference channels with confidential mes-
sages (BC-CM and IC-CM) [29], BC-CM defined by Csiszár
and Körner [30], [31], bidirectional broadcast channel (BBC)
with common and confidential messages [32], and multiple
access wiretap channel (MAC-WTC) [29], [33].
To the best of our knowledge, no polar coding scheme
has been designed for wiretap channels with secret keys
available at legitimate users. In this paper, we propose a
low-complexity polar coding scheme to achieve the secrecy
capacity of WTC-K under the strong secrecy condition. The
proposed scheme relies on previous polar coding techniques,
and consists of choosing the sets of information indices and
using the shared key properly. The presence of the shared
key in the model affects the coding construction such that
polar coding schemes for wiretap channels without shared
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