Wireless Communications and Mobile Computing

Volume 2018, Article ID 7963451, 10 pages

https://doi.org/10.1155/2018/7963451

## PHY-Aided Secure Communication via Weighted Fractional Fourier Transform

^{1}Graduate School, Air Force Engineering University, Xi’an 710077, China^{2}College of Information and Navigation, Air Force Engineering University, Xi’an 710077, China

Correspondence should be addressed to Lei Ni; moc.361@dgkielin

Received 23 December 2017; Accepted 10 April 2018; Published 15 May 2018

Academic Editor: Gonzalo Vazquez-Vilar

Copyright © 2018 Lei Ni et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A weighted fractional Fourier transform (WFRFT) based on channel state information (CSI), aiming to safeguard the physical (PHY) layer security of wireless communication system, is proposed. With the proposed scheme, WFRFT is first applied to satellite communications such that the transmitted signal is distorted and can only be neutralized by inverse-WFRFT with the same parameter. Moreover, by exploiting the physical properties of wireless channels, the CSI matrix is transformed as a secret key. In addition, by adding phase rotation (PR) factors to each branch of the WFRFT system, a new unitary matrix with the encryption properties is constructed, and hence, the satellite communications secrecy is reliably guaranteed due to the variation in signal characteristics. Finally, the efficacy of the security enhancement is evaluated in terms of the average bit error rate (BER) and the secrecy capacity. Simulation results show that the proposed encryption method can make the detection and demodulation more difficult for the eavesdropper.

#### 1. Introduction

The satellite communications network plays an indispensable role in both civil and military applications. However, due to the openness of wireless communications, malicious receivers within the cover range can illegitimately access the spectrum bands and analyze the transmission without being detected, which makes the confidential message transferred through a satellite communications network vulnerable to eavesdropping attacks [1]. To handle this issue, conventional approaches for data privacy implemented at upper layer protocol stack [2], which mainly focus on the computational complexity of cryptographic algorithms, have been widely used for guaranteeing communication security [3]. However, most of the existing cryptography-based strategies are computationally secure and thus could be compromised when the eavesdropper has powerful computing capability; for example, quantum computing is available [4, 5]. In addition, it is difficult for legitimate user to design an applicable secret key distribution and management protocol to satisfy the great diversity of wireless scenarios.

Without using any encryption, an emerging technique termed as physical (PHY) layer security has attracted considerable interest in recent years [6, 7]. By exploiting the coding techniques as well as physical properties of wireless channels, PHY layer security has a great potential to achieve “perfect secrecy” and is identified as a significant complement to traditional cryptographic techniques [8]. As introduced in [9, 10], the weighted fractional Fourier transform (WFRFT) is performed as a new paradigm for PHY layer security enhancement. WFRFT, known as a novel time-frequency mathematical tool, is proposed by Shih in 1995 for the first time [11]. The WFRFT-based communication system developed in [12, 13], which can be understood as a fusion of single-carrier (SC) and multicarrier (MC) communication system, not only possesses hybrid carrier characteristics with an optimal fractional order over selective fading channels, but also has a good performance of anti-interception [14]. Moreover, WFRFT signal is encrypted with rotation characteristics, which is widely applied to secret communication system in recent years [15, 16].

According to Wyner’s wiretap channel model, there exist three participants: the sender Alice, the purpose receiver Bob, and the eavesdropper Eve, the same as the traditional cryptology. Under this wiretapping model, we assume that the baseband signal modulation scheme is preknown to Eve, and once Eve has unlimited computational capabilities, a potential issue might arise that Eve is likely to get the correct WFRFT parameter through scanning the parameter of the whole period [17]. Therefore, it is necessary to take more effective measures to protect the WFRFT signal from being cracked. Motivated by the aforementioned aspects, this paper proposes an encryption scheme based on channel state information (CSI) and WFRFT. In general, the WFRFT-based communication system is reconstructed with a new encrypted unitary matrix, which is determined and dynamically encrypted by the CSI, whereby the proposed scheme renders itself with the following advantages: on the one hand, the unique communication nodes of legitimate channel between Bob and Alice can reduce the possibility of secret leakage in the upper layer. On the other hand, the channel needed for transferring the secret key is saved; thus the channel’s efficiency is able to be improved.

Numerical simulation results show that the proposed WFRFT-based PHY layer security scheme is able to overcome the shortcomings of the traditional WFRFT system, and the BER of the eavesdropper is kept above 50%. The remainder of this paper is organized as follows. Section 2 reviews the mathematical concepts of WFRFT. Physical implementation process of order WFRFT as well as the proposed system model is described in Section 3. Simulation results are revealed and discussed in Section 4. Finally, conclusions are addressed in Section 5.

*Notations*. Vectors and matrices are denoted by lower and upper case boldface letters, respectively. represents an identity matrix with appropriate size. and stand for the inverse and Hermitian transpose operations, respectively. Additionally, is the complementary distribution function of the standard Gaussian and .

#### 2. Preliminary

##### 2.1. Weighted Fractional Fourier Transform

Known as a generalized Fourier transform, the discrete Fourier transform (DFT) of order WFRFT is expressed as [18]where are the 1~3 times DFT of complex vector , respectively. Furthermore, the DFT can be defined as a form of a matrixwhere denotes the DFT matrix, and the elements can be written as . Here, is equal to ; besides, the weighting coefficients in (1) are generated aswhere , and parameter is with a cycle of 4, commonly selected in the real interval . Note that there is only one WFRFT parameter in (1), which can be called single parameter WFRFT [19].

In addition, (1) can be represented as the following form:where denotes times of FFT operation. According to the symmetry of discrete Fourier transform, we can obtain and , where represents a permutation matrix and can be denoted by

It can be readily verified that in WFRFT-based communication system is a unitary matrix; besides, the DFT matrix utilized in the orthogonal frequency division multiplexing (OFDM) system is also a unitary matrix. Motivated by this property, in this paper, we consider constructing a new encrypted unitary matrix in WFRFT system to improve the secrecy performance.

##### 2.2. Encrypted Unitary Matrix

According to the operational properties of unitary matrix, a new unitary matrix can be derived by multiplying a unitary matrix and a permutation matrix. The construction method of our scheme is described as follows.

First of all, we denote as the unitary matrix that needs to be extended.

*Definition 1 (permutation matrix). *The necessary and sufficient condition for a permutation matrix is that each row or column of has only one element equal to one, and the other elements are all zeros.

Theorem 2. *Let be the upper (lower) triangular unitary matrix, then must be a diagonal matrix satisfying .*

*Definition 3 (PR matrix). *The necessary and sufficient condition for a PR matrix with size is that the element equals , where can be chosen in , and the other elements of are all zeros. It can be observed that the total number of rotation matrices is [20].

Secondly, creating a new unitary matrix , the original unitary matrix can be either a square or a nonsquare matrix theoretically. Since the matrix in WFRFT system is a unitary matrix, we consider reconstructing to form a new unitary matrix to enhance the secrecy performance.

In (4), can be obtained by multiplying a block matrix with a coefficients matrix ; since both the block matrix and are unitary matrices, we can derive that is also a unitary matrix. To make it easier for practical implementation, a PR matrix with size and a permutation matrix with size are designed, respectively.

Finally, the new unitary matrix can be calculated asmoreover, the inverse matrix is expressed bywhere the PR matrix is composed of four diagonal elements .

Considering that Bob needs to calculate the transpose of , without loss of generality, we adopt the unit matrix as this permutation matrix for simplicity. Therefore, the generation of the PR matrix becomes the key to the construction of this new encrypted unitary matrix.

#### 3. Encrypted Unitary Matrix Based on CSI

##### 3.1. Channel State Information Encryption

Due to the inherent nature of wireless communications, the uniqueness and short-time reciprocity of the channel make it possible for legitimate user to distribute secret key reliably. In addition, the secure transmission between legitimate users can be guaranteed by exploiting the spatial difference between Bob and Eve. Therefore, we can employ these properties to protect the confidential messages that need to be transmitted, and the specific conversion methods are described in the following subsections.

Firstly, Bob sends a training array to Alice, and Alice takes advantage of it to estimate the CSI of legitimate channel. Meanwhile, Bob estimates the CSI through the pilot sequence sent by Alice. It is assumed that the CSI is perfect at the legitimate transmitter; thus the CSI matrices estimated by Alice and Bob are completely identical. Under this assumption, it is necessary to convert the estimated legitimate CSI into a secret key by certain means. In this paper, the single Hash function [21], as a frequently used method in the encryption algorithms, is introduced. The single Hash function can be generalized aswhere is the fixed-length Hash value, is a one-way Hash function, and denotes the input sequence of arbitrary length.

The mapping of the compression function can help us to convert any variable length input string into a fixed-length output string. Moreover, the Hash function has a good performance in collision-resistance and the uniformity of mapping distribution. According to the previous analyses, we can utilize the Hash function to map the estimated CSI matrix into a bit string with fixed length, and due to the one-way property of the Hash function, even if the key sequence is known by Eve, the correct CSI matrix cannot be obtained.

In this paper, two main Hash algorithms are proposed: MD5 and SHA-1 algorithm [22], then we can convert the CSI matrix into a binary sequence of 128 bits or 160 bits. Moreover, the four diagonal elements of the PR matrix can be derived by transforming the binary key sequence into the decimal numbers; these four decimal numbers , , , are of paramount importance for the PR matrix and the corresponding phase factors can be calculated by the following equation:

However, there are many ways to convert binary bits into four decimal numbers, considering that the receiver needs to obtain encrypted unitary matrix with the same size; the 128 bits are divided into four groups in our work, then the binary bits of each group are converted to four decimal numbers. Figure 1 demonstrates the generation method for the secret key.