Abstract

-The paper proposes an application of a quadrature amplitude modulation (QAM) scheme using orthogonal frequency division multiplexing (OFDM) transmission for data protection. Our approach bases on the symmetric encryption and uses a natural or induced noise in a wireless channel as a random part of a secret key. The main idea for this is the sensitivity of the signal decoding model to variations of measured values. This kind of protection does not require additional computational costs and does not affect the bit rate of the channel.

Keywords

• QAM OFDM wireless channel
• physical layer security
• data protection
• Ill-conditioned problem

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Copyright © 2020 for this paper by its authors.

Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0)   

I. INTRODUCTION

Contemporary wireless communications are vulnerable to various attacks because of the open nature of radio propagation [ 1, 2 ]. One of the main ones is an eavesdropping the content of the transmission [3, 4]. And in consequence, the conventional wireless security mechanisms are cryptographic techniques [5, 6]. Currently, there are a lot of various cryptographic methods to secure the wireless transmissions [7]. Nevertheless, we are trying to design an easy-to-use additional cryptographic tool in this paper. The proposed tool operates at the physical (PHY) layer security of wireless networks [ 8-11 ]. PHY layer is the bottom layer of the network and deals with carrier frequency generation, modulation, signal detection etc.

Everyone knows how important it is to generate a truly random encryption key [ 12-14 ]. At the same time, everyone knows that natural noises in the wireless channel are random and unpredictable. If we add noises to the encryption protocol, we will improve communication security. To do this, we are taking into account some mathematical features of the digital channel formed using OFDM technology.

OFDM is a promising technique for wideband digital communication. The method it is based on is encoding digital data on multiple carrier frequencies. The basic principle of multicarrier transmission is to translate high rate serial data stream into several slower parallel streams and then modulate it on subcarriers. By reducing the symbol transmission rate in the subcarrier the effect of the delay spread of the multipath channel is also greatly reduced. In addition to that, inter-symbol interference can be also decreased [ 15-17 ].

A simple and reliable scheme of modulating subcarriers is provided by QAM. In this case, the amplitudes of two waves of the same frequency, 90° out-of-phase with each other are modulated to represent the multi-leveled digital signal. The number of modulation levels depends on the signal-to-noise ratio (SNR). In practice, it is chosen based on worst-case forecast for each subcarrier.

If we use QAM scheme to modulate subcarriers, the modulation scheme is called QAM OFDM [ 17, 18 ]. = {{ | = 〈 0, 1, … , −1〉} ⊂ , (1) where z k  x k  I  y k

,  x k , y k   q hm , I 2   1 , ℎ = {(ℎ ∙ 1, ℎ ∙ 2)| 1, 2 ∈ 0, … ,2 ∙ − 1, ∈ } and the Z T symbol denotes the transpose operation on Z .

Then, we map the N-tuple Z N  Q mN to the signal

N 1 I 2 kt s ( t )   z k  e T on the interval  0 , T  , where T is the k  0 useful symbol duration. We send s ( t ) to the channel and we need to extract Z N from s ( t ) in the receiver side.

The values T, N, m determine the bit rate of the channel 2 N  lo g  m   1 

T

bps. We note that the subcarriers as C B 

I 2 kt e T

 L2  0 , T  are orthogonal to each other on the interval  0 , T  . This property allows using the discrete Fourier transform (DFT) to demodulate the signal on the receiver N 1 side. Indeed, let us define the set  tn  

T N

  n   n  0

and the  I 2 kn  square matrix F N N   Nkn    e N  , where k , n  0 .. N  1 .

  If we denote S NT  s 0 , s1 , .., s N 1  s  t0  , s  t1  , .., s  t N 1  and Z NT  z 0 , z1 , .., z N 1 then F N N Z N  S N 

1 Z N 

F N*N S N 

N   I 2 kn  where F N*N   N nk    e N  is a Hermitian transpose of   F N N .

It is easy to show that

F M N Z N  S MM , Z N 

F N*M S MM  1 M where

 I 2 kn  F M N   Mkn    e M  ,  

  I 2 kn  F N*M   M nk    e M  ,   S MM T  s 0 , s1 , .., s M 1  s  t0  , s  t1  , .., s  t M 1  , t n   n , n  0 .. M  1 , k  0 .. N  1 , M  N .

T M

Now we note that we actually send s ( t ) to the noisy channel and we need to extract ∈ from s ( t )  s ( t )   ( t ) in the receiver side. We assume that  ( t ) L2 0 ,T    T  2 ( t ) d t   . 0 Similar to the previous one, we can define the set

N 1   tn  

T M

Problems with a low condition numbers are called wellconditioned, while problems with a high condition numbers are called ill-conditioned. An ill-conditioned problem is one where, for a small change in the RHS of an equation there is a large change in the solution.

It is easy to show that   F N N   1 . If we choose h (see (1)) to correspond with SNR, we can use nearest neighbor search (NNS) to define Z N from (5) as Z N  aZr gNmQ mNin Z N  Z N N

.

C M  bps.

In the following sections, the QAM OFDM scheme in the wireless noisy channel will be used to protect transmitted data from unauthorized access.

T M N 

Technically, we have constructed a new channel with the useful symbol duration T M N and the bit rate 2 N  log  m   1  2 M  log  m   1 

2 N  log  m   1  

 T

T  C B and try to find Z N by solution of the matrix equation  F NMN Z N  S NM  

This problem is complicated by the fact that, if ∈ it becomes ill-conditioned. Calculations have shown that the condition number   F NMN  depends exponentially on M . For example, if N  1 6 then   F NNN s   e 5.4s  7.8 , and if N  3 2 then   F NNN s   e 11.1s 16.6 , were s  1 ..2  N . Because of this, small values of  ( t ) L2 0 ,TM N    leads to large differences between the solutions of (7) and (8), and so we cannot use NNS to define Z N by Z N . not orthogonal to each other on the interval  0 , T M N  , so the QAM OFDM scheme becomes the QAM FDM scheme. III. WELL-CONDITIONED PROBLEM FOR A QAM FDM SCHEME

AND A SECRET KEY bit C M 

Let us set N  2  N 0 and M  2  N . Note that now the rate of the QAM FDM channel is 2 N  log  m   1 

4 N  log  m   1  

 2  C B bps.

TMN T

In this section we are going to reduce the ill-conditioned problem (7), (8) with matrix F NMN to the well-conditioned one in the subspace 0 ⊂ .

First of all we define a submatrix F of the matrix 1 N 0 N 0 F NMN , which is obtained by deleting all odd rows and all odd  columns. In this case F N 0 N 0   4 2Nk0 2 n     Nkn0    e N 0  , 1

M where  S N 0 N 0

M   S N N

.

Z

N0 0 , and where therefore, Z N , as Z channel is equal to the bit rate of QAM OFDM channel, i.e. C M0  2 N 0  lo g  m   1 

4 N 0  lo g  m   1  

 TM N

T which is obtained by deleting all odd rows and all even , where because e  F N20 N 0   1 0 be a unique solution of the matrix equation

F

N 0 N 0 Z N0 0  S 1

N0 0  therefore,

Now we are ready to define Z

N  Z

N0   0N  z 0 , 0 , z1 ,  1 , .., z N 0 1 ,  N 0 1 a

combination and rewrite (8) in the form

F NMN Z N   S NM  F NMN Z N0  F NMN  0N   S NM  S NM   where  S NM   0M ,  1M , ..,  NM1 .

And at last we can set a secret key Г 0 = 〉 ∈ = 2 0 for the sender and regularization.

N0  z 0 , 0 , z1 , 0 , .., z N 0 1 , 0 .

F NMN Z N0   S NM  S N0  F NMN 0 Z N0 0   S NM  S N0 

FKMN 0 Z N0 0   S KM  S K0       

This results can be generalized to the cases where N  m  N 0 and M  m  N . If N 0  2 k then we can use radix-2 fast Fourier transform (FFT) to demodulate the signal on the receiver side.

IV. CONCLUSION

In this paper we have configured a QAM OFDM modulation scheme for data protection in wireless noisy channel. Our approach is based on symmetric cryptography and uses composite secret key. The regular part of the key is used to modulate odd subcarriers. Natural or induced noise in the wireless channel is used as a random part of the key. This scheme is very sensitive to signal variations without knowing the regular part of the secret key. We called this scheme QAM FDM NSK.

Note that after syncing the QAM FDM NSK session, similar secret keys can be generated along with encoding and decoding in the sender and receiver sides.

The QAM FDM NSK scheme does not affect the bit rate of the channel and does not require extra computations to decode the transmitted data. It can be customized for using radix-2 FFT algorithm in its calculations.

Following by [ 20 ], we could adapt the QAM FDM NSK scheme to discrete orthogonal transforms associated with some recursive self-similar processes, regardless of the transmission medium. For example, we could customize this tool for visible light communications (VLC) [ 21-25 ] or underwater wireless communications (UWC) [ 26-28 ]. pp. 1-10, 2009. [2] H.K. Kalita and A. Kar, “Wireless sensor network security analysis,”

International Journal of Next-Generation Networks (IJNGN), vol. 1, [3] V. Aggarwal, L. Sankar, A. Calderbank and H. Poor, “Secrecy capacity of a class of orthogonal relay eavesdropper channels,” Information Theory and Applications Workshop, pp. 295-300, 2009. [4] J. Vilela, J.B.M. Bloch and S. McLaughlin, “Wireless secrecy regions with friendly jamming,” IEEE Transactions on Information Forensics and Security, vol. 1.6, no. 2, pp. 256-266, 2011.

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