SECURE DIGITAL COMMUNICATIONS BY MEANS OF STOCHASTIC PROCESS SHIFT KEYING: PRINCIPLES AND PROPERTIES Arnt-Børre Salberg and Alfred Hanssen University of Tromsø, Physics Department Applied Physics Group, N-9037 Tromsø, Norway Email:
[email protected] and
[email protected] ABSTRACT A digital communications keying concept that applies stochastic processes rather than deterministic waveforms is introduced and demonstrated. The technique implies a higher degree of security that conventional digital communications systems, reducing the risk of eavesdropping. 1. INTRODUCTION Conventional digital communications is the process of transmitting base band digital information over a channel by altering certain aspects of a deterministic carrier signal [4]. Well-known examples are amplitude shift keying, frequency shift keying, and phase shift keying. The receiver estimates the parameters of the deterministic information carrying signal, and invokes some decision rule to classify the received waveform as belonging to one of the possible discrete parameter signals. Conventional digital communication methods provides no protection against eavesdroppers. Spread spectrum techniques (e.g. [4]) are often applied to increase the privacy during information transmission by means of a quasi stochastic spreading sequence in the transmitter. To decode the signal, the receiver must have complete knowledge about the spreading sequence, and it must be strictly synchronous with the transmitter. Recently, also chaotic digital encoding [11, 7, 10] has been proposed as a way of achieving secure communications. Chaotic communication systems are related to spread spectrum in the sense that a chaotic carrier signal is applied in the transmitter. The decoding takes place by exploiting a self synchronization property that certain chaotic systems possess [7]. If synchronization is achieved, one may subtract the chaotic carrier to reconstruct the transmitted information signal. In this paper, we demonstrate a new digital communications shift keying concept that is based solely on the use of stochastic processes. By nature, this method has an inherent resistance against eavesdroppers, and the resulting technique is significant simpler that spread spectrum and chaotic encoding. This novel technique was recently introduced by on of us in [6].
2. STOCHASTIC PROCESS SHIFT KEYING We propose the following method for encoding digital information. Transmit a (noise like) stochastic process X0 (t), 0 t < T to represent bit zero, and another stochastic process X1 (t), 0 t < T to represent bit one. A proper name for this digital encoding technique is Stochastic Process Shift Keying (SPSK). Thus, rather than altering aspects of a deterministic carrier signal, we propose to transmit realizations of two different stochastic processes. This has the effect that two subsequent equal source bits have entirely different transmitted waveforms. It is obvious that this signaling method adds an extra (physical) layer of security in digital communication, thus reducing the risk of eavesdropping. Any relevant (deterministic) set of parameters describing some property of the transmitted stochastic process may be applied in SPSK communications. E.g., parameters connected to the correlation, or power spectral properties of the two different processes may be exploited for SPSK communication. For SPSK, the relevant digital information is generally contained in a set of attributes characterizing the stochastic processes applied. We will refer to the collection of attributes of a given stochastic process as the attribute vector. Furthermore, the K -dimensional attribute vector is a point in a K -dimensional attribute space. For M -ary SPSK communication, we must have M distinct points in a K -dimensional attribute space. At this abstract level of description, we observe a strong resemblance to the notion of signal space or signal constellation applied in the detection of deterministic signals in noise (e.g., [4]). Usually, channel coding, diversity, and/or automatic repeat-request is applied in conventional digital communication systems to achieve reliable transmission of digital information over noisy channels [4]. For SPSK, one may want to use error detecting/correction schemes even in the absence of additive noises, because of the finite probability of receiver error, due to the inherently random nature of the transmitted signals. There are several natural choices of prototype processes
Bit Sequence
Stochastic Process Modulator
Modulator
Channel
Demod.
Stochastic Process Demod.
Bit Sequence
Figure 1: Block diagram of SPSK communication system.
that could be used for SPSK communications. Clearly, the use of two different autoregressive/moving-average (ARMA) processes is an attractive choice, mainly due to the simplicity of ARMA-processes. Another viable choice is to use two different flicker-noise (1=f -noise) processes, with different spectral exponents 1 and 2 . Bilinear and nonlinear processes are other attractive options. In practice, one would realize the transmitter be means of a stochastic process modulator in addition to the conventional modulator. The stochastic process modulator consists of two different noise generators with known characteristics, and a fast low-noise switch to alternate between the generators according to the source bit stream. The block diagram shown in Fig. 1 illustrates the basic elements of a stochastic process shift keying communication system with a binary information sequence at its input at the transmitting end and at its output at the receiving end. A stochastic process modulator converts the binary information sequence into a sequence of stochastic processes. This process is then modulated with a carrier and transmitted over the channel. The modulator may be an amplitude modulator (AM), frequency modulator (FM) or a phase modulator (PM), depending of the application of our communication system. If we use broad band stoastic processes, the information containing part of the transmitted physical signal covers a broad frequency band, thus reducing the effect of a possible jammer. A very interesting and intriguing possibility is to let the two processes X0 (t) and X1 (t) have identical power spectral densities, but different bispectral densities (e.g., [9]). Thus, one may code the digital information by means of an attribute connected to the bispectral densities of the process. A bispectral attribute would be completely insensitive to additive colored noise with symmetrical amplitude probability density function. Thus, bispectrumbased SPSK provides an inherent protection against additive Gaussian noise. 3. DECODING Since X0 (t) and X1 (t) are stochastic processes, the decoding cannot be carried out by means of ordinary matched
filters. In general, a Neyman-Person hypothesis test (e.g, [14]) may be applied, since we assume that all relevant probability densities are known to the detector. If flickernoise is applied at the transmitter, one or more of the estimation methods reviewed in [12] may be used to derive a detector. Alternatively, the wavelet based hypothesis tests for 1=f -processes discussed in [15] may be applied to decode the received signals. For communication systems based on bilinear or nonlinear time series, estimation methods described in [13] may be applied by the decoder, and if the information sequence is encoded by means of higher-order statistics, detectors discussed in [3] may be suitable for decoding. 3.1. Detection of Gaussian Stochastic Signals in AWGN Since the information bearing signals in a SPSK communication system are stochastic, a problem rises in designing an optimum receiver. An important special case is that in which both signals and noise are independent Gaussian random vectors. Assume that measurements of the stochastic process are given in the data vector = [x1 ; x2 ; :::; xN ]T . Assume further that = + , where is a vector of samples from a stochastic process with zero mean value and covariance matrix s;i = E f T j!i g. Furthermore, is a vector of white noise samples independent of the stochastic signal, with zero mean value, and covariance matrix n2 . This means that the mean value of is equal to the mean value of , and the covariance matrix i is equal to 2 s;i + n . In order to communicate with a minimum error probability Pe , the receiver must choose class 0 when
x x s n R SS
I
R
I
s
n
X
S
R
P ( 0 jx) > P ( 1 jx);
(1)
otherwise choose class 1 . Using Bayes rule and assuming equal a priori probabilities, i.e P ( 0 ) = P ( 1 ), the decision rule can be written as
L(x) = xT (R0 1
R1 1)x + ?1 0
0
(2)
R R
where the threshold is = logf 0 j=j 1 jg. If the SPSK communication system uses autoregressive stochastic processes, there exists several distance measures that may be used as a detector. Let i ; i = 0; 1 denote the attribute vector consisting of the AR-parameters. Then the Itakura distance
a
aT R^ a0 d = log T0 a1 R^ a1 "
#
(3)
is a suitable distance measure because it approximates the log likelihood ratio for a large sample size [8, 5, 1]. In Eq. is the estimated autocovariance matrix of dimen(3), sion p p of the received data, where p is the order of the
R^
AR-process. The final decoding step is simply a straightforward threshold test: Choose bit 0 if d < 0, otherwise choose bit 1. 4. POWER SPECTRAL DENSITY OF SPSK PROCESSES A statistical model for an infinite duration SPSK signal can be written as
X (t) =
1
X
n=
[an X0 (t) + (1
1
an )X1 (t)]u(t
)
nT
(4) where u(t)is a unit amplitude rectangular pulse of duration T , an is a wide-sense stationary stochastic sequence with 0 and 1 as possible outcomes, mean value E [an ] = a , correlation sequence E [an an+k ] = Ra (k ), and is a uniformly distributed random variable independent of an . The mean value of the total SPSK process defined in Eq. (4) is trivially zero if X0 (t) and X1 (t) are assumed to have zero mean values. In this case, one can show that the correlation function is given by
RXX ( )
= T1
1
X
k=
1
Q(
kT )[Ra (k )RX0 X0 ( )
+(1 2a + Ra (k))RX1 X1 ( )]
(5)
where (
j j < T (6) 0; j j T As expected, the correlation properties of X (t) depend Q( ) =
T 2 (1
j =T j);
critically on the correlation function for the amplitude sequence Ra (k ) and on the correlation functions for the two signaling processes. If we now assume that the a priori probability of an = 1 is p1 (so that the a priori probability of an = 0 is 1 p1) we find that a = p1 and a2 = p1 (1 p1 ). The power spectral density can then be shown to be given by
SXX (! )
=
p21 SX0 X0 (! ) + (1
p1 )2 SX1 X1 (! )
!T =2) + p1 (1 p1 ) 2T sin( (!T =2) [SX0 X0 (!) + SX1 X1 (!)]
2
5. BIT-ERROR PROBABILITY It is obvious that the detector’s ability to discriminate between the two different processes depends strongly on how “similar” the processes are. If the processes are very “dissimilar”, then the decoder would produce few bit errors. On the other hand, we want the processes to be as “similar” as possible, to make it difficult for eavesdroppers to decode our transmitted bit stream. We thus face a classical trade-off problem of balancing between an acceptable bit-error probability and the possibility of interceptors to decode our message. Furthermore the bit-error probability decreases with increasing pulse length N . In fact, one may obtain an arbitrarily low bit-error probability by increasing N sufficiently, for a fixed non-zero attribute separation s = jj1 2 jj, where i ; i = 0; 1 denotes a parameter vector describing the process Xi (t). We thus face another tradeoff problem of balancing between an acceptable bit-error probability and a satisfactory transmission rate. 5.1. Calculation of the Bit-Error Probability
x
The log likelihood ratio is given in Eq. (2). If L( ) > 0 we choose class 1 , otherwise we choose class 0 . Let Lk = L( ) when belongs to class k . Now the biterror probability can be written simply as
x
Pe
= ProbfL0 > 0gP ( 0) + ProbfL1 < 0gP ( 1 )
(8)
Since the bit-error probability is now expressed in terms of the one-dimensional variable Lk , it can be readily calculated if the probability distribution of Lk is obtained. Following [2] we calculate the characteristic function of Lk , which includes all information concerning the distribution of Lk . Then the inverse Fourier transformation and basic transform theorems for integrals are applied, and we have that the bit-error probability is
Pe
= P ( 0 )[1
D0 ] + P ( 1 )D1
(9)
where Dk are given as
Dk (7)
where the asterisk denotes a convolution. From Eq. (7) we see that for long pulses it is safe to assume that the power spectrum of the SPSK-signal is the average of the two spectral densities weighted by their a priori probabilities. For short pulse lengths, however, this is no longer true, and the modifications caused by the convolutional terms may broaden the total spectrum significally. This effect is of course due to the rapid switching between the two processes, and the frequency characteristics of the switching itself.
x
Z 1 Qn 1 1 i=1 jFki (! )j = 2+ ! 0 !
sin
n X i=1
ki (!)
d!
(10)
Here jFki (! )j is given as
jFki (!)j = [1 + (2aki !)2 ]
1=4
(11)
and
ki (!) = 21 tan 1 (2aki !)
! log i
(12)
POWER SPECTRAL DENSITY
18
18
16
16
14
14
12
12
10
10
(dB)
(dB)
POWER SPECTRAL DENSITY
8
8
6
6
4
4
2
2
0
0
−2
−2 0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0
0.05
0.1
f
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
f
Figure 2: Power spectral density of the SPSK process with pulse length N = 11. The dotted line is the mean spectrum of the two processes, while the dashed line is the the estimated spectrum of the SPSK process by means of Welch’s method. The solid line is the power spectral density calculated from a discrete time version of Eq. (7).
Figure 3: Power spectral density of the SPSK process with pulse length N = 75. The dotted line is the mean spectrum of the two processes, while the dashed line is the the estimated spectrum of the SPSK process by means of Welch method. The solid line is the power spectral density calculated from a discrete time version of Eq. (7).
where
a1 = 0:6 and 02 = 12 = 1. The a priori probabilities are P ( 0 ) = P ( 1 ) = 0:5. The dotted line is the average of the two spectral densities weighted by their a priori probabilities, while the dashed line is the the estimated spectrum of the SPSK process by means of Welch’s method. The solid line is the power spectral density calculated from a discrete time version of Eq. (7).
a1i a2i
= 1 1 ; i = i 1
(13) (14)
Furthermore, i ; i = 1; :::; n are the eigenvalues obtained from the determinant equation
jR1
R0 j = 0:
(15)
A
Thus the ith row of the n n transformation matrix is a normalized eigenvector corresponding to the ith eigenvalue i , which satisfies the following relations:
AR0AT = I
and
AR1AT = ;
(16)
is a diagonal matrix whose diagonal terms are where the eigenvalues 1 ; 2 ; :::; n . Performing numerical integration of Eq. (10), the bit-error probability can then be found exactly. 6. NUMERICAL SIMULATIONS 6.1. Power Spectral Density of SPSK Processes In Fig. 2 we show the power spectral density of SPSK process based on two AR(1)-processes, with a pulse length N = 11. Fig. 3 shows the power spectral density when a pulse length N = 75 is used. In both examples the parameters of the two AR(1)-processes is equal to a0 = 0:9,
6.2. SPSK Communication System To demonstrate the SPSK concept, we in this section present results from numerical simulation of a digital communication system based on digitally generated source signals. Fig. 4 shows an example of a transmitted baseband signal for the message ’1100’ encoded by means of two different autoregressive processes of order 3 (AR(3)). The vertical lines indicate the length of each information carrying pulse. The two processes (indexed by k = 0 and k = 1) are defined by
xk (n)
=
ak (1)xk (n ak (3)xk (n
1) ak (2)(n 2) 3) + k (n) (17)
for n = 0; :::; N 1, where k (n) is a Gaussian driving noise with zero mean value and variance k2 , and N is the number of discrete samples in each pulse. The parameters for bit zero (k = 0) and three choices of bit one (k = 1) are given i Table 1. Note that the driving noise variance of
TRANSMITTED BASEBAND SIGNAL
POWER SPECTRAL DENSITIES
10
20
"1"
"1"
"0"
"0"
8 15 6 10 4 5
x[n]
(dB)
2
0
0
−2
(iii) −5
−4
(ii) −10
(i)
−6
−8
0
50
100
150
200 n
250
300
350
400
1(i) 1(ii) 1(iii)
a1 (k ) -1.67 -1.33 -1.23 -1.13
a2 (k ) 1.01 0.45 0.45 0.45
a3 (k ) -0.2 0.04 0.04 0.04
k2 0.64 0.98 1.55 2.09
Table 1: Parameters for the two AR(3)-processes used in this paper. Note that three choices of k = 1 are applied. process k
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
f
Figure 4: Message ’1100’ encoded by two different AR(3)-processes.
k 0
−15
= 1 has been chosen according to r0 (0) 12 = 02 00 r1 (0)
Figure 5: Exact power spectral densities of the AR(3)processes with parameters from Table 1. Solid line: k=0, broken line: k=1.
tion of the pulse length N for three different values of the attribute separation s = jj1 0 jj. From Fig. 6 we clearly see that an increasing value of the attribute separation s leads to a smaller bit-error probability in the receiver. Note also that the bit-error probability decreases as a function of increasing pulse length N . In Fig. 7 we show a plot of Monte Carlo simulated and the exact bit-error probability given in Eq. (9) as a function of signal-to-noise ratio (SNR). The SNR is defined as
E fjX (t)j2 g SN R = 10 log10 E fjn(t)j2 g
(18)
to ensure that the power of the two AR-processes are equal. Here r00 (0) and r10 (0) the variance of X0 (t) and X1 (t), respectively, if the driving noise variance of each process was equal to one. This is a natural requirement since it further decreases the possibility for an eavesdropper to recognize the information keying. Observe from Fig. 4 that two subsequent equal source bits have totally different waveforms, since they are realizations of a stochastic process. In Fig. 5 we show the power spectral densities functions for the two processes applied in this example. The solid line is the power spectral density of X0 (t), and the broken lines is the power spectral density of three different processes representing X1 (t). Note that the two spectra are quite similar, yet there are visible differences due to the different parameter choices for the process generators. 6.2.1. Bit-Error Probability In Fig. 6 we show a plot of Monte Carlo simulated and the exact bit-error probability given in Eq. (9) as a func-
(19)
From Fig. 7 we see, as expected, that the bit-error probability decreases as a function of increasing SNR. 7. CONCLUSIONS From the general description and the numerical example given above, we see that the proposed SPSK digital communication method has the potential of introducing a high degree of security at a low receiver complexity. Any set of parameters (attributes) characterizing the chosen stochastic process may be used to encode the message in SPSK. The receiver need only to know which attributes that are used in addition to an accurate test to distinguish the two processes. We do believe that the concept of SPSK suggested in this paper will become a useful complement to spread spectrum and chaotic encoding techniques. Further scientific work on SPSK method should concentrate on its properties related to synchronization, multiple access, and hardware based real time implementations.
10
BIT−ERROR PROBABILITY
0
BIT−ERROR PROBABILITY
0
10
−1
10
10
−1
(i) −2
10 −2
−3
10
P
P
e
e
10
(ii)
−4
10
10
(i)
−3
(ii) (iii)
−5
10
10
−4 −6
10
10
−5
(iii)
−7
1
10
2
10
N
Figure 6: Bit-error probability as a function of N for the AR(3)-processes with parameters from Table 1. Crosses are Monte Carlo simulations, and curves are theoretical results.
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3
10
10 −30
−20
−10
0
10
20
30
40
50
60
SNR (dB)
Figure 7: Bit-error probability as a function of SN R for the AR(3)-processes with parameters from Table 1. Crosses are Monte Carlo simulations, and curves are theoretical results.
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