Security of data is an issue that is of significant interest. There are two fundamental issues:
Security of data during transmission;
Security of stored data.
The first issue, in our case, security during wireless transmission, has been studied widely [1-4]. The second issue, security of stored data is less studied and of greater relevance. If sensitive data are stored in the database, then we need to find a way to preserve the security of user's data. A security measure needs to be in place so that even if the data are stolen, the eavesdropper cannot make sense of them.
Encryption is the perfect technique to solve this problem. In this work, for the first time, we employ compressive sensing, which is usually treated as a data compression framework, to encrypt data with good secrecy.
One aspect of encryption is the granularity of data to be encrypted or decrypted. Small sample group may appear to be the best choice, because it would minimize the number of bytes encrypted. However, as we have discovered, practical methods of embedding encryption within relational database entail a significant start up cost for an encryption operation. Large sample group encryption amortizes this cost over larger data. Because the encryption process of compressive sensing is quite simple, this problem could be easily solved by increasing the number of information bits.
In this approach, the sender of the encrypted data supplies the key, and the database provides the encryption function. Only authorized users who are given the key can decrypt the data using the decryption algorithm. Because the key is owned by the sender and not stored in the database, an unauthorized user who may get hold of data cannot get hold of the key. The full security provided by the encryption algorithm is inherited by the data in the database. Note generic functions named encrypt and decrypts are used in the query. In fact, one could implement the two functions with any encryption algorithm. Also note that users can easily specify and use encryption algorithms of their choice. From the previous investigation, it is clear that there is a great need for a new approach to encryption that is both highly secured and efficient to avoid the high penalty over the encrypted data.
The remainder of this paper is organized as follows. In Section 2, we summarize characteristics of ultra-wideband (UWB) noise radar signal and its time-frequency representation. In Section 3, we introduce principles of compressive sensing and its application in data compression. Section 4 presents the algorithm used in compressive sensing. We propose compressive sensing-based data communication system with encryption and theoretic analysis of the performance of compressive sensing as an encryption algorithm in Section 5. Experiment setup that is carried out at the Villanova University is illustrated in Section 6. Finally, simulation results and conclusions are given in Sections 7 and 8, respectively.
2 UWB Noise Radar Fundamentals
Sensing-through-wall techniques have attracted so many interests due to a broad range of military and civilian applications. It will be useful in locating weapons caches during military operations, searching and rescuing people from natural disasters such as earthquakes and providing substantiality assessment of bridges and buildings. Current sense-through-wall systems are mainly based on short-pulse waveforms that require special antennas to avoid unwanted signal coupling. In addition, periodically transmitted pulses will be easily intercepted by others.
In recent years, UWB waveforms are frequently employed for indoor wireless propagation systems because of its exceptional range resolution and strong penetrating capability. Because ultra-wideband (UWB) radars used for sense-through-wall applications are usually in ultra-high frequency (UHF) range (500–1000 MHz) for good penetration through walls and building materials. The 500-MHz bandwidth will yield a 30-cm range resolution. The major challenge is to develop a simplified architecture that will reduce cost without sacrificing performance. Systems building on the existing technology for through-wall sensing are typically heavy and inconvenient to deploy. Chip-based arbitrary signal generators can be used as noise sources and field programmable gate array technology can be used for time delay and cross-correlation in UWB noise radar implementation.
Immunity from jamming and interference: Unwanted signals caused by jammers or other interfering transmitters will not correlate with the time-delayed transmit replica and hence yield zero at the correlate output.
Immunity from detection: Because the waveform is not repeatable and spreads over a wide band, it cannot be recognized as the intentional signal on other's receiver.
Because UWB signal is usually in UHF range (500–1000 MHz) [6, 7], sampling rate should be more than 1 GHz (Nyquist rate) for alias-free signal sampling that requires plenty of storage for the received signal. In this paper, we apply compressive sensing to UWB noise radar signal, which asserts that one can recover it from far fewer samples than the conventional Nyquist rate method. It also enhances the operational feasibility of UWB noise radar.
Because UWB noise radar signal is stochastic, it can only be described by its statistics. The UWB noise radar signal x(t) can be described by the following properties:
Auto-correlation Rxx(τ) is an impulse or impulse-like at τ = 0.
Probability density function (PDF) px(X) can be characterized as Gaussian distribution.
Power spectral density (PSD) Sxx(f) is assumed uniform and band limited.
From the aforementioned statistics, we can propose the time-frequency model for the UWB noise radar signal as follows:
where ω0 is the center radian frequency of the signal, a(t) is the Rayleigh distributed amplitude which describes amplitude fluctuations and δω(t) is the uniformly distributed frequency fluctuations over the ± Δω range, i.e., [−Δω ≤ δω ≤ + Δω] . Assuming that the random variables a(t) and δω(t) are uncorrelated, we can show that the average power of the signal x is 〈a2(t)〉/2R0, where 〈 ⋅ 〉 denotes time average and R0 is the system impedance. The center frequency f0 and the bandwidth B can be derived as ω0/2π and Δω/π, respectively.
An alternative time–frequency representation of UWB noise radar signal is given by
where sI(t) and sQ(t) are zero-mean Gaussian processes and f0 = ω0/2π is the center frequency. This can also be written as
where is the Rayleigh distributed amplitude and ϕ(t) = tan − 1(sQ(t)/sI(t)) is the uniformly distributed phase.
3 Compressive Sensing Background
Compressive sensing provides a framework for integrated sensing and compression of discrete-time signals that are sparse or compressible in a known basis or frame [9-17]. Many natural signals have concise representations when expressed in the proper basis . Mathematically speaking, consider a discrete signal f ∈ RN that can be expanded in an orthonormal basis Ψ = [ψ1ψ2 …ψn] as follows:
where x is the coefficient sequence of f that can be computed from signal f:
It will be convenient to express f as Ψx (where Ψ is the n × n matrix with ψ1, ψ2, …, ψn as columns). We can say the discrete signal f is K-sparse in the domain Ψ, K ≪ N, if only K out of N coefficients in the sequence x are nonzero. Sparsity of signal is a fundamental principle used in the compressive sensing as well as in most modern lossy coders such as JPEG-2000 and many others, because a simple way for image compression would be to compute x from f and then only encode the values and locations of the largest K coefficients. Examples in  show that perceptual loss is hardly noticeable from a megapixel image to its approximation obtained by throwing away 97.5% of the coefficients. Unfortunately, this compression process requires computing all N coefficients of signal f and the locations of the significant coefficients, which may not be known in advance.
Note that there are only K coefficients that are nonzero, so we can remove this “sampling redundancy” by acquiring only M samples of the signal f, where K < M ≪ N. The new M-length observation vector y can be represented as follows:
where Φ is an M × N measurement matrix. The aforementioned equation can be written as
where Θ is given by
The signal f can be perfectly recovered from M equals to or a little bit more than measurements K, if Θ satisfies the so-called restricted isometry property (RIP) . It suggests that Θ is sufficiently incoherent and Φ cannot sparsely represent basic vectors of matrix Ψ and vice versa.
It has been shown in  that choosing an independent and identically distributed Gaussian random matrix as sensing matrix Φ, Θ is also an independent and identically distributed Gaussian for various orthonormal bases Ψ such as spikes, sinusoids, wavelets, Gabor functions, curvelets and so on. Θ is shown to have satisfied RIP with high probability, if M ≥ cK log(N/K), where c is a small constant and hence stable reconstruction is possible with high probability. Note that it is not known in advance which coefficients of f are zeroes, or which samples are not needed.
With the new observation matrix y, we decide to recover the signal f by -norm minimization; the proposed reconstruction f* is given by f∗ = Ψx∗, where x* is the solution to the convex optimization program
That is, among all the objects consistent with the data, we choose the one whose coefficient sequence has minimal -norm .
4 Recovery Algorithm
In this section, we propose an alternative method for dealing with absolute values in linear program problem in Equation (9), which introduces new variables x+, x–, constrained to be non-negative, and let . (Our intention is to have or , depending on whether xi is positive or negative.) We then replace the occurrence of |x| with and obtain the alternative formulation
where and .
The relations , x+ ≥ 0, x− ≥ 0, are not enough to guarantee that , and the validity of this reformulation may not be entirely obvious. At an optimal solution to the reformulated problem, and for each i, we must have either or , because otherwise we could reduce both and by the same amount and preserve feasibility, while reducing the cost, in contradiction of optimality. Having guaranteed that either or , the desired relation now follows.
Let A be the m × 2n matrix [Θ − Θ], Θ is defined in Equation (8). Equation (11) can be written as
It has a solution , which can be partitioned as z∗ = [u∗v∗]; then x∗ = u∗ − v∗ solves Equation (9). The reconstruction f∗ = Ψx∗. This linear program is typically considered computationally tractable . Intuitively, we can represent UWB noise radar signal in cosine basis based on its representation in Equation (1).
5 Compressive Sensing-Based Data Security
In any data transmission system, security is a major concern. The sender of data wants to ensure that only authorized users may gain access to the data, but not unauthorized users. As shown in Figure 1, which illustrates the conventional data communication system with encryption scheme, data are transmitted by a sender to an authorized user coupled to a receiver while ensuring that the data remain unavailable to other users who are coupled to the same receiver. The data are encrypted by the sender using a key. The encrypted data are then scrambled and transmitted by the sender. All users who are coupled to the receiver will have access to the descrambled data, but the data are unreadable because they are encrypted. Only the authorized user with a proper key may decrypt the encrypted data to obtain clear data.
In most cryptographic methods, the security of the message is based on the fact that we have transformed the message into something that is not immediately decipherable. When attempts are made to break the code, if a message is extracted that makes sense, i.e., is not just gibberish, it is assumed that the message has been successfully decoded. When using compressive sensing as the encryption scheme, this is not necessarily a valid assumption.
Using compressive sensing, we compress the message x into a coded message y by y = Φx, where Φ is the sensing matrix. In order to decrypt the message, an eavesdropper would need to find both Φ and x that give y. So, the question becomes does there exist Φ′ and x′ such that y = Φ′x′. Assuming that any sensing matrix is possible, we see that
is a possible solution for any other message x′. This means that it is possible to encrypt data and an attempt to break the encryption will result in decrypting different data. This is possible because the system is under-determined, i.e., we know fewer variables than we need to find. The key part of this is Φ′ being a valid sensing matrix. This is certainly the case if we assume that we transmit the entire sensing matrix.
Because it is impossible to decrypt the original message without the entire sensing matrix, the next problem we should handle is how we could ensure that only authorized user can have access to it. In order to solve this problem, direct-sequence spread spectrum is adopted to modulate the sensing matrix and the information sequence. The modulated message is then transmitted over the air. Direct-sequence spread spectrum modulates the information bit pseudo-randomly with a continuous string of pseudo-noise (PN) code symbols called “chips,” each of which has a much shorter duration than the information bit. That is, each information bit is modulated by a sequence of much faster chips. This chip looks like a noise signal. It is a pseudo-random sequence of 1 and −1 values at a frequency much higher than that of the original signal. PN code is generated by a pseudo random generator with an initial random seed that is shared between transmitter and receiver. The receiver can then use the same PN sequence to counteract the effect of the PN sequence on the received signal in order to reconstruct the information bit. The modulated signal resembles white noise. However, this noise-like signal can be used to exactly reconstruct the original data at the receiving end by multiplying it by the same pseudo-random sequence (because 1 × 1 = 1 and −1 × −1 = 1). This process, known as “de-spreading,” mathematically constitutes a correlation of the transmitted PN sequence with the PN sequence that the receiver believes the transmitter is using. The resulting effect of enhancing signal to noise ratio on the channel is called process gain. This effect can be made larger by employing a longer PN sequence and more chips per bit, but physical devices used to generate the PN sequence impose practical limits on attainable processing gain.
We further investigate the secrecy of compressive sensing from the information theory point of view. Shannon introduces the idea of perfect secrecy in . An encryption scheme achieves perfect secrecy if the probability of a message conditioned on the cryptogram is equal to the a priori probability of the message, i.e., P(X = x|Y = y) = P(X = x). Alternatively, this condition can be interpreted as I(X;Y) = 0. If the eavesdropper does not have the exact PN sequence, he will only have partial information of the original message x which can be described as yp = h ⊗ y, where h is the impulse response of the bandpass filter. Because Φ in Equation (6) is linear dependent, we can conclude that the original message x and yp are dependent, which also means I(X;Y) ≠ 0. Therefore, compressive sensing is not an encryption scheme with perfect secrecy.
It is proved in  that sensing matrix Φ is said to satisfy a RIP of order k if there exists a δk such that
holds for all x with sparsity k. In , Candés verified that if , one can recover the original message x with high probability. If the length of PN sequence is long enough, the energy of the original message that falls in certain bandwidth would decrease so that . The probability to recover the original message would be quite small. In other words, compressive sensing would be an encryption scheme with good secrecy.
In the worst case, unauthorized users are assumed to decrypt the entire sensing matrix successfully. They still cannot decrypt the original message because they do not have information of the size of the original sensing matrix. For example, 1200 information bits could construct matrices with different sizes, such as 12 × 100, 20 × 60, 30 × 40, etc. There are so many possible combinations for the choice of sensing matrix.  also shows that ck measurements (typically c≈ 3 or 4) are required to reconstruct the message. We assume the original sensing matrix with size of M × N and the size of estimated sensing matrix is M′ × N′. If M′ < ck, the probability of recovering the original message x is zero. If M′ ≥ ck, Theorem 1 in  demonstrates that the sparsity of estimated message is M′ with probability 1 over the set of Φ′. Hence, the original message could never be reconstructed because the sparsity of the original message x is k, not M′. It is a great advantage of compressive sensing to be implemented as an encryption algorithm.
Figure 2 demonstrates our proposed data communication system. Transmitting part is quite simple. Compressive sensing is employed to do joint data compression and encryption that simplify the hardware complexity and reduce the processing time as well. PN code can be easily generated by a pseudo-random generator with an initial random seed that is shared between transmitter and receiver. At the receiving part, the authorized user will access to clear data using the entire sensing matrix. Because the sensing matrix is almost random and the dimension is large, it is impossible for other users to decrypt it even if they have access to the encrypted data. The performance of compressive sensing as an encryption algorithm is shown in the next section.
6 Experiment Setup and Data Collection
Our simulation is based on the sense-through-wall noise UWB radar data from the Center for Advanced Communication (CAC), Villanova University. It is carried out in 6 June 2008. The goal of this experiment is to detect the existence of targets, i.e., human, on the other side of the wall based on the reflected signal. The experiment setup and two students who stood still behind the wall are illustrated in Figure 3. Two students stood at 1 and 2 m behind the wall, respectively. The total moving length for UWB noise radar is 1.8 m. Six data sets were collected at locations that are equally spaced at 30 cm. The length of the blue cable is 4 m. The distance between the antenna of UWB radar and wall is only 8 cm, which can collect the reflected signal efficiently.
7 Simulation Results
The UWB noise radar signal used in this simulation was collected in the Radar Imaging Lab at Villanova University. The frequency of the transmitted signal is 400–700 MHz and the sampling rate is 1.5 GHz. The property and waveform of UWB noise radar signal are illustrated in Figure 4. Figure 5(a) illustrates the sparse form of UWB noise radar signal in cosine basis. It is clear that only a small number of coefficients are nonzero. In other words, we can say UWB signal is sparse when expressed in cosine basis (K ≪ N). Hence, we can apply the compressive sensing to UWB noise radar signal. Figure 5(b) shows exactly its reconstruction from a new observation vector y with length of 377 samples. It suggests that we can use only one-third samples of the original data to represent the UWB noise radar signal without any information loss.
In the second experiment, we study the dependence of probability of recovery (Pr) on the number of random measurements M. For each value of M, we performed a Monte Carlo simulation involving 5000 realizations of uniformly random measurements. The role played by probability in compressive sensing is demonstrated in Figures 6 and 7. When the random measurements M are less than 100, no algorithm whatsoever would of course be able to reconstruct the signal. The probability of perfect reconstruction increases whereas the size of M increases. The probability of exact recovery that does not occur is truly negligible when M ≥ 370. The performance of Bernoulli sensing matrix is better than that of Gaussian sensing matrix.
The performance of compressive sensing as an encryption algorithm is illustrated in Figures 8-10. In Figure 8, we assume that unauthorized user only knows the size of sensing matrix. In Figure 9, we assume unauthorized user has half of the information of the sensing matrix. In Figure 10, only the last row of sensing matrix is unknown to unauthorized user. From these figures, we can conclude that compressive sensing provides good secrecy as an encryption algorithm and it is impossible to retrieve clear data without the entire sensing matrix.
We have applied the novel concept of compressive sensing on a practical problem of sampling UWB noise radar signal. We could draw the following conclusions: (1) The UWB noise radar signal is sparse when expressed in cosine basis Ψ. (2) Random Gaussian matrices are largely incoherent with any fixed basis, which can efficiently acquire the information from the original signal. (3) With 377 random measurements, we can reconstruct the signal with negligible probability to fail. (4) Our proposed compressive sensing-based data communication system provides good secrecy in terms of data security, and it is impossible to retrieve clear data without the entire sensing matrix.
This work was supported in part by U.S. Office of Naval research (ONR) under Grant N00014-11-1-0071, U.S. National Science Foundation (NSF) under Grant CNS-0964713, and Tianjin Natural Science Foundation of China under Grant 10JCYBJC00400.