Secure Massive IoT Using Hierarchical Fast Blind Deconvolution
Gerhard Wunder, Ingo Roth, Rick Fritschek, Benedikt Groß, Jens Eisert
11 Secure Massive IoT Using Hierarchical Fast BlindDeconvolution
Gerhard Wunder ∗ , Ingo Roth † , Rick Fritschek ‡ , Benedikt Groß § , Jens Eisert ¶ ,Freie Universit¨at BerlinEmail: { ∗ g.wunder, † i.roth, ‡ rick.fritschek, § benedikt.gross } @fu-berlin.de, ¶ [email protected] Abstract —The Internet of Things and specifically the TactileInternet give rise to significant challenges for notions of security.In this work, we introduce a novel concept for secure massiveaccess. The core of our approach is a fast and low-complexityblind deconvolution algorithm exploring a bi-linear and hier-archical compressed sensing framework. We show that blinddeconvolution has two appealing features: 1) There is no need tocoordinate the pilot signals, so even in the case of collisions in useractivity, the information messages can be resolved. 2) Since allthe individual channels are recovered in parallel, and by assumedchannel reciprocity, the measured channel entropy serves as acommon secret and is used as an encryption key for each user.We will outline the basic concepts underlying the approach anddescribe the blind deconvolution algorithm in detail. Eventually,simulations demonstrate the ability of the algorithm to recoverboth channel and message. They also exhibit the inherent trade-offs of the scheme between economical recovery and secretcapacity.
Keywords —5G, massive IoT, physical layer security, blinddeconvolution, compressed sensing, hierarchical sparsity
I. I
NTRODUCTION
Over the last decades, major developments in communica-tion technologies have radically altered the way we communi-cate. This entails difficult network challenges from the techno-logical side. As the sheer volume of data being transmitted isgrowing, these challenges are concomitant with new demandson the security of the communication channels. To accompanythe significant challenges of security of communication inthe realm of big data, novel physical layers of security willhave to be identified and developed. This seems particularlyrelevant in the context of the Internet of Things (IoT) andthe Tactile Internet (TI). In this work we will show thatsparse signal processing can be incorporated naturally withinthe concept of massive IoT, including the TI and embeddedsecurity. We go on to demonstrate that it indeed exhibits anew degree of freedom in the design of (low-complexity)algorithms, naturally entailing new interesting trade-offs suchas compressibility versus secrecy [1], [2].Our specific innovations are as follows: We propose a fast , scalable , and secure access procedure with low complexity[1], [2]. At the heart of our approach is a new fast blinddeconvolution algorithm based on bilinear compressed sensing(CS) and hierarchical sparsity frameworks [3], [4], [5], [6]. Theproposed algorithm has the additional advantageous feature ofbeing inherent to low-complexity by avoiding semi-definiteprogramming techniques. Using blind deconvolution for theuncoordinated massive access has two appealing features: i) There is no need to coordinate the pilot signals, so even incase of collisions user activity and information messagescan be resolved.ii) Since all the individual channels can be recovered in par-allel, and by assumed channel reciprocity, the measuredchannel entropy serves as a common secret and is usedas an encryption key for each user [7].In this work, we will outline the underlying basic concepts, anddescribe the proposed blind deconvolution algorithm in detail.Eventually, simulations demonstrate the (not at all obvious)ability of the algorithm to recover both channel and message,and also nicely reveal the inherent trade-offs. If a channelis sparser, the recovery is improved but at the same timeless entropy for key generation is available. Hence, whilethe recovery can be achieved more economically, the secrecyproperties are degraded. Basic notations • The circular convolution of two vectors f, g ∈ C n willbe denoted by f (cid:126) g and is defined as ( f (cid:126) g ) j := n (cid:88) i =1 f j g ( i − j +1) mod n . (1) • (cid:107)·(cid:107) will denote either the (cid:96) -norm of a vector or theFrobenius-norm of a matrix depending on the context. • For a set S let |S| denote its cardinality. For any positive N ∈ N we define [ N ] := { , , . . . , N } . • For a vector x ∈ C n we denote by | · | the function thatreturns the number of non-zero elements of x , i.e. | x | := |{ i ∈ [ n ] : x i (cid:54) = 0 }| . (2) • The transpose/Hermitian of a matrix A with complexentries will be denoted A T and A H , respectively. • The Kronecker product of the matrices A ∈ C M × N and B ∈ C O × P is denoted by A ⊗ B and is defined as the C MO × NP block matrix A ⊗ B := a , B a , B . . . a ,N Ba , B . . . . . . ...... . . . . . . a M − ,N Ba M, B . . . a
M,N − B a
M,N B , (3)where a i,j is the ( i, j ) -entry of A . • The map vec : C M × N → C MN is the column-wisevectorization of a matrix, i.e. it stacks the columns ofa matrix into a long vector. a r X i v : . [ c s . I T ] J a n • circ( v ) denotes the circulant matrix of a vector v ∈ C n ,which is defined as circ( v ) := v v n − . . . v v v v v n − v ... v v . . . ... v n − . . . . . . v n − v n − v n − . . . v v . (4)II. S YSTEM MODEL
We consider a secure random access scenario where ac-cess point “Alice” with N t antennas communicates with N r “Bobs”, which are low-complexity devices, equipped witha single antenna each. Furthermore, we assume an OFDMsignal model, so that essentially all wireless channel oper-ations become cyclic, acting by the circ( · ) operation. Thecommunication is bi-directional and TDD in T ≥ time slots t , t , . . . , t T − in the following fashion: • First, Alice sends out multiple beacon OFDM symbols sothat the Bobs can synchronize and measure the channelsto each of Alice’s antennas. From the measured channelseach Bob generates a key and encrypts its message. • Subsequently all Bobs transmit in an uncoordinated fash-ion their encrypted messages in the same slot while nopilot signaling is used. Alice uses a blind deconvolutionalgorithm to simultaneously estimate the channels and the signals “in one shot”.
A. Wireless channel properties
The most important random entity is the wireless channelfrom Alice to all the Bobs and from the Bobs to Alice perantenna. We use the following convention for the bi-directionalcommunication: p is the index of the transmitting antenna, q of the receiving antenna, and i represents the delay domainin some time slot. Hence, the matrix H a → bp = ( h a → bp,q,i ) thatrepresents the wireless channels from Alice’s p -th antenna to all Bobs is given by H a → bp = ... h a → bp,q, ... h a → bp, ,i . . . ... . . . h a → bp,N r ,i ... h a → bp,q,N d ... ∈ C N d × N r (5)for p = 1 , . . . , N t . In addition, the matrices H b → ap = ( h b → apqi ) representing the wireless channels from p th Bob to Alice aregiven by H b → ap = ... h b → ap,q, ... h b → ap, ,i . . . ... . . . h b → ap,N t ,i ... h b → ap,q,N d ... ∈ C N d × N t (6)for p = 1 , . . . , N r . Notably, we impose a typical structuralassumption for wireless channels: Each column vector h p,q =( h p,q,i ) contains only N d (cid:28) N coefficients, where N d iscalled delay spread of the Channel Impulse Response (CIR) of any p -th/ q -th pair that gets transmitted/received. This is acommon assumption, e.g. for OFDM systems.Now, the received time-space signal (represented by rowsand columns, respectively) in some time slot for Alice is givenby Y a ( t i ) ∈ C N × N t and for all the Bobs by Y b ( t i ) ∈ C N × N r , where N (cid:29) is the signal space dimension and N t and N r are the numbers of antennas that transmit and receive.We assume that the channel coherence time is essentially largerthan the slot time and shall henceforth drop the dependency onthe time slot to ease the notation. On each transmit antenna p with ≤ p ≤ N t , both some known and some unknown transmitted sequences s a/bp , x a/bp ∈ C N are broadcast. Thesignals for Alice and Bob in one time slot then becomeAlice → Bob: Y b = N t (cid:88) p =1 (cid:0) S ap + X ap (cid:1) H a → bp + Z b , (7a)Alice ← Bob: Y a = N r (cid:88) p =1 (cid:0) S bp + X bp (cid:1) H b → ap + Z a . (7b)Here, S a/bp = circ( s a/bp ) , X p = circ( x a/bp ) ∈ C N × N d are thecirculant matrices of the transmitted sequences as defined in(4). The matrices Z a/b denote additive white Gaussian noisewith variance η . We will impose the following structuralproperties: • Reciprocity property : If not stated otherwise, we assumethe reciprocity property, i.e., if we change the roles ofthe transmitting antenna p and the receiving antenna q , the channel coefficients are conjugate complex, i.e., h a → bp,q,i = (cid:0) h b → aq,p,i (cid:1) ∗ . Note that this assumption is by far notunrealistic today, as it is already possible to verify withoff-the-shelf WiFi devices [8]. • Natural structural properties : We assume that out of the N d channel coefficients, in each column of H a → bp , H b → ap only σ > of the CIR coefficients are actually non-zeroand the exact positions of the coefficients within H p areunknown, i.e., the channel is σ -sparse (in the canonicalbase). • Imposed structural properties : Our final structural as-sumption is that the unknown signals x p are s -sparse bydesign in some known subspaces with bases Q , Q , . . . such that x p = Q p b p , where b p is a binary vector with | b | = s . The rate delivered by this approach is R = 1 N log (cid:18) Ns (cid:19) [bits] . In the sequel, we will propose an algorithm that is ableto exploit these structural assumptions to recover both theunknown channels and the unknown signals, given only thesuperposition of their convolutions.
B. Inherent security of the scheme
We briefly describe the information theoretic secrecy stem-ming from the envisioned scheme. It builds on the reciprocityproperty of the channel and exploits randomness of the channelgain to generate a key and encrypt the message. We refer to Which is due to fading in the wireless channel. the work [9] for an in-depth analysis regarding the use ofchannel gains for keys, as that was the first rigorous work onthe subject.
Phase 1 : • Alice sends a predefined pilot signal to all Bobs. • Each Bob q can measure the complex-valued channelgains H a → bp,q,i = h b → aq,p,i ∀ p, i . • Each Bob encrypts his message m with c = f ( m, { h a → bp,q,i } ) , effectively using the channel as a sourceof randomness for key generation. Phase 2 • All the Bobs p send their encrypted cipher texts c p toAlice in an uncoordinated way. • Alice receives the superposition of all the convolutions ofthe cipher text with the respective channels. Now she hasa blind de-mixing/de-convolution problem and receivesthe cipher-texts and complex-valued channel gain pairs ( H b → ap , c p ) = ( H a → bp , c p ) ∀ p, q, i of every Bob by usingour algorithm. • Since Alice knows H b → ap , which is the same as H a → bp due to reciprocity, she can generate the key herself anddecrypt the cipher-texts.We note that small variations between both channels, i.e.small violations of reciprocity do not matter, since we canadjust the key generation process. One can for examplequantize the channel gain coarse enough to equalize the keys.This would lower the achievable key rate, but would not impactthe security of the scheme, due to the assumed independencebetween the channel gains from Alice to Eve and Alice to Bob.However, a detailed analysis shall be carried out in follow-upwork.III. F ORMULATION AS BLIND DE - CONVOLUTION PROBLEM
A. Single user case
For the purpose of exposition, we first consider the case ofa single user and a single antenna. Bob sends the signal x b over the channel h b → a to Alice, who receives y a = h b → a (cid:126) x b = circ( x b ) h b → a . (8)Using the so-called lifting trick, which was introduced in thecontext of phase retrieval [10], [11] and later generalized toblind deconvolution problems [12], this bi-linear equation canbe transformed into a linear one as y a = B vec (cid:0) x b ( h b → a ) T (cid:1) + z a . (9)Here, B is a suitable matrix with ( B ) i, ( j,k ) = δ i,j + k mod N ( ( j, k ) is a double index notation), which is composed as B = ... ...
01 0 ... ... ... ... ... ... ... ...... ... ... ... ...... ... ... ... ... ∈ (0 , N × N , (10)and z a is a Gaussian noise vector. The sparse signal model x b = Qb b with the random coding matrix Q ∈ C N × E and s -sparse binary vector b a ∈ {− , } E of length E can beincorporated in the formulation to yield y a = B ( I N d ⊗ Q ) (cid:124) (cid:123)(cid:122) (cid:125) =: M vec( b b ( h b → a ) T ) . (11)By this procedure, the blind deconvolution problem of recov-ering h b → a and x b from the measurement y a is turned into amatrix recovery problem in X = b b ( h b → a ) T , given the linearmeasurement operator A : C N d × E → C N , defined by (11).The factors h b → a and b b can be obtained from X as the firstleft and right singular vectors of the SVD of X . B. Multi-user case
In the more general case of multiple Bobs, each of Alice’santennas receives a superposition of signals, each convolvedwith its respective channel, y q = N r (cid:88) p =1 h p,q (cid:126) Q p b p + z q for q = 1 , . . . , N t , (12)where we have dropped the superscripts indicating the senderand receiver to simplify the notation. The lifting trick can beapplied to each summand, resulting in y q = N r (cid:88) p =1 B ( I N d ⊗ Q p ) vec( b p h Tp,q ) + z q . (13)In comparison to (11), this is a (more challenging) problem ofsimultaneous blind deconvolution and blind de-mixing. Workon this problem has been done in ref. [13]. Problem (13) canbe brought into the form y q = M vec( X q ) + z q for q = 1 , . . . , N t , (14)with the big system matrix M = B I N d ⊗ Q ... I N d ⊗ Q N r T ∈ C N × N d · E · N r , (15)and the unknown X q = [ X ,q X ,q . . . X N r ,q ] with X p,q = b p h Tp,q . With the structural assumptions that each channel h q is σ -sparse, each b q is s -sparse and only µ of the N r usersare active at a time, the vectorization vec( X q ) ∈ C N d · E · N r becomes a hierarchically ( s, σ, µ ) -sparse vector. The finalequation for the multi-user, multi-antenna setup is then y T ... y Tq ... y TN t T (cid:124) (cid:123)(cid:122) (cid:125) = Y ∈ C N × Nt = B I N d ⊗ Q ... I N d ⊗ Q N r T (cid:124) (cid:123)(cid:122) (cid:125) = M ∈ C N × Nd · E · Nr · vec( X ) T ... vec( X q ) T ... vec( X N t ) T T (cid:124) (cid:123)(cid:122) (cid:125) = X ∈ C Nd · E · Nr × Nt . (16)It is worth noting that the columns of X are jointly sparse,since the antennas are close to each other, and hence for each p , the channels h p,q have the same support for all q . IV. F
AST BLIND DE - CONVOLUTION ALGORITHM
A. Prior work
There exists a number of recent works on solution strategiesfor the blind deconvolution problem and the extended blinddeconvolution and blind de-mixing problem using the differentapproaches. Convex approaches use the formulation min X ϕ ( X ) s.t. A ( X ) = y, (17)where X is the unknown matrix variable, A is the linearmeasurement operator and y the given data. The objectivefunction ϕ ( · ) is used to incorporate structural assumptionson X that can be exploited to find a unique solution to theunder-determined system A ( X ) = y . In ref. [12], the nuclearnorm ϕ ( X ) = (cid:107) X (cid:107) ∗ is used, exploiting the fact that X asan outer product of b and h is a rank one matrix. Instead ofsparsity priors for b and h , in ref. [12] the authors assume thatboth vectors are in known low-dimensional subspaces. Thissetting was generalized to include the de-mixing of multipleconvolutions in ref. [13], [14].To relax the subspace assumption to sparse vectors, it seemsnatural to linearly combine the regularizers promoting low-rankness and sparsity of the matrix, i.e. ϕ ( X ) = (cid:107) X (cid:107) ∗ + λ (cid:107) X (cid:107) . But in fact one can show that the linear combinationdoes not yield an improved sampling complexity, compared tojust using one of the regularizers [15]. Furthermore, convexformulations including the nuclear norm are semidefinite pro-grams and can be solved by popular interior-point solvers suchas SDPT3 [16] or SeDuMi [17]. These SDP-solvers have thedrawback of being prohibitively slow and memory consumingfor large scale problems, as their computational and storagecomplexity typically scales cubically in the system size.For this reason, subsequent convex approaches focused onexploiting the sparsity of X and structured versions thereof.Ling and Strohmer minimize ϕ ( X ) = (cid:107) X (cid:107) assuming that atleast one of the factors h , b is sparse and hence also X . Inthis setting the sparsity of X is structured since each columnis either vanishing or dense. This block-sparse structure mo-tivated the use of the objective function is ϕ ( X ) = (cid:107) X (cid:107) , ,which is defined as the sum of the column norms of X , in ref.[18]. The current work follows in this line of research, furtherincorporating the sparsity structures inherent to the problem,if both vectors h and b are assumed to be sparse.Following a different approach, a number of non-convexalgorithms, mostly based on alternating minimization, existthat deal with blind-deconvolution and related problems. Forexample, the blind deconvolution and blind de-mixing prob-lem, where low-dimensional subspaces for both vectors areknown, is tackled in [19], [20] and the sparse setting is handledin [21]. For these to work properly, a good initial guess forthe unknown factors of X is crucial. Therefore, in [19] a basinof attraction is constructed, and a spectral method is used toobtain an initialization close to the solution. The algorithm of[21] uses a hard thresholding algorithm to compute a suffi-ciently close initial guess and only then proceeds with theiralternating minimization algorithm. This algorithm, however,involves the projection onto a complicated, non-convex setwhose success can not be guaranteed. B. Proposed algorithm
Motivated by the application in mMTC, the recovery ofhierarchical sparse signals from linear measurements wasstudied in ref. [4]. In this work, the HiHTP algorithm wasextended to solve the outlined 3-dimensional problem min z ∈ C Nd · E · Nr (cid:107) y − M z (cid:107) s.t. z is hierarchically ( s, σ, µ ) -sparse.(18)A hierarchically sparse vector z ∈ C N d · E · N r has the followingstructure. z = ( z , z , . . . z r (cid:122) (cid:125)(cid:124) (cid:123) ( z r , z r , . . . , z rj (cid:122) (cid:125)(cid:124) (cid:123) ( z r ,j , z r ,j , . . . , z ri,j , . . . , z rD,j ) , . . . , z rE ) , . . . , z N r ) T (19)As described above, only µ of the vectors z r are differentfrom zero, or “active”. The active vectors only have σ non-zero blocks, and each of these blocks is s -sparse.HiHTP tries to find such a structured solution to (18) byrepeating the following steps:i) Perform one gradient step on the current iterate z ( k ) .ii) Determine the support S ( k +1) of the next iterate viahierarchical hard thresholding.iii) Solve a least squares problem on S ( k +1) to obtain thenew iterate z ( k +1) .The details of each step are explained below. Gradient step:
The gradient of the objective functionfrom (18) at z ( k ) is given by M T ( M z ( k ) − y ) . Hence, theintermediate point is given by ˜ z ( k +1) = z ( k ) + M T ( y − M z ( k ) ) . (20) Hierarchical hard thresholding:
In this step, the support ofthe next iterate is found by thresholding the intermediate pointdefined in (20) with the algorithm explained below. Define thehard thresholding operator T s : C n → [ n ] s applied to a vector g ∈ C n as T s ( g ) = argmax { i ,...,i s }⊂ [ n ] s (cid:88) k =1 | g i k | . (21)The hierarchical hard thresholding operator with three layers,denoted by T ( s,σ,µ ) , is given by the following algorithm. Algorithm 1
Support via hierarchical hard thresholding
Require:
Structured vector g ∈ C D · E · N r as above, sparsity ( s, σ, µ ) for k = 1 , . . . , N r dofor j = 1 , . . . , E do I kj = T s ( g rj ) v kj = (cid:80) i ∈ I j | g kij | end for J k = T σ ( v k ) u k = (cid:80) j ∈ J k v kj end for K = T µ ( u ) S = (cid:83) k ∈ K (cid:83) j ∈ J k I kj Ensure: ( s, σ, µ ) -sparse support set S Hence, the support in step k + 1 is computed as S ( k +1) = T ( s,σ,µ ) ( z ( k +1) ) (22) Least-squares problem
The entries of the next iterate z ( k +1) are then computed by solving a least squares problem withsupport constraints, i.e. z ( k +1) = argmin z ∈ C NdENr (cid:110) (cid:107) y − M z (cid:107) s.t. supp( z ) ⊆ S ( k +1) (cid:111) . (23)The algorithm is stopped, if S ( k +1) = S ( k ) or the maximumnumber of iterations is reached. The whole algorithm issummarized below. Algorithm 2
HiHTP - multi-user case
Require:
Measurement matrix M ; data y ; channel- , signal-and user-sparsities s, σ, µ Set z (0) = 0 , k = 0 repeat Compute support via hierarchical hard thresholding: S ( k +1) = T s,σ,µ (cid:0) z ( k ) + M H ( y − M z ( k ) ) (cid:1) Compute the corresponding entries by solving the least-squares problem: z ( k +1) = argmin z ∈ C NdENr (cid:8) (cid:107) y − M z (cid:107) s.t. supp( z ) ⊆ S ( k +1) (cid:9) and set k = k + 1 until stopping criterion is met Ensure:
Hierarchical sparse solution z ∗ V. S
IMULATIONS
To test the efficiency of the HiHTP-algorithm in the mul-tiuser setting, the following tests were conducted: We assumefor simplicity that Alice only consists of one antenna and thatthere are N r Bobs, from which only µ < N r are active. Themulti-antenna setting will offer further possibilities to improvethe performance, since the correlations between the antennaswill introduce more structure into the model. The completionof this model and the design of an efficient algorithm for it iscurrently investigated by the authors. For each of the N r usersa σ -sparse channel h k ∈ R N d was drawn with the locationsof the non-zeros distributed uniformly and entries drawn fromthe standard normal distribution. The signals were computedas x k = Q k b k were Q ∈ R N × E is a random matrix withentries Q i,j ∼ N (0 , and b ∈ R E is s -sparse with values in {− , } if the user is active, and if the user is not active.This results in the data y ∈ R N , y = N r (cid:88) k =1 h k (cid:126) Q k b k . (24)The measurement matrix M ∈ R × N d · E · N r is such that y = M vec( X ) , (25)with X = [ b h T . . . b N r h TN r ] . The experiments were con-ducted with N = 1024 , N d = E = 128 and N r = 10 .The number of active users varied from 2 to 5 and thesparsity levels of h and b varied from 2 to 15. An experimentwas classified successful, if the support of X was recoveredcorrectly and the residual was below − . The graphics below s Fig. 1. Recovery rate for 2 of 10 active users s Fig. 2. Recovery rate for 3 of 10 active users show the rate of successful recovery for varying number ofactive users, averaged over 20 runs per setup. The x- and y-axis show the channel sparsity µ and the signal sparsity s ,respectively. VI. C ONCLUSIONS
We have proposed a new access scheme for IoT applicationsin which many low complexity devices spontaneously senddata to a base station in an uncoordinated fashion and includeda physical layer security scheme. The base station is able torecover the signals as well as the channels by employing afast, scalable blind deconvolution algorithm called HiHTP.The benefit of this novel approach is that it requires nopilot signaling to measure the channels, thus greatly reducingthe overhead. This is crucial for next generation wirelesscommunication, where the number of devices will increasedramatically. We have provided numerical experiments thatshow the feasibility of our approach and illustrate the trade-offbetween the number of active users, the required sparsity of thesignals and the channel sparsity. The adaptation of our HiHTPalgorithm to the multi-user, multi-antenna case, its robustness s Fig. 3. Recovery rate for 4 of 10 active users s Fig. 4. Recovery rate for 5 of 10 active users to noisy measurements and the proof of rigorous performanceguarantees will be a topic of future research.VII. A
CKNOWLEDGEMENTS
We would like to thank the DFG within grants WU 598/7-1, WU 598/8-1, and EI 519/9-1 (DFG Priority Programon Compressed Sensing) and the Templeton Foundation forsupport. This work has also been performed in the frameworkof the Horizon 2020 project ONE5G (ICT-760809) receivingfunds from the European Union. The authors would liketo acknowledge the contributions of their colleagues in theproject, although the views expressed in this contribution arethose of the authors and do not necessarily represent theproject. R
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