Physical Layer Security of Generalised Pre-coded Spatial Modulation with Antenna Scrambling
aa r X i v : . [ c s . I T ] J un Physical Layer Security of Generalised Pre-codedSpatial Modulation with Antenna Scrambling
I. I
NTRODUCTION
In the last decade or so, physical layer security had beenan active research area drawing substantial attentions andcollaborative efforts across different societies. Complementaryto the conventional application layer security approach domi-nantly based on cryptography that is potentially vulnerable toquantum computationally capable eavesdropper, physical layersecurity was envisioned of providing the ultimate securityfrom the information theoretical point of view. This areaof research was pioneered by Shannon’s foundation workon secrete communications [1], where the ’one-time pad’was proposed in order to achieve the perfect security. Thislandmark finding inspired further intriguing research on thephysical layer security of ’wire-tap’ channel [2], where theeavesdropper’s channel between Alice and Eve constitutesa degraded version of the main channel between Alice andBob. In particular, the important notion of security capacitywas formally defined as the capacity difference between themain channel and the eavesdropper’s channel, where a positivesecurity capacity enables physical layer security.The research of physical layer security became much morecomplicated yet interesting when wireless communications isconsidered, where the eavesdropper’s channel becomes notnecessarily degraded [3], [4], [5] in comparison to the mainchannel owing to the fast variations of the wireless channelpropagation. As a benefit, the classic solution exploited thefast varying Channel State Information (CSI) of the mainchannel as the security key [6], which was assumed to beonly privately known between Alice and Bob. This solutionbecame naturally inappropriate in the graver condition, wherethe CSI of the main channel became also publicly known toEve. In this challenging scenario, various signal processingadvances appeared particularly in the so-called Multiple InputMultiple Output Multiple Eavesdropper (MIMOME) physicallayer security model [7], [8]. More explicitly, when eavesdrop-per’s channel is optimistically assumed to be known to Alicehaving multiple antennas, intelligent beam-forming constitutedeffective solutions. On the other hand, when eavesdropper’schannel is unknown to Alice but its dimension (number ofantennas) is known, the technique of generating artificialnoise [9] by Alice was found to be powerful, where theartificial noise was pre-processed to lie in the null space of the
The financial support of the EPSRC under the India-UK Advanced Tech-nology Centre (IU-ATC), that of the EU under the Concerto project as wellas that of the European Research Council’s (ERC) Advanced Fellow Grant isgratefully acknowledged. main channel so as to only affect Eve adversely . However,one common limitation of the above two distinct approacheswas the stringent dimensionality requirement, where Alicemust have more antennas than Eve.In contrast to the above two design philosophies, we proposea physical layer security solution, without the necessity ofobeying the dimensionality requirement, based on our pre-viously proposed Generalised Pre-coded Spatial Modulation(GPSM) scheme [12], [13]. Let us now briefly introduce thebackground of GPSM scheme before embarking on our noveldesign. Spatial Modulation (SM) constituted novel MultipleInput Multiple Output (MIMO) technique, which conveysextra information by appropriately activating the Transmit
Antenna (TA) indices, in addition to the classic modulationschemes used on each activated TAs, as detailed in [14], [15].By contrast, the GPSM scheme is capable of conveying extrainformation by appropriately selecting the
Receive
Antenna(RA) indices in addition to the classic modulation schemesloaded on each activated RAs, as detailed in [13]. To elaboratea little more, in [13], comprehensive performance comparisonswere carried out between the GPSM scheme as well asthe conventional MIMO scheme and the associated detectioncomplexity issues were discussed. Furthermore, a range ofpractical issues were investigated, namely the detrimentaleffects of realistic imperfect Channel State Information at theTransmitter (CSIT), followed by a low-rank approximation in-voked for large-dimensional MIMOs. As a further exploitation,error probability, achievable rate and capacity analysis wereconducted in [16]. When physical layer security is consid-ered, early work on SM considered its straightforward use,where the security capacity is firstly reported under variousSNRs [17] and then compared to that of the single input singleoutput fading channel [18].We now advocate a novel physical layer security solutionthat is unique to our previously proposed GPSM scheme withthe aid of the proposed antenna scrambling . The novelty andcontribution of our paper lies in three aspects: • principle : we introduce a ’security key’ generated atAlice that is unknown to both Bob and Eve, where thedesign goal is that the publicly unknown security keyonly imposes barrier for Eve. • approach : we achieve it by conveying useful informationonly through the activation of RA indices, which is in Similarly, exploiting helper to generate artificial interference towards Evein the cooperative communications scenario is also found effective [10], [11]. urn concealed by the unknown security key in terms ofthe randomly scrambled symbols used in place of theconventional modulated symbols in GPSM scheme. • design : we consider both Circular Antenna Scrambling(CAS) and Gaussian Antenna Scrambling (GAS) in detailand the resultant security capacity of both designs arequantified and compared.As a design merit, we find that the proposed physical layersecurity solution imposes great challenge on Eve and providespositive security capacity under both the conventional settingof having less antennas at Eve than at Alice as well as thedisadvantageous setting that Alice is less equipped than Eve,which is largely neglected in the literature.The organisation of the paper is as follows. In Section II, weintroduce the physical layer security model considered, alongwith the brief introduction of GPSM for self completenessin addition to the straightforward use of GPSM scheme interms of physical layer security. This motivates our proposedantenna scrambling in Section III, where both CAS and GASare discussed in detail together with their security capacityexpressions. We present numerical results in Section IV andconclude in Section V.II. P HYSICAL L AYER S ECURITY OF
GPSMConsider a typical physical layer security scenario with N t TAs at Alice and N r RAs at Bob in the presence of N e RAsat Eve, where we assume N t ≥ N r with no constraints put on N e . Throughout the paper, we further assume that the MIMOchannel between Alice and Bob HHH B ∈ C N r × N t is knownbetween each other and is also publicly available at Eve, whilethe MIMO channel between Alice and Eve HHH E ∈ C N e × N t isonly known at Eve, where both HHH B and HHH E are frequency-flat Rayleigh fading with each entry being independent. Inthis MIMOME set-up, we have the following typical systemof equations constituted by the coexistence of legal and illegalcommunications yyy B = HHH B xxx + B , (1) yyy E = HHH E xxx + E , (2)where B ∈ C N r × and E ∈ C N e × are the circularlysymmetric complex Gaussian noise vector with each entryhaving a zero mean and a variance of σ B and σ E , i.e. wehave E [ || B || ] = N r σ B and E [ || E || ] = N e σ E . The noise B and E are independent between each other and are alsoindependent from the transmit signal xxx ∈ C N t × . Finally, thesecurity capacity C S of the above MIMOME set-up is definedas the capacity difference between Alice and Bob C B andAlice and Eve C E , which is formerly expressed as C S = C B − C E . (3) A. Transceiver Structure of GPSM
Let us now elaborate more on our previously proposedGPSM scheme.
1) Conceptual Description:
In our GPSM scheme, a totalof N a < N r RAs are activated at Bob so as to facilitate thesimultaneous transmission of N a data streams between Aliceand Bob, where the particular pattern of the N a RAs activatedconveys extra information in form of so-called spatial symbolsin addition to the information carried by the conventionalmodulated symbols mapped to them. Hence, the number ofbits in GPSM conveyed by a spatial symbol becomes k ant = ⌊ log ( |C t | ) ⌋ , where the set C t contains all the combinationsassociated with choosing N a activated RAs out of N r RAs.As a result, the total number of bits transmitted by the GPSMscheme is k eff = k ant + N a k mod , where k mod = log ( M ) denotes the number of bits per symbol of a conventional M -ary modulation scheme and its alphabet is denoted by A .Finally, it is plausible that the conventional MIMO schemebetween Alice and Bob with multiplexing of N a = N r datastreams conveys a total of k eff = N r k mod bits altogether.For assisting further discussions, we also let C ⊂ C t be theset of selected activation patterns, C ( k ) and C ( k, i ) denote the k th RA activation pattern and the i th activated RA in the k thactivation pattern, respectively.
2) Mathematical Description:
Let sss km be an explicit repre-sentation of a so-called super-symbol sss ∈ C N r × , indicatingthat the RA pattern k at Bob is activated and N a conventionalmodulated symbols bbb m = [ b m , . . . , b m Na ] T ∈ C N a × aretransmitted, where we have b m i ∈ A and E [ | b m i | ] = 1 , ∀ i ∈ [1 , N a ] . In other words, we have the relationship sss = sss km = ΩΩΩ k bbb m , (4)where ΩΩΩ k = III [: , C ( k )] is constituted by the specificallyselected columns determined by C ( k ) of an identity matrixof III N r . Following pre-coding of PPP ∈ C N t × N r , the resultanttransmit signal xxx from Alice may be written as xxx = p β/N a PPPsss, (5)where β is designed for maintaining E [ || xxx || ] = 1 . Theconventional design of GPSM allows no energy leaks intothe unintended RA patterns at Bob. Hence, the classic linearChannel Inversion (CI)-based pre-coding [19], [20] may beused, which is formulated as PPP = HHH HB ( HHH B HHH HB ) − , (6)where the power-normalisation factor of the output power afterpre-coding is β = N r / Tr[(
HHH B HHH HB ) − ] .As a result, the signal observed at the N r RAs of Bob maybe written as yyy B = p β/N a HHH B PPPsss km + B . (7)Consequently, the joint detection of both the conventionalmodulated symbols bbb m and of the spatial symbol k obeys theMaximum Likelihood (ML) criterion, which is formulated as [ ˆ m , . . . , ˆ m N a , ˆ k ] = arg min sss ℓn ∈S || yyy B − r βN a HHH B PPPsss ℓn || , (8)where S = C × A N a denotes the super-alphabet. Alternatively,decoupled or separate detection may also be employed, whichtreats the detection of the conventional modulated symbols bbb m nd the spatial symbol k separately. In this reduced-complexityvariant, we have ˆ k = arg max ℓ ∈ [1 , |C| ] N a X i =1 | y B C ( ℓ,i ) | , (9) ˆ m i = arg min n i ∈ [1 ,M ] | y B C (ˆ k,i ) − r βN a hhh B C (ˆ k,i ) ppp C (ˆ k,i ) b n i | , (10)where hhh B C (ˆ k,i ) is the C (ˆ k, i ) th row of HHH B representing thechannel between the C (ˆ k, i ) th RA at Bob and the transmitter atAlice, while ppp C (ˆ k,i ) is the C (ˆ k, i ) th column of PPP . Thus, correctdetection is declared, when we have ˆ k = k and ˆ m i = m i , ∀ i . B. Security Capacity of GPSM
Since the MIMO channel
HHH E is intentionally and practi-cally assumed unknown to Alice, we now investigate the secu-rity capacity of GPSM scheme dispensing with the knowledgeof HHH E at Alice.
1) DCMC Capacity:
The security capacity as expressed in(3) for GPSM scheme relies on the Discrete input Continuousoutput Memoryless Channel (DCMC) capacity calculations of C B and C E , respectively. It is known that Shannon’s channelcapacity is obtained, when the input signal obeys a Gaussiandistribution [21]. Our GPSM scheme is special in the sensethat the spatial symbol conveys integer values constitutedby the RA pattern index, which does not obey the shapingrequirements of Gaussian signalling. This implies that thechannel capacity of the GPSM scheme depends on a mixture ofa continuous and a discrete input. Hence, for simplicity’s sake,we discuss the DCMC capacity in the context of discrete-inputsignalling for both the spatial symbol and for the conventionalmodulated symbols mapped to it.Before calculating C B and C E for (1) and (2) employingGPSM scheme, we first provide a general expression ofcalculating the DCMC capacity that will be used throughoutthe paper. The DCMC capacity is obtained by maximizingthe mutual information between discrete input uuu ∈ B andcontinuous output zzz , where we have C = max p ( uuu τ ) , ∀ τ I ( uuu ; zzz )= max p ( uuu τ ) , ∀ τ |B| X τ =1 ∞ Z −∞ p ( zzz, uuu τ ) log p ( zzz | uuu τ ) P |B| ǫ =1 p ( zzz, uuu ǫ ) ! dzzz, (11)furthermore, we have log p ( zzz | uuu τ ) P |B| ǫ =1 p ( zzz, uuu ǫ ) ! = log p ( zzz | uuu τ ) P |B| ǫ =1 p ( zzz | uuu ǫ ) p ( uuu ǫ ) ! = − log |B| |B| X ǫ =1 p ( zzz | uuu ǫ ) p ( zzz | uuu τ ) = log ( |B| ) − log |B| X ǫ =1 Θ ǫ,τ , (12) Although our security capacity discussion later does not rely on a particulardetection scheme, we include them for self-completeness. where we let Θ ǫ,τ = p ( zzz | uuu ǫ ) /p ( zzz | uuu τ ) . By substituting (12)into (11) and exploiting that (11) is maximized when we haveequal probable p ( uuu τ ) , we have C = log ( |B| ) − |B| |B| X τ =1 E log |B| X ǫ =1 Θ ǫ,τ . (13)
2) Security Capacity:
By substituting (5) into (1) and (2),we have the explicit relation yyy B = p β/N a HHH B PPPsss + B , (14) yyy E = p β/N a HHH E PPPsss + E . (15)Hence, with perfect knowledge of GGG B = p β/N a HHH B PPP and
GGG E = p β/N a HHH E PPP being the equivalent channels, theconditional probability of receiving yyy B and yyy E given that sss τ was transmitted are formulated as p ( yyy B | sss τ ) ∝ exp (cid:26) −|| yyy B − GGG B sss τ || σ B (cid:27) , (16) p ( yyy E | sss τ ) ∝ exp (cid:26) −|| yyy E − GGG E sss τ || σ E (cid:27) . (17)As a result, we have Θ Bǫ,τ = exp (cid:26) −||
GGG B ( sss τ − sss ǫ ) + B || + || B || σ B (cid:27) , (18) Θ Eǫ,τ = exp (cid:26) −||
GGG E ( sss τ − sss ǫ ) + E || + || E || σ E (cid:27) , (19)and hence the DCMC capacity of C B and C E for (14) and(15) may be obtained upon substituting (18) and (19) into(13), respectively and also by letting |B| = |C| M N a in (13).Finally, the security capacity is obtained by evaluating (3) asa difference between C B and C E .
3) Performance:
Fig 1 shows the security capacityof GPSM scheme employing CI based pre-coding when [ N t , N r ] = [8 , and using QPSK modulation having differentnumber of activated RAs N A = [1 , , when N e = N r = 4 (dash) and N e = N t = 8 (solid). It is clearly shown inFig 1 that when N e = N r = 4 , the security capacity ispositive for all N a considered and activating more RAs resultsinto higher security capacity. On the other hand, no positivesecurity capacity is found when N e = N t = 8 for all N a considered as shown by solid curves. Indeed, N e = N t constitutes a challenge setting in the physical layer securityliterature, where typically N e < N t was considered. This isbecause even though the eavesdropper’s channel is known byAlice mysteriously, the celebrated strategies of transmittingin the null space of Eve or of minimizing the signal leakagetowards Eve leave no degree of freedom for communicatingbetween Alice and Bob if N e < N t was violated. This isalso true for the typical strategy of generating artificial noiseby Alice, where the underlining assumption of N e < N t wasalso explicitly imposed.III. S ECURITY V IA A NTENNA S CRAMBLING
We now propose a novel physical layer security scheme exclusive to GPSM scheme with the aid of antenna scrambling.
20 −10 0 10 20 30−1−0.500.511.5 SNR [dB] S e c u r i t y C apa c i t y ( b i t s / s / H z ) [Na,Ne] = [1,8][Na,Ne] = [2,8][Na,Ne] = [3,8][Na,Ne] = [1,4][Na,Ne] = [2,4][Na,Ne] = [3,4][Nt,Nr] = [8,4] Fig. 1: Security capacity of GPSM scheme employing CI basedpre-coding when [ N t , N r ] = [8 , and using QPSK modula-tion having different number of activated RAs N A = [1 , , when N e = N r = 4 (dash) and N e = N t = 8 (solid). A. Motivation and Principle
In the security capacity of GPSM analysed in Section II-Band showed in Fig 1, we found that the explicit and completeknowledge of B known to both Bob and Eve constitutes onepotential source of confidentiality leakage, which allows Eveto perform exhaustive search over B when Eve has unboundedcomputational capability.Based on this backdrop, we propose to only convey usefulinformation through the RA patterns in terms of spatial sym-bols, while the conventional modulated symbols bbb m belongingto a finite alphabet of A N a are purposely replaced by randomlygenerated scrambled symbols, which are changed at symbolrate and are unknown to both Bob and Eve. In other words, theuseful information in spatial domain is concealed by the time-domain scrambled symbols with infinite possibilities, whichare in turn acted as security key.Thus, the aim is to investigate proper scrambled symbolsthat result in significant barriers only for Eve to retrievethe useful information in spatial domain, but not for Bob.In this light, we consider both Circular Antenna Scrambling(CAS) and Gaussian Antenna Scrambling (GAS), where thescrambled symbols in the former and in the latter are drawnfrom a unit circle with uniformly distributed phases andfrom a complex Gaussian distribution with zero-mean and(normalised) unity variance, respectively. B. Security Capacity of GPSM with CAS
When CAS is considered, we rewrite (4) with conven-tional modulated symbols bbb m replaced by random symbols eee = [ e jθ , . . . , e jθ Na ] T ∈ C N a × drawn from a unit circlewith uniformly distributed phase θ i , ∀ i ∈ [1 , N a ] , namely wehave sss = ΩΩΩ τ eee, (20) when the τ th RA pattern is activated at Bob. It is thus clearthat the useful information is contained in ΩΩΩ τ only and isconcealed by the phase rotations of eee .Now, if we followed the same procedure as in Section II-B2and adopted the same equations as expressed in (14) and (15)by treating sss as a whole in order to obtain the DCMC capacityof C B and C E , the conditional probability calculations of (16)and (17) become problematic. This is because the symbol sss of (20) belongs to an infinite alphabet in CAS, where itsenumeration is impossible.On the other hand, we may only consider the spatialdomain information contained in ΩΩΩ τ with finite alphabet of C . However, the unknown randomly scrambled symbols of eee ,which is also changed at symbol rate, analogously contributeto strong channel uncertainty to both Bob and Eve. Thismakes the conditional probability calculations of (16) and(17) become again intractable since coherent detection is notallowed with unknown eee .We thus discuss the security capacity of GPSM scheme withCAS replying on non-coherent DCMC capacity calculations of C B and C E based on the non-coherent transformation of (14)and (15) with (20) been substituted, which are given as follows rrr B = | GGG B ΩΩΩ τ eee + B | = | X N a v =1 ggg B C ( τ,v ) e jθ v + B | , (21) rrr E = | GGG E ΩΩΩ τ eee + E | = | X N a v =1 ggg E C ( τ,v ) e jθ v + E | , (22)where we have ggg B C ( τ,v ) and ggg E C ( τ,v ) stand for the C ( τ, v ) thcolumn of GGG B and GGG E respectively. Let us further elaboratea bit more on two cases as follows. N a = 1 : In this special case, we have a simplifiedexpression of (21) and (22) given as rrr B = | ggg B C ( τ, e jθ + B | , (23) rrr E = | ggg E C ( τ, e jθ + E | , (24)Hence, by normalising with respect to σ B, = σ B / and σ E, = σ E / , it is clear that for the i th entry of rrr B and rrr E , we have r B,i /σ B, ∼ χ h λ B,i = | g B C ( τ, ,i | /σ B, i , (25) r E,i /σ E, ∼ χ h λ E,i = | g E C ( τ, ,i | /σ E, i , (26)where χ ( λ B,i ) and χ ( λ E,i ) stand for the non-central chi-square distribution with degree of freedom of two and non-centrality given by | g B C ( τ, ,i | /σ B, and | g E C ( τ, ,i | /σ E, asa result of | e jθ | = 1 , respectively. In particular, when CIpre-coding of (6) is employed, where GGG B = p β/N a HHH B PPP = p β/N a III N r , (27)we have λ B,i = 0 , ∀ i
6∈ C ( τ ) for inactivated RAs and λ B,i = β/N a σ B, , ∀ i ∈ C ( τ ) for activated RAs.Thus the conditional probability of receiving rrr B and rrr E given that ΩΩΩ τ was transmitted and subjected to the knowledgef equivalent channels GGG B and GGG E but dispensing with theknowledge of eee are formulated as p ( rrr B | ΩΩΩ τ ) = N r Y i =1 f χ h | g B C ( τ, ,i | /σ B, i r B,i σ B, ! , (28) p ( rrr E | ΩΩΩ τ ) = N e Y i =1 f χ h | g E C ( τ, ,i | /σ E, i r E,i σ E, ! . (29)As a result, we have Θ Bǫ,τ = Y N r i =1 f χ h | g B C ( ǫ, ,i | /σ B, i (cid:0) r B,i /σ B, (cid:1)Y N r i =1 f χ h | g B C ( τ, ,i | /σ B, i (cid:16) r B,i /σ B, (cid:17) , (30) Θ Eǫ,τ = Y N e i =1 f χ h | g E C ( ǫ, ,i | /σ E, i (cid:0) r E,i /σ E, (cid:1)Y N r i =1 f χ h | g E C ( τ, ,i | /σ E, i (cid:16) r E,i /σ E, (cid:17) , (31)and hence when N a = 1 , the non-coherent DCMC capacityof C B and C E for (23) and (24) may be obtained by letting |B| = |C| and substituting (30) and (31) into (13). Finally, thesecurity capacity is obtained by evaluating (3). N a > : Now we consider (21) and (22) when N a > ,where in this case we have r B,i /σ B, ∼ χ (cid:20) λ B,i = | X N a v =1 g B C ( τ,v ) ,i e jθ v | /σ B, (cid:21) , (32) r E,i /σ E, ∼ χ (cid:20) λ E,i = | X N a v =1 g E C ( τ,v ) ,i e jθ v | /σ E, (cid:21) . (33)In contrast to N a = 1 , since eee is unknown to both Bob andEve, the non-centrality of both λ B,i and λ E,i of (32) and (33)become also unavailable in general. This is because the impactof the unknown eee adds extra constructive and destructivefading. As a result, the conditional probability calculationssimilar to (28) and (29) may be intractable.In particular, when the CI pre-coding of (6) is employedwith the beneficial structure of (27), we have λ B,i = 0 , ∀ i ( τ ) for inactivated RAs and λ B,i = β/N a σ B, , ∀ i ∈ C ( τ ) for activated RAs. Hence, the knowledge of eee becomes un-necessary for Bob in order to evaluate (32). However, thebeneficial structure similar to (27) is not seen for Eve as HHH E PPP ∈ C N e × N r , so the knowledge of eee becomes crucial for Eve in order to evaluate (33), resulting into a barrier.Consequently, with the knowledge of both HHH B and HHH E ,Eve is forced to be ’pretended’ as Bob by conducting post-processing on yyy E of (15) as ˜ y ˜ y ˜ y E = HHH B ( HHH HE HHH E ) − HHH HE yyy E , = GGG B ΩΩΩ τ eee + HHH B ( HHH HE HHH E ) − HHH HE E , (34)where by letting ˜ w ˜ w ˜ w E = HHH B ( HHH HE HHH E ) − HHH HE E , the non-coherent transformation of (34) is ˜ r ˜ r ˜ r E = | GGG B ΩΩΩ τ eee + ˜ w ˜ w ˜ w E | = | X N a v =1 ggg B C ( τ,v ) e jθ v + ˜ w ˜ w ˜ w E | . (35)Hence, by being forced to equip higher number of antennasthan Alice ( N e ≥ N t ) in order to perform the above post-processing of (34), Eve finally becomes also capable of retriev-ing the information concealed in the spatial domain dispensing with the knowledge of eee at the cost of noise amplificationprovided σ E > . More explicitly, by normalising withrespect to ˜ σ E, = ˜ σ E / , we have ˜ r E,i / ˜ σ E, ∼ χ (˜ λ E,i ) with ˜ λ E,i = 0 , ∀ i
6∈ C ( τ ) for inactivated RAs and ˜ λ E,i = β/N a ˜ σ E, , ∀ i ∈ C ( τ ) for activated RAs.Thus, when the CI pre-coding of (6) is employed and N e ≥ N t , the conditional probability of receiving rrr B and ˜ r ˜ r ˜ r E given that ΩΩΩ τ was transmitted and subjected to the knowledgeof equivalent channels GGG B and GGG E but dispensing with theknowledge of eee are formulated as p ( rrr B | ΩΩΩ τ ) = N r Y i =1 f χ [ [ i ∈C ( τ )] β/N a σ B, ] r B,i σ B, ! , (36) p (˜ r ˜ r ˜ r E | ΩΩΩ τ ) = N r Y i =1 f χ [ [ i ∈C ( τ )] β/N a ˜ σ E, ] ˜ r E,i ˜ σ E, ! , (37)where [ · ] is the indicator function that returns one (zero)when the entry is true (false). As a result, we have Θ Bǫ,τ = Y N r i =1 f χ [ [ i ∈C ( ǫ )] β/N a σ B, ] (cid:0) r B,i /σ B, (cid:1)Y N r i =1 f χ [ [ i ∈C ( τ )] β/N a σ B, ] (cid:16) r B,i /σ B, (cid:17) , (38) Θ Eǫ,τ = Y N r i =1 f χ [ [ i ∈C ( ǫ )] β/N a ˜ σ E, ] (cid:0) ˜ r E,i / ˜ σ E, (cid:1)Y N r i =1 f χ [ [ i ∈C ( τ )] β/N a ˜ σ E, ] (cid:16) ˜ r E,i / ˜ σ E, (cid:17) , (39)and hence when N a > and CI pre-coding is employed with N e ≥ N t , the non-coherent DCMC capacity of C B and C E for (21) and (35) may be obtained by letting |B| = |C| andsubstituting (38) and (39) into (13). Then, the security capacityis obtained by evaluating (3). Finally, note that when N e < N t ,Eve becomes not capable of gaining any information containedin the RA patterns without knowing eee even though σ E . C. Security Capacity of GPSM with GAS
When GAS is considered, we rewrite (4) with conventionalmodulated symbols bbb m replaced by random symbols nnn =[ n , . . . , n N a ] T ∈ C N a × drawn from a complex Gaussiandistribution with zero-mean and (normalised) unity variance,i.e. n i ∈ CN (0 , , namely we have sss = ΩΩΩ τ nnn, (40)when the τ th RA pattern is activated at Bob. Now, whencompared with the CAS case, the uncertainties arise from bothphase and amplitude. From certain perspective, we may treatGAS as introducing multiplicative artificial noise.Similar to CAS, since the symbol sss belongs to an infinitealphabet in GAS, we may consider the finite alphabet of C forthe spatial domain information contained in ΩΩΩ τ . But unlike inCAS, now coherent DCMC capacity calculations of C B and C E are possible. By substituting (40) into (14) and (15), wehave yyy B = GGG B ΩΩΩ τ nnn + B = X N a v =1 ggg B C ( τ,v ) n v + B , (41) yyy E = GGG E ΩΩΩ τ nnn + E = X N a v =1 ggg E C ( τ,v ) n v + E . (42)hen it is clear that for the i th entry of yyy B and yyy E as expressedin (41) and (42), we have y B,i ∼ CN (0 , X N a v =1 | g B C ( τ,v ) ,i | + σ B ) , (43) y E,i ∼ CN (0 , X N a v =1 | g E C ( τ,v ) ,i | + σ E ) , (44)where without the explicit knowledge of nnn , the conditionalprobability of receiving yyy B and yyy E given that ΩΩΩ τ was trans-mitted and subjected to the knowledge of equivalent channels GGG B and GGG E are formulated as p ( yyy B | ΩΩΩ τ ) ∝ exp ( − N r X i =1 | y B,i | P N a v =1 | g B C ( τ,v ) ,i | + σ B ) , (45) p ( yyy E | ΩΩΩ τ ) ∝ exp ( − N e X i =1 | y E,i | P N a v =1 | g E C ( τ,v ) ,i | + σ E ) . (46)As a result, we have Θ Bǫ,τ = exp ( N r X i =1 | y B,i | / N a X v =1 | g B C ( τ,v ) ,i | + σ B !) exp ( N r X i =1 | y B,i | / N a X v =1 | g B C ( ǫ,v ) ,i | + σ B !) , (47) Θ Eǫ,τ = exp ( N e X i =1 | y E,i | / N a X v =1 | g E C ( τ,v ) ,i | + σ E !) exp ( N e X i =1 | y E,i | / N a X v =1 | g E C ( ǫ,v ) ,i | + σ E !) , (48)and hence dispensing with the knowledge of nnn , the DCMCcapacity of C B and C E for (41) and (42) may be obtainedby letting |B| = |C| and substituting (47) and (48) into (13).Finally, the security capacity is obtained by evaluating (3).However, a further look results into the following Lemma Lemma III.1.
The DCMC capacity calculation of C B basedon (47) for Bob is ≤ C B ≤ log ( |C| ) , but the DCMCcapacity calculation of C E based on (48) returns zero capacityfor Eve, i.e. we have C E = 0 .Proof. Let us rewrite (13) here by setting |B| = |C| as C = log ( |C| ) − |C| |C| X τ =1 E log |C| X ǫ =1 Θ ǫ,τ , (49)and we further expand the following term as E log |C| X ǫ =1 Θ ǫ,τ = E " − log P |C| ǫ =1 Θ ǫ,τ ≥ − log E " P |C| ǫ =1 Θ ǫ,τ ≥ − log P |C| ǫ =1 E [Θ ǫ,τ ]= log |C| X ǫ =1 E [Θ ǫ,τ ] = log |C| X ǫ =1 E (cid:2) e A − B (cid:3) ≥ log |C| X ǫ =1 ,ǫ = τ e E [ A − B ] , (50) where the inequities involved during the above derivationsare followed from Jensen’s inequality. Further, A and B areshorthand for the inner term in the exponential operator ofboth the numerator and denominator, respectively, either of(47) or of (48). Substituting (50) into (49), we have C ≤ log ( |C| ) − |C| |C| X τ =1 log |C| X ǫ =1 ,ǫ = τ e E [ A − B ] (51)1) When C B is considered, let us first define C c = C ( τ ) ∩C ( ǫ ) as the set containing the commonly activated RAs oftwo different activation patterns and thus C d = ¯ C c denotes thecomplementary set containing the activated RAs in difference.We further define C u ( τ ) = C ( τ ) ∩ C d and C u ( ǫ ) = C ( ǫ ) ∩C d as the set containing the activated RAs uniquely found in C ( τ ) and C ( ǫ ) , respectively. With the above definitions and byexploiting the beneficial property of (27), we have E [ A − B ] = |C d | / X i =1 E [Υ i ] β/N a + σ B − E [Υ i ] σ B . (52)where we have Υ i = | y B C u ( τ,i ) | − | y B C u ( ǫ,i ) | and E [Υ i ] = β/N a . Now, when σ B , the summation of (52) approaches −∞ , so (50) approaches zero and (51) results into C B ≤ log ( |C| ) . On the other hand, when σ B
7→ ∞ , the summationof (52) approaches zero, so (50) approaches log ( |C| ) and (51)results into C B ≤ . Since we also must have C B ≥ , weend up with C B = 0 . Overall, we have ≤ C B ≤ log ( |C| ) .2) To see C E = 0 , it is simple to see from (48) that E [ A − B ] = N e X i =1 E (cid:20) | y E,i | A ′ + σ E (cid:21) − E (cid:20) | y E,i | B ′ + σ E (cid:21) = 0 , (53)where A ′ = P N a v =1 | g E C ( τ,v ) ,i | and B ′ = P N a v =1 | g E C ( ǫ,v ) ,i | andwe have E [ A ′ ] = E [ B ′ ] . As a result, (50) approaches log ( |C| ) and hence (51) results into C E ≤ . Since we also must have C E ≥ , we end up with C E = 0 .Lemma III.1 suggests that Eve cannot follow the same ap-proach to retrieve the useful information as Bob. Thus, sim-ilarly to the context of CAS, by being forced to equip moreantennas than Alice ( N e ≥ N t ) and at the cost of noiseamplification provided σ E > , Eve is again tried to be’pretended’ as Bob by conducting post-processing on yyy E of(42) to arrive at ˜ y ˜ y ˜ y E = GGG B ΩΩΩ τ nnn + HHH B ( HHH HE HHH E ) − HHH HE E . (54)Hence, we have ˜ y E,i ∼ CN (0 , X N r v =1 | g B C ( τ,v ) ,i | + ˜ σ E ) , (55)and after a few manipulations, we arrive at Θ Eǫ,τ = exp ( N r X i =1 | ˜ y E,i | / N a X v =1 | g B C ( τ,v ) ,i | + ˜ σ E !) exp ( N r X i =1 | ˜ y E,i | / N a X v =1 | g B C ( ǫ,v ) ,i | + ˜ σ E !) , (56) PSM CAS GAS Θ Bǫ,τ , Θ Eǫ,τ (18),(19) N a = 1 N a > (47), (56)(30), (31) (38), (39) TABLE I: Summarized findings of { Θ Bǫ,τ , Θ Eǫ,τ } .and hence dispensing with the knowledge of nnn , the DCMCcapacity of C E for (54) may be obtained by letting |B| = |C| and substituting (56) into (13). Then, the security capacityis obtained by evaluating (3). Finally, again note that when N e < N t , Eve becomes not capable of gaining any informationcontained in the RA patterns without knowing eee even though σ E , since the operation of (54) is not allowed. D. Further Discussions1) Example:
We now provide an example to better aidthe conceptual understanding of our proposed physical layersecurity of GPSM with antenna scrambling. Fig 2 shows theplot of 1000 samples of the received signal yyy B at Bob and yyy E at Eve when CI pre-coding aided GPSM is employed at Alicewith [ N t , N r , N e ] = [8 , , under a particular MIMO channelrealisation of HHH B and HHH E when SNR is set to 30dB to removethe disturbance of noise. Thus we focus on the effect of CASand GAS on the realisations of yyy B and yyy E . In Fig 2, we includethree different cases correspond to three rows of subplots inpair for yyy B (left) and yyy E (right), respectively. On each subplot,the received signal samples of each individual RA are detailedin four miniplots, each with horizontal axis and vertical axisrepresent real and imaginary part of y B,i , ∀ i ∈ [1 , N r ] or y E,i , ∀ i ∈ [1 , N e ] .The first row of Fig 2 illustrate the scenario, where N a = 1 and employing CAS with first RA been activated. It can beseen from Fig 2a that the received signal samples of the firstRA at Bob is clearly distinguishable with respect to that ofthe rest RAs. Moreover, being a circle at the first RA impliesthat its non-coherent transformation (magnitude) constitutes aconstant value. On the other hand, the four distinguishablecircles observed at Eve from Fig 2b imply that Eve is alsounaffected by CAS, but failing to capture the correct patternas compared to Bob.When the second row of Fig 2 is considered, where N a > and employing CAS with first and second RAs been activated.It can be seen from Fig 2c that the received signal samplesof the first two RAs are stand-out clearly from the rest, beingunaffected by the unknown knowledge of eee . However, Eveis completely messed up by the confusion of CAS as seenfrom Fig 2d, where a large overlap can be found in these fourminiplots. Finally, we consider the third row of Fig 2 whenGAS is employed. Again, clear patterns are observed at Bobas seen in Fig 2e but indistinguishable clouds are found at Eveas seen in Fig 2f.
2) Remarks:
We infer that the effect of CAS is analogouslyto introduce artificial phase ambiguity at Eve, while the effectof GAS is analogously to introduce artificial multiplicativenoise. It is also worth noting that the proposed antennascrambling is different from the so-called directional modu-lation [22], [23], [24], although the useful information is also −1 0 1−101 Re(y
B,1 ) I m ( y B , ) −1 0 1−101 Re(y B,2 ) I m ( y B , ) −1 0 1−101 Re(y B,3 ) I m ( y B , ) −1 0 1−101 Re(y B,4 ) I m ( y B , ) (a) N a = 1 CAS (Bob) −1 −0.5 0 0.5 1−1−0.500.51 Re(y
E,1 ) I m ( y E , ) −1 −0.5 0 0.5 1−1−0.500.51 Re(y E,2 ) I m ( y E , ) −1 −0.5 0 0.5 1−1−0.500.51 Re(y E,3 ) I m ( y E , ) −1 −0.5 0 0.5 1−1−0.500.51 Re(y E,4 ) I m ( y E , ) (b) N a = 1 CAS (Eve) −2 −1 0 1 2−2−1012 Re(y
B,1 ) I m ( y B , ) −2 −1 0 1 2−2−1012 Re(y B,2 ) I m ( y B , ) −2 −1 0 1 2−2−1012 Re(y B,3 ) I m ( y B , ) −2 −1 0 1 2−2−1012 Re(y B,4 ) I m ( y B , ) (c) N a > CAS (Bob) −1 0 1−101 Re(y
E,1 ) I m ( y E , ) −1 0 1−101 Re(y E,2 ) I m ( y E , ) −1 0 1−101 Re(y E,3 ) I m ( y E , ) −1 0 1−101 Re(y E,4 ) I m ( y E , ) (d) N a > CAS (Eve) −4 −2 0 2−4−2024 Re(y
B,1 ) I m ( y B , ) −4 −2 0 2−4−2024 Re(y B,2 ) I m ( y B , ) −4 −2 0 2−4−2024 Re(y B,3 ) I m ( y B , ) −4 −2 0 2−4−2024 Re(y B,4 ) I m ( y B , ) (e) GAS (Bob) −5 0 5−505 Re(y E,1 ) I m ( y E , ) −5 0 5−505 Re(y E,2 ) I m ( y E , ) −5 0 5−505 Re(y E,3 ) I m ( y E , ) −5 0 5−505 Re(y E,4 ) I m ( y E , ) (f) GAS (Eve) Fig. 2: Plot of 1000 samples of the received signal yyy B at Boband yyy E at Eve when CI pre-coding aided GPSM is employedat Alice with [ N t , N r , N e ] = [8 , , under a particular MIMOchannel realisation of HHH B and HHH E when SNR is set to 30dB.Three different cases correspond to three rows of subplots inpair for yyy B (left) and yyy E (right), respectively. On each subplot,the received signal samples of each individual RA are detailedin four miniplots, each with horizontal axis and vertical axisrepresent real and imaginary part.carried only the antenna level. Without using antenna scram-bling as we proposed at the baseband, by varying the antennanear-field electromagnetic boundary in analogy domain, thedesired phase and amplitude of the far-filed electromagneticboundary is modulated at the intended direction, which isindistinguishable to undesired directions.Finally, for clarity, we summarize the main findings of { Θ Bǫ,τ , Θ Eǫ,τ } for calculating the DCMC capacity { C B , C E } and security capacity C S of our physical layer security forGPSM scheme with both CAS and GAS in Table I. We alsoemphasize that our proposed design imposes tough require-ment on Eve by enforcing it to equip with N e ≥ N t antennasin order to start intercepting and even in this challenging set-ting, substantial positive security capacity can still be observedas shown in next Section.V. N UMERICAL R ESULTS
Let us now illustrate the numerical results of our advocatedphysical layer security of GPSM scheme with antenna scram-bling. Throughout the simulations, we let σ B = σ E > andalso N e = N t without loss of generosity. As discussed, when N e < N r , Eve is not capable of intercepting the communi-cations between Alice and Bob, resulting into C E = 0 andhence perfect security may be guaranteed by our design evenwhen σ E . Thus, we focus on the challenging scenarioof N e = N t that was largely neglected in the literature. Notethat this is the least number of antennas required for Eve toperform (34) and (54). A. Performance with Antenna Scrambling1) N a = 1 : The upper subplot of Fig 3 shows the securitycapacity of GPSM scheme with both CAS (solid) and GAS(dash) employing CI based pre-coding when N a = 1 havingdifferent combinations of [ N t , N r ] and N e = N t . It is clearthat the security capacity is higher when employing GAS forall different combinations considered, although the higher theratio of N t /N r , the lower the difference of security capacity.Also, as expected, along with increasing the ratio of N t /N r ,the security capacity increases in general for both CAS andGAS schemes. N a > and CAS: The lower subplot of Fig 3 showsthe security capacity of GPSM with CAS employing CI basedpre-coding when [ N t , N r ] = [16 , having different numberof activated RAs N a = [2 , , , , , and N e = N t . Firstly,when including N a = 1 for bench-marker, the lower subplotof Fig 3 shows that activating more RAs provides higher andwider security capacity region than opting for larger ratio of N t /N r as indicated by the apparent comparisons between solidcurves and dashed curves, despite that they are operated ondifferent SNR ranges. Further, we can also see that along withincreasing N a = [2 , , , the security capacity increases. Inparticular, the security capacity region of activating N a = 2 and N a = 3 RAs at Bob constitute an approximately hori-zontal shifted security capacity region of activating N a = 6 and N a = 5 RAs. This is intuitively true since activating N a = 6 ( N a = 5 ) RAs at Bob is conceptually equivalent todeactivating N t − N a = 2 ( N t − N a = 3 ) RAs at Bob, wherein both cases, we have the maximum security capacity being C S = k eff = 4 bits/s/Hz ( C S = k eff = 5 bits/s/Hz). Finally,we observe that the highest security capacity is achievedwhen N a = 4 with the maximum security capacity being C S = k eff = 6 bits/s/Hz. N a > and GAS: Fig 4 shows the security capacityof GPSM with GAS employing CI based pre-coding when [ N t , N r ] = [16 , (upper) having different number of activatedRAs N a = [2 , , , , , and N e = N t as well as its securitycapacity difference with respect to that of CAS (lower). Itcan be seen from the upper subplot of Fig 4 that the securitycapacity offered by GAS is also substantial, which is similarto the case of CAS. Again, along with increasing N a =[1 , , , , the security capacity increases and the mirroringeffects also appear between N a = [1 , , and N a = [7 , , accordingly. It is thus interesting to investigate the security −20 −15 −10 −5 0 5 10 15 20−0.500.511.522.5 SNR [dB] S e c u r i t y C apa c i t y ( b i t s / s / H z ) [Nt,Nr,Ne] = [16,8,16][Nt,Nr,Ne] = [24,8,24][Nt,Nr,Ne] = [32,8,32]−20 −10 0 10 20 30 40 500246 SNR [dB] S e c u r i t y C apa c i t y ( b i t s / s / H z ) Na = 2Na = 3Na = 4Na = 5Na = 6Na = 7Na = 1Solid: CASDash: GAS[Nt,Nr,Ne] = [16,8,16] Na = 1Nt/Nr = 4,3
Fig. 3: (Upper) Security capacity of GPSM scheme with bothCAS (solid) and GAS (dash) employing CI based pre-codingwhen N a = 1 having different combinations of [ N t , N r ] and N e = N t . (Lower) Security capacity of GPSM withCAS employing CI based pre-coding when [ N t , N r ] = [16 , having different number of activated RAs N a = [2 , , , , , and N e = N t . −20 −10 0 10 20 30 40 500246 SNR [dB] S e c u r i t y C apa c i t y ( b i t s / s / H z ) Na = 2Na = 3Na = 4Na = 5Na = 6Na = 7−20 −10 0 10 20 30 40 50−2−1012 SNR [dB] C s D i ff e r en c e ( b i t s / s / H z ) Na = 2Na = 3Na = 4Na = 5Na = 6Na = 7
Fig. 4: Security capacity of GPSM with GAS employing CIbased pre-coding when [ N t , N r ] = [16 , (upper) havingdifferent number of activated RAs N a = [2 , , , , , and N e = N t and its security capacity difference with respect tothat of CAS (lower).capacity difference between CAS and GAS, which is shown inthe lower subplot of Fig 4, where we can see that the securitycapacity differences when N a > alternate around zero. Thisindicates no dominant superiority between CAS and GAS,where both offer substantial security capacity with differentregions of operation under the setting of N e = N t . B. Further Investigations1) Robustness to CSIT Error:
Like in all pre-codingschemes, an important aspect related to GPSM is its resilience
20 −10 0 10 20 30 40 500123 SNR [dB] S e c u r i t y C apa c i t y ( b i t s / s / H z ) no CSIT errorerror var = 0.3error var = 0.4error var = 0.5−20 −10 0 10 20 30 40 500123 SNR [dB] S e c u r i t y C apa c i t y ( b i t s / s / H z ) no Correlation ρ = 0.3 ρ = 0.4 ρ = 0.5 Fig. 5: Security capacity of GPSM with GAS employing CIbased pre-coding when [ N t , N r , N a ] = [16 , , and N e = N t under CSIT error of σ i = [0 . , . , . (upper) and antennacorrelations of ρ = [0 . , . , . (lower).to CSIT inaccuracies. In this paper, we let HHH B = HHH
B,a + HHH
B,i , where
HHH
B,a represents the matrix hosting the averageCSI, with each entry obeying the complex Gaussian distribu-tion of h B,a ∼ CN (0 , σ B,a ) and HHH
B,i is the instantaneousCSI error matrix obeying the complex Gaussian distributionof h B,i ∼ CN (0 , σ B,i ) , where we have σ B,a + σ B,i = 1 . Asa result, only
HHH
B,a is available at Alice for pre-processing,while we assume perfect
HHH B at Bob and Eve. The upper sub-plot of Fig 5 shows the security capacity of GPSM with GASemploying CI based pre-coding when [ N t , N r , N a ] = [16 , , and N e = N t under CSIT error of σ i = [0 . , . , . . It isclear from the upper subplot of Fig 5 that, as expected, thehigher the CSIT error imposed, the lower security capacityexhibited.
2) Robustness to Channel Correlations:
Another typicalimpairment is antenna correlation. The correlated MIMO chan-nels are modelled by the widely-used Kronecker model [25],which are written as
HHH B = ( RRR / A,t ) HHH B, ( RRR / B,r ) T and HHH E =( RRR / A,t ) HHH E, ( RRR / E,r ) T , with HHH B, and HHH E, representing theoriginal MIMO channel imposing no correlation, while RRR
A,t , RRR
B,r and
RRR
E,r represent the correlations at the transmitterof Alice and receiver of Bob and Eve, respectively, with thecorrelation entries given by R A,t ( i, j ) = ρ | i − j | A,t , R
B,r ( i, j ) = ρ | i − j | B,r , R
E,r ( i, j ) = ρ | i − j | E,r . In this paper, we assume ρ A,t = ρ B,r = ρ E,r = ρ . The lower subplot of Fig 5 shows thesecurity capacity of GPSM with GAS employing CI basedpre-coding when [ N t , N r , N a ] = [16 , , and N e = N t underantenna correlations of ρ = [0 . , . , . . We can see from thelower subplot of Fig 5 that the security capacity of GPSM withGAS is not necessarily reduced or improved by increasing thedegree of channel correlations. Rather, the positive securitycapacity region is shifted towards higher SNR range, whilstincreasing the value of channel correlations. N e > N t : Fig 6 shows the security capacity (upper)and outage security capacity (lower) when
SN R = 10 dB −20 −10 0 10 20 30 40 500123 SNR [dB] S e c u r i t y C apa c i t y ( b i t s / s / H z ) Ne = 16Ne = 18Ne = 20Ne = 22Ne = 240 1 2 3 4 500.20.40.60.81 Security Capacity (bits/s/Hz) P r ob ( C s < X a x i s ) Ne = 16Ne = 18Ne = 20Ne = 22Ne = 24[Nt,Nr,Na] = [16,8,2] [Nt,Nr,Na] = [16,8,2]
Fig. 6: Security capacity (upper) and outage security capacity(lower) when
SN R = 10 dB of GPSM with GAS employingCI based pre-coding when [ N t , N r , N a ] = [16 , , and N e =[16 , , , , .of GPSM with GAS employing CI based pre-coding when [ N t , N r , N a ] = [16 , , and N e = [16 , , , , . It isclear from the upper subplot of Fig 6 that the higher thenumber of receive antennas N e at Eve, the lower and narrowerpositive security capacity exhibited, as Eve’s capability isnaturally improved. Also, we can see from the lower subplotof Fig 6 that the higher the number of receive antennas N e atEve, the lower the span of positive security capacity values.For example, when N e = 22 , the security capacity ranges upto1 bits/s/Hz, while when N e = 16 , the security capacity rangesupto nearly 4 bits/s/Hz. Both figures suggest that our proposedphysical layer security of GPSM with antenna scramblingworks reasonably well even under N e > N t .V. C ONCLUSION
VI. L
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