Quantization for Physical Layer Security

Transcription

1 Quntiztion for Physicl Lyer Security On Grur, Nzi Islm, nd Werner Henkel Jcobs University Bremen Electricl Engineering nd Computer Science Bremen, Germny Emils: {o.grur, Abstrct We propose multi-level CSI quntiztion nd key reconcilition scheme for physicl lyer security. The noisy wireless chnnel estimtes obtined by the users first run through trnsformtion, prior to the quntiztion step. This enbles the definition of gurd bnds round the quntiztion boundries, tilored for specific efficiency nd not compromising the uniformity required t the output of the quntizer. Our construction results in n better key disgreement nd initil key genertion rte trde-off when compred to other level-crossing quntiztion methods. I. INTRODUCTION AND MOTIVATION Due to the inherently rndom nture of the brodcst wireless chnnel, extensive reserch efforts hve been directed, over the lst decde, towrds the genertion of encryption keys t the physicl lyer. As opposed to cryptogrphic security tht relies on limited computtionl resources of potentil evesdroppers, s well s key exchnge mechnism, physicl lyer security ims t providing users shring wireless chnnel with symmetric keys tht cn be used for encryption/decryption, without relying on preexistent key distribution infrstructure. The originl principle dtes bck to the one-time pd, described by Vernm [] in 96, where plintext is encrypted with previously known secret key through modulr ddition. If the key is truly rndom nd s long s the plintext messge, n evesdropper without ccess to the key cnnot decrypt the messge. Shnnon, lter on, formulted this finding in informtion-theoretic terms [], i.e., the vilbility of the ciphertext t the evesdropper does not provide ny id in guessing the plintext messge if the entropy of the key is lrger or equl to the entropy of the messge. Although this finding hs been widely known, the key distribution, long with the requirement of the lrge key length, constituted the prcticl limittions for which much of the reserch focus hs shifted towrds clssic cryptogrphic schemes over the lst century. The interest in informtiontheoretic security hs risen gin, long with the dvncements in d-hoc networks, such s wireless sensor networks (WSN), nd Internet of Things (IoT), where the ssumption of key distribution infrstructure, on which computtionl security relies, is unfesible. A thorough survey on recent physicl lyer key genertion dvncements cn be found in [3]. Our contribution in this pper focuses on the quntiztion nd reconcilition of correlted chnnel estimtes obtined by two users. Unlike other works [4], [5], we use the complex chnnel stte informtion (CSI) for key genertion, nd quntize the rel nd imginry prts jointly. In obtining the complex CSI, prsitic ntenn rry is used t one of the users, in order to generte rtificil fding nd to ensure high degree of chnnel rndomness, even in lineof-sight (LOS) or sttic environments. We lso propose trnsformtion tht mps the complex chnnel estimtes to the unit squre, prior to vector quntiztion step, whose purpose is multifold. First, the trnsformtion we propose fcilittes the use of well-known, low-complexity vector quntizers, such s Linde-Buzo-Grey (LBG) [6], by providing uniformly distributed quntizer output, even for higher level quntiztion. Second, we show how such trnsformtion enbles the strightforwrd construction of gurd bnds round the quntiztion boundries, without scrificing the uniformity of the quntizer output. Such gurd bnds re introduced s key reconcilition method, with the purpose of excluding the mesurement smples most likely to be erroneous due to independent noise on both sides or circuitry imperfections. Our findings show tht in the cse of very strong correltion between the legitimte user mesurements, for fixed key genertion rte, higher level quntiztion with wider gurd bnds is preferred to lower level quntiztion with nrrow gurd bnds. Section IV show the performnce improvement of our scheme to previous methods described in literture. A. Requirements for Physicl Lyer Key Genertion Three min requirements need to be fulfilled for physicl lyer key genertion, chnnel reciprocity, rndomness, nd sptil decorreltion. If we denote by h AB the forwrd chnnel from Alice to Bob, nd by h BA the reverse chnnel from Bob to Alice, s shown in Fig., the chnnel reciprocity principle implies h AB h BA c. This is true to some extent for TDD systems, such s 8., WiMAX, LTE, etc.. As long s the coherence time of the chnnel is lrger thn the mesurement time, the two users, Alice nd Bob, cn send previously known pilot signls in consecutive time slots in order to obtin their vectors of chnnel estimtes, â c + n, ˆb c + n b, where â nd ˆb denote the estimte vectors t Alice, nd Bob, respectively. n, nd n b represent the independent noise vectors ffecting the user estimtes, rising from different trnsceiver circuitry, nd possible vritions of the chnnel during the non-simultneous mesurement slots. Although they re correlted, strightforwrd quntiztion of the mesurement vectors will likely

2 result in key mismtch between the users, nd n dditionl step of key reconcilition needs to be performed. The chnnel mesurements cn represent either chnnel stte informtion (CSI), or received signl strength informtion (RSSI). The CSI mesurements cn refer to either the chnnel impulse response or frequency response, contining both mplitude nd phse informtion. RSSI, in contrst, refers to n verge of the received power for the whole pcket nd cn be seen s more corse-grined, compred to the instntneous CSI vlues. Although CSI-bsed key genertion hs been shown to significntly outperform RSSI-bsed key genertion [7], RSSI dt hs been used in most of the previously proposed key genertion schemes bsed on its vilbility from most off-the-shelf WiFi crds. Sptil decorreltion. If pssive evesdropper, Eve, is locted few wvelengths wy from ny of the legitimte users, the chnnel smples estimted by Eve, ê, will exhibit low correltion to the forwrd nd reverse chnnel smples estimted by Alice nd Bob. Hence, most of the rndomly generted key bits will be secure with respect to Eve. In order to support this clim, we refer the reders to the work in [8] [], where theoreticl nd experimentl pproches hve been tken in deriving the mount of key informtion tht leks to the evesdropper, depending on its seprtion distnce. Even smll mount of informtion leked is of serious concern nd is highly dependent on Eve s position nd reltive distnce to the legitimte users, thus further processing methods such s dvntge distilltion nd privcy mplifiction [], [], re required. An initil key K N of length N, obtined by Alice nd Bob, is reduced to shorter key of length R, i.e., K R, through the use of hsh function g, known by ll prties, such tht the lekge H(K R ê, g) is diminished. In here, we hence ignore the effect of pssive Eve, by ssuming sufficient evesdropper seprtion, s well s privcy mplifiction fter the quntiztion step. We re lso not concerned here with the possibility of ctive jmming ttcks. Rndomness. A lrge degree of rndomness ensures the existence of lrge key pool fter quntiztion, mking it more computtionlly prohibitive for the evesdropper to resort to brute-force ttcks. If the reciprocl wireless chnnel does not exhibit high degree of fding, rtificil fding cn be induced through the use of prsitic ntenn rrys, s we discuss in Section II-A. It should be noted tht the uncertinty t the evesdropper is mximum when the key is uniformly distributed over its lphbet. For compring different pproches, we follow the frmework of [3], with the following metrics. ) Correltion Coefficient: We use the Person correltion coefficient, ρ, s mesure of the degree of reciprocity of two sequences, â [â â N ] T nd ˆb [ˆb ˆb N ] T. ) Key Disgreement Rte (KDR): The key disgreement rte is defined s the verge rtio between the key symbols found in disgreement between Alice nd Bob to the overll number of symbols. We shll distinguish ccordingly between symbol nd bit disgreement rte in subsequent sections. 3) Initil Key Genertion Rte (IKGR): We define the symbol IKGR s the verge of the rtio between the number of symbols used for key genertion fter quntiztion to the totl number of symbols fed to the quntizer. For multilevel quntiztions, i.e., N q, the bit IKGR is obtined by multiplying the symbol IKGR with the number of bits per symbol. II. SYSTEM DESCRIPTION A. RECAP Antenns for Chnnel Rndomiztion The wireless chnnel between the users does not only include the propgtion chnnel, but lso the rdition chrcteristics of the trnsmit/receive ntenns. The chnnel cn be seen s time-vrying if the propgtion chnnel is multipth rich nd one of the nodes is moving. However, in the cse of line-of-sight (LOS) chnnel, reconfigurble perture ntenn rrys (RECAPs) cn be used. A RECAP ntenn consists of reconfigurble elements (REs), confined to physicl perture [8], [4]. Ech RE cn be in number of sttes, e.g., by vrying cpcitive lods. B. System Model Figure depicts the system model. We consider one of the legitimte users, e.g., Alice, to be equipped with RECAP ntenn with 4 prsitic reconfigurble elements (REs), nd one feed element, shown in ornge. Bob nd Eve re ech equipped with single dipole ntenn. The perceived chnnel distribution, s seen by the users, when RECAP ntenn is used, depends on the number of REs tht re ctive, s well s on the number of sttes. Although using RECAP ntenns for creting rtificil fding does not lwys led to norml chnnel distribution, when the number of reconfigurble elements of the rry is lrge, the number of sttes is lrge, nd the reflection coefficient is controlled, s discussed in [5], the distribution is very close to complex Gussin, with the rel nd imginry prts i.i.d.. The chnnel distributions obtined for the RECAP configurtions in Fig., re shown subsequently, in Fig. 3. For the rest of this work, we ssume the chnnel to follow normlized complex Gussin distribution, with the rel nd imginry prts i.i.d., s obtined for the N RE 4 RECAP configurtion. Additionl detils on how the chnnel mesurements used in this work were obtined, long with study on the number of secure bits s function of multipth nd evesdropper ntenn seprtion, cn be found in [6], [7]. Fig.. System Model

3 Fig.. RECAP configurtions;, 4, 8, 6, 4 ctive reconfigurble elements (REs); feed element in ornge, ctive REs in blue, inctive REs in green Fig. 3. Complex chnnel pdfs given different RECAP configurtions;, 4, 8, 6, 4 ctive reconfigurble elements (REs) The first step in the key genertion nd reconcilition process is the chnnel probing phse, in which both Alice nd Bob obtin their length N vectors of complex CSI estimtes, nmely nd b. Note tht throughout this pper, bold lowercse nottions re used for vectors, while the noisy estimtes re indicted by theˆ symbol. If c x + jy denotes complex chnnel smple, x A + j y A nd b x B + j y B represent the noisy estimtes of the chnnel smple c t Alice, nd Bob, respectively, with x A, y A, x B, y B denoting the estimtes of the rel nd imginry components. All nottions re summrized in Tble I, for convenience. TABLE I N OTATIONS Nottion c x + jy x [x xn ]T y [y yn ]T x A + jy A b x B + jy B x A x + nx y A y + ny x B x + nxb y B y + nyb u u x + ju y u b u xb + ju yb σ σn σn b σ σ + σn σb σ + σn b Ri (u x, u y ) (u xb, u yb ) (nux, nuy ) Description vector of chnnel smples (complex) vector of chnnel smples (rel prt) vector of chnnel smples (imginry prt) vector of complex chnnel estimtes t Alice, [ N ]T vector of complex chnnel estimtes t Bob, b [b b N ]T mesurement estimtes of x t Alice mesurement estimtes of y t Alice mesurement estimtes of x t Bob mesurement estimtes of y t Bob trnsformed vector (uniform domin) trnsformed vector b (uniform domin) vrince of x, y (chnnel vrince) noise vrince t Alice noise vrince t Bob vrince of (t Alice) vrince of b (t Bob) number of quntiztion regions ith quntiztion region rel nd imginry components of Alice s smple fter trnsformtion rel nd imginry components of Bob s smple b fter trnsformtion to uniform domin rndom vribles describing the rel nd imginry noise components, respectively, ffecting complex smple c x + jy, fter trnsformtion to uniform domin C. Uniform Trnsformtion Hving obtined their sequences of nlog complex chnnel estimtes, both Alice nd Bob first pply trnsformtion of the dt, before quntizing, s follows: u x x A y A erfc ; u y erfc. ) (σ + σn ) (σ + σn () u x nd u y correspond to the rel nd imginry components of smple fter the trnsformtion, t Alice, likewise u xb nd u yb for Bob. Since the estimtes x A nd y A re ech distributed ccording to zero-men Gussin pdf with, the trnsformed smples u x nd vrince σ σ + σn u y will ech follow uniform distribution in (, ). Thus, the vector of complex estimtes, in the originl domin, will now be mpped into vector u, uniformly distributed in the unit squre, fter trnsformtion. We refer to the domin fter trnsformtion s the uniform domin. Performing such trnsformtion of the dt before quntiztion is motivted by the reduced complexity of the vector quntiztion, s well s the computtion of the gurd bnd widths. Our gol now is to find n equl-re prtitioning of the unit squre into quntiztion regions, while mintining low probbility tht chnnel smple is quntized to different regions by Alice nd Bob, due to independent noise effects. The equl-re constrint is required to ensure uniform distribution of the quntized key symbols. We first turn our ttention to the noise distribution fter trnsformtion. Since nx nd ny were normlly distributed before trnsformtion with mens x nd y, nd vrinces σnx σny σn, their mrginl pdfs fter trnsformtion re found to be (x+σ erfc (nux )) σ (erfc (nux )) σn f (nux x) e (), σn The mrginl pdf for the imginry noise component is similr to the one in (). The noise t Alice, round the point of

4 coordintes (x, y) will hve distribution, fter the trnsformtion, which will depend on the (x, y) positions in the originl domin. The rel nd imginry noise pdfs t Bob re obtined by substituting σ with σ b in (). Figure 4 shows the mrginl pdf of the noise fter trnsformtion (rel dimension only), from (), for vrious positions of (x, y) in the originl domin. As (x, y) vry from the inside of the circulrly symmetric Gussin towrds the periphery, the mrginl noise pdfs fter trnsformtion hve nrrower vrince towrds the edges of the unit squre. reltive frequency of n ux Uniform domin noise distribution fter trnsformtion n ux x.3 (simultion) x.3 (theoreticl) x.5 (simultion) x.5 (theoreticl) x (simultion) x (theoreticl) x.5 (simultion) x.5 (theoreticl) x.3 (simultion) x.3 (theoreticl) Fig. 4. Mrginl PDF of the noise fter trnsformtion to uniform domin; rel component only. The circulr mrkers correspond to the theoreticl curves, s given in (), while the lines correspond to the pdf estimtes from simultions. After the trnsformtion phse, both Alice nd Bob re redy to proceed with the vector quntiztion. We will first proceed with describing the vector quntiztion scheme chosen nd then describe the computtion of the gurd bnds. III. QUANTIZATION One of the most widely used vector quntiztion schemes is the Linde-Buzo-Gry (LBG) lgorithm [6]. Its widespred is consequence of the fct tht is hs very low implementtion complexity nd it outperforms other vector quntizers in prcticl situtions. LBG is the discrete version of the Lloyd- Mx quntizer, nd, unlike Lloyd-Mx, it does not require closed form expression of the input density, but n initil trining vector tht is itertively prtitioned into N q clusters. We show in Fig. 5 the difference in output distribution when we pply the LBG quntiztion to circulrly symmetric Gussin input nd when we pply it to uniformly distributed input, s the one obtined fter our trnsformtion in (). With n increse in the number of quntiztion regions, the uniformity of the output is not preserved when LBG quntiztion is pplied directly to the estimte vectors, without first performing the trnsformtion. Hence, the proposed trnsformtion is importnt for key genertion. A. Gurd Bnd Construction The second rgument for such trnsformtion is tht once the coordintes of the vertices of the Voronoi cells re obtined, the computtion of the gurd bnds is strightforwrd. Since the quntiztion cells re now equl-re polygons tessellting the unit squre, we cn design gurd bnds round the quntiztion boundries strting from certin efficiency. We define efficiency η to represent the rtio of points tht fll outside the gurd bnds to the overll number of points. Due to the uniform distribution of the smples Fig. 5. Output distribution cross quntiztion regions for uniform nd normlly distributed LBG trining sets; () N q 3; (b) N q 6 fter trnsformtion, the efficiency η A useful A Asqure gb, where A gb is the totl re of the gurd bnds. Since we re interested in preserving the uniformity of the output symbols, even fter the construction of the gurd bnds, we require tht n re of η N q is ssigned to the gurd bnds within ech quntiztion region. It should be noted, however, tht lthough the quntiztion regions should hve equl res in the unit squre, this does not imply equl widths of the gurd bnds cross the regions. If we let (v x, v y ) denote the vectors holding the rel nd imginry coordintes of the vertices of certin region R i, the coordintes of the inner polygon representing the useful region of cell R i cn be computed using the theorem of similr polygons, s follows: v xrescled η(v x µ vx ) + µ vx, v yrescled η(v y µ vy ) + µ vy, (3) where µ v is the men of vector v. For illustrtion purposes, we show in Fig. 6 () the LBG quntiztion regions, long with the corresponding gurd bnds, for N q 8, nd n efficiency η.8. Fig. 6 (b) shows the sme points tht hve been quntized in the uniform domin, fter being trnsformed bck to the originl domin. This is shown to illustrte the fct tht n equivlent quntiztion in the originl domin would hve non-liner gurd bnds. Note tht our method does not require such n inverse trnsformtion of the points, nd both Alice nd Bob generte their keys by quntizing their trnsformed complex vectors û nd û b, respectively. The gurd bnd construction is lso possible in the originl domin, however, more complicted. For the simplest cse of the binry quntiztion, when the dt is quntized in the originl domin, with the quntiztion boundry consisting of stright line going through the origin, the gurd bnds of width r round the quntiztion line re computed by solving η π(σ + σn) r r e ˆx A +ŷ A (σ +σ n ) dˆx A dŷ A. (4) However, for lrger number of Voronoi cells, with rbitrry quntiztion boundries, (4) becomes much more complex, mking even individul numericl solutions for the gurd bnd

5 widths in the originl domin prcticlly unfesible. Note tht regrdless of the domin chosen, neither the quntiztion nor the gurd bnd reconcilition should ffect the uniformity t the output of the quntizer. Previous quntiztion nd 8; LBG VQ with GBs (uniform domin) P b 8; points fter inverse trnsformtion points outside GBs points inside GBs points outside GBs points inside GBs.6.5 uy.7 y x ux Fig. 6. () Linde-Buzo-Gry (LBG) quntiztion of the uniformly distributed squre; points tht fll in the gurd bnds in blue, points outside gurd bnds in green. This plot ws generted for n efficiency η.8; (b) equivlent positions in the originl domin. reconcilition schemes hve tken different pproches. We briefly describe some of the more relevnt previous results, since we will use them s bsis of comprison to our method in Section IV. However, most of the results vilble in the literture discuss binry quntiztion schemes nd do not tke dvntge of quntizing the rel nd imginry components of the complex chnnel jointly, s we propose. Aono [4] drops RSSI vlues below threshold, chosen s the medin, nd communictes the positions of the discrded smples to the other prty. In [8], Tope computes the difference in RSSI vlues nd relies on two thresholds to exclude points tht will likely result in key mismtch. Similrly, in the work of Mthur et l. [5], Alice nd Bob check for successive blocks of m smples tht fll bove or below two thresholds, q+, nd q. Jn et l. [4] proposed both multibit nd single bit dptive secret bit genertion (ASBG) lgorithm, which is modifiction of the work of Mthur [5]. In contrst, the quntizer proposed by Jn, however, divides the mesurements into smller block lengths nd clcultes the thresholds for ech block seprtely, resulting in fster dpttion to shifts in the RSSI men vlue over time. Unlike ll of the bove methods, we strt from fine-grined CSI mesurements, following complex circulrly symmetric distribution, s explined in Section II-A. B. Key Disgreement Rtes The symbol mismtch probbility P ( Ri b / Ri ) between Alice nd Bob cn be computed s follows: P b c i j,j i c i j,j i p(b Rj, Ri, c)dc P (b Rj, Ri c)p(c)dc P (b Rj c)p ( Ri c)p(c)dc.5.3 i j,j i.5.8 c.9 P (b Rj, Ri ) i j,j i (5) Note tht P denotes probbility, while the lowercse p refers to probbility density function. For the simplest cse, with two quntiztion regions, with the quntiztion boundry t x, the bit mismtch probbility, or bit disgreement rte (BDR), is derived in (6)-(9), where R is from (, ), R is from (, ), nd R R R. Note tht, b, nd c re ll complex vribles, so ny integrtion is in rel nd imginry dimensions. For binry quntiztion,, due to the symmetry of the quntiztion, it is sufficient to integrte only the long the rel components of the vribles, nmely x A, x B, nd x, s shown in (9). P b N P b P ( R, b R ) + P ( R, b R ) P ( R c)p (b R c)p(c)dc N P b N c R (6) (7) p( c)p(b c)p(c)d db dc (8) c R b R R IV. R ESULTS For fir evlution of the quntiztion nd reconcilition performnces, we show simultion results for the bit disgreement rte (BDR) versus the correltion coefficient. We provide results for both the binry quntiztion,, s well s multibit quntiztion, 4 nd 8. One interesting spect is whether ny dvntge cn be chieved by using higher level quntiztion, e.g., extrcting multiple bits from single CSI smple, while fixing the initil key genertion rte. In Fig. 7, we show the bit disgreement rtes (BDR) versus correltion for vrious efficiency curves, for two quntiztion regions, nd the sme BDR plotted ginst the SNR in Fig. 8. For binry quntiztion we investigte three scenrios: Method quntizes the user estimtes with line through the middle of the originl domin, i.e., no trnsformtion is pplied. Method splits the originl domin into two eqully probble regions by using circle of pproprite rdius. Method 3 trnsforms the user estimtes to the unit squre, before quntizing with line t ux.5. A mximum key genertion rte of, for the binry quntiztion cse, is chieved for the cse when the gurd bnd widths re zero. As confirmed by simultions, for the binry quntiztion, methods nd 3 hve, of course, the sme performnce, nd surpss Method. For four quntiztion regions, 4, we hve considered four options. Method quntizes the originl dt by splitting the originl domin with four squres intersecting in the origin. Method uses 4 concentric circles to quntize the Gussin estimtes. The rdii of the concentric This is equivlent to LBG quntiztion in the uniform domin for

6 Initil Key Genertion Rte (IKGR) Bit Disgreement Rte (BDR) Initil Key Genertion Rte (IKGR) Bit Disgreement Rte (BDR) Pâ ˆb σn σ nb σ Method : line (originl domin) Method : circle (originl domin) Method 3: line (uniform domin) correltion coefficient Method : line (originl domin) Method : circle (originl domin) Method 3: line (uniform domin) correltion coefficient Fig. 7. Bit disgreement rte versus correltion for different initil key genertion rtes; N q 4 6 Method 3: line (uniform domin) Method : line (originl domin) Method : circle (originl domin) Method : line (originl domin) Method : circle (originl domin) Method 3: line (uniform domin) 3 4 Fig. 8. Bit disgreement rtes nd the corresponding IKGRs for N q circles re computed such tht the output of the quntiztion is uniformly distributed. Methods 3 nd 4 first trnsform the dt to the uniform domin. Method 3 is similr to Method, however, the quntiztion with four squres is done in the uniform domin, insted of the originl one. Method 4 is the LBG quntiztion in the uniform domin, tht we hve described in the previous sections. Figure 9 shows the BDR versus the correltion coefficient for different key genertion rtes. Note tht since we now consider bits/symbol, n e (ˆx A x) σn e (ˆx B x) σn b e x σ dˆx A dˆx B dx (9) IKGR of corresponds to n efficiency η when no points re discrded, i.e., gurd bnd widths re zero. Our findings show identicl performnce for the quntiztion with four squres, whether it is done in the uniform domin, nd the Linde-Buzo-Gry quntiztion cse. Agin, the concentric quntiztion shows the worst performnce. Figure shows the BDR versus the SNR for N q 4 for different key genertion rtes. Bit Disgreement Rte (BDR) Initil Key Genertion Rte (IKGR) Method : 4 squres GB (originl domin). Method : 4 concentric circles GB (originl domin) Method 3: 4 squres GB (uniform domin) Method 4: LBG GB (uniform domin) Method : 4 squres GB (originl domin) Method : 4 concentric circles GB (originl domin) Method 3: 4 squres GB (uniform domin) Method 4: LBG GB (uniform domin) correltion coefficient ρ correltion coefficient ρ Fig. 9. Bit disgreement rtes versus correltion for different initil key genertion rtes; N q 4 We illustrte, in Fig., performnce comprison between our work nd previous quntiztion nd reconcilition schemes, following the frmework discussed in [3]. We compre our LBG results for the cse of N q nd N q 3 with the multibit scheme proposed by Jn [4]. As shown in Fig., with the initil key genertion fixed, IKGR, our -bit LBG quntiztion shows n improvement over the corresponding -bit quntiztion of Jn, regrdless of the vlue of the correltion coefficient. However, by incresing the number of bits extrcted per CSI symbol, i.e., 3 bits/symbol, performnce increse in terms of BDR is chievble only for high correltion vlues, compred to the cse of extrcting only bits/symbol. The schemes proposed by Tope nd Murer exhibit low BDR, however, they lso hve very low initil key genertion rtes. Nevertheless, if we fix our IKGR to similr vlues s shown of the other schemes in Fig., our LBG methd is dvntgeous is terms of BDR when compred to ll the other quntiztion schemes. V. CONCLUSION Insted of simply pplying the quntiztion to the CSI estimte vectors tht Alice nd Bob obtin, we first pply trnsformtion of the dt tht leds to uniform distribution. The dvntge of performing such trnsformtion

7 Bit Disgreement Rte (BDR) Initil Key Genertion Rte (IKGR) Method 4: LBG GB (uniform domin) Method : 4 squres GB (originl domin) Method : 4 concentric circles GB (originl domin) Method 3: 4 squres GB (uniform domin) Method : 4 squres GB (originl domin) Method : 4 concentric circles GB (originl domin) Method 3: 4 squres GB (uniform domin) Method 4: LBG GB (uniform domin) 3 4 Fig.. Bit disgreement rtes versus SNR for different initil key genertion rtes; N q 4 Fig.. Bit disgreement Rte (BDR) nd Initil Key Genertion Rte (IKGR) versus correltion - comprison to previous works of the complex dt, enbles us to use well known vector quntizer, such s Linde-Buzo-Gry [6], to prtition the uniform domin into equl-re regions nd to esily construct the gurd bnds round the quntiztion boundries, for fixed key genertion rte. Our proposed method tkes dvntge of the joint quntiztion of the rel nd complex CSI components nd discrds the smples tht re most likely to result in key mismtches, without disturbing the uniformity requirement t the output of the quntizer. Our method shows better performnce in terms of bit disgreement nd initil key genertion rtes compred to other previous quntiztion nd reconcilition schemes currently vilble, while lso benefiting from low implementtion complexity. REFERENCES [] G. Vernm, Cipher Printing Telegrph Systems for Secret Wire nd Rdio Telegrphic Communictions, Journl of the AIEE, vol. 45, p. 95, 96. [] C. Shnnon, Communiction Theory of Secrecy Systems, Bell System Technicl Journl, vol. 8, no. 4, pp , 949. [3] J. Zhng, T. Duong, A. Mrshll, nd R. Woods, Key Genertion from Wireless Chnnels: A Review, IEEE Access, vol. 4, pp., 6. [4] S. Jn, S. N. Premnth, M. Clrk, S. Kser, N. Ptwri, nd S. Krishnmurthy, On the Effectiveness of Secret Key Extrction from Wireless Signl Strength in Rel Environments, in Mobile Communictions (MobiCom9), (Beijing), pp., 9. [5] S. Mthur, W. Trppe, N. Mndym, C. Ye, nd A. Reznik, Rdio- Telepthy: Extrcting Secret Key from n Unuthenticted Wireless Chnnel, Proceedings of the 4th ACM interntionl conference on Mobile computing nd networking - MobiCom 8, p. 8, 8. [6] Y. Linde, A. Buzo, nd R. M. Gry, An Algorithm for Vector Quntizer Design, IEEE Trnsctions on Communictions, vol. 8, pp , jn 98. [7] Y. Liu, S. C. Drper, nd A. M. Syeed, Exploiting Chnnel Diversity in Secret Key Genertion from Multipth Fding Rndomness, IEEE Trnsctions on Informtion Forensics nd Security, vol. 7, no. 5, pp ,. [8] R. Mehmood nd J. W. Wllce, Experimentl Assessment of Secret Key Genertion using Prsitic Reconfigurble Aperture Antenns, Proceedings of 6th Europen Conference on Antenns nd Propgtion, EuCAP, no., pp. 5 55,. [9] R. Mehmood, J. W. Wllce, nd M. A. Jensen, Experimentl Chrcteriztion of Chnnel-Bsed Key Estblishment using Reconfigurble Antenns, 8th Europen Conference on Antenns nd Propgtion, EuCAP 4, no. EuCAP, pp , 4. [] A. J. Pierrot, R.. Chou, nd M. R. Bloch, Experimentl Aspects of Secret Key Genertion in Indoor Wireless Environments, in IEEE 4th Workshop on Signl Processing Advnces in Wireless Communictions (SPAWC), pp , 3. [] U. Murer nd S. Wolf, Secret-Key Agreement over Unuthenticted Public Chnnels- Prt III: Privcy Amplifiction, IEEE Trnsctions on Informtion Theory, vol. 49, pp , 3. [] C. H. Bennett, G. Brssrd, C. Crkpeu, U. M. Murer, nd S. Member, Generlized Privcy Amplifiction, vol. 4, no. 6, pp , 995. [3] R. Guillume, C. Zenger, A. Mueller, C. Pr, nd A. Czylwik, Fir Comprison nd Evlution of Quntiztion Schemes for PHY-bsed Key Genertion, OFDM 4, 9th Interntionl OFDM Workshop 4 (InOWo 4); Proceedings of, pp. 5, 4. [4] T. Aono, K. Higuchi, T. Ohir, B. Komiym, nd H. Ssok, Wireless Secret Key Genertion Exploiting Rectnce-Domin Sclr Response of Multipth Fding Chnnels, IEEE Trnsctions on Antenns nd Propgtion, vol. 53, pp , nov 5. [5] R. Mehmood nd J. W. Wllce, Wireless Security Enhncement using Prsitic Reconfigurble Aperture Antenns, in Europen Antenns Propgtion Conference, (Rome, Itly), pp , pr. [6] J. W. Wllce nd R. K. Shrm, Automtic Secret Keys From Reciprocl MIMO Wireless Chnnels: Mesurement nd Anlysis, IEEE Trnsctions on Informtion Forensics nd Security, vol. 5, pp , sep. [7] R. Mehmood, J. W. Wllce, nd M. A. Jensen, Key Estblishment Employing Reconfigurble Antenns: Impct of Antenn Complexity, IEEE Trnsctions on Wireless Communictions, vol. 3, pp , nov 4. [8] M. A. Tope nd J. C. Mcechen, Unconditionlly Secure Communictions over Fding Chnnels, Militry Communictions Conference, vol., no. c, pp ,.