Anti-Jamming Message-Driven Frequency Hopping: Part I System Design

Ati-Jammig essage-drive Frequecy Hoppig: Part I System Desig Lei Zhag Huahui Wag Togtog Li Abstract This is Part I of a two-part paper that cosiders atijammig system desig i wireless etworks based o messagedrive frequecy hoppig DFH, a highly efficiet spread spectrum techique. I this paper, we first aalyze the performace of DFH uder hostile jammig. It is observed that while DFH is robust uder strog jammig, it experieces cosiderable performace losses uder disguised jammig from sources that mimic the true sigal. To overcome this limitatio, we propose a ati-jammig DFH AJ-DFH system. The mai idea is to trasmit a secure ID sequece alog with the iformatio stream. The ID sequece is geerated through a cryptographic algorithm usig the shared secret betwee the trasmitter ad the receiver, it is the exploited by the receiver for effective sigal extractio. It is show that AJ-DFH ca effectively reduce the performace degradatio caused by disguised jammig, ad is also robust uder strog jammig. I additio, we exted AJ- DFH to the multi-carrier case, which ca icrease the system efficiecy ad jammig resistace sigificatly through jammig radomizatio ad frequecy diversity, ad ca readily be used as a collisio-free multiple access system. Part II of the paper focuses o the capacity aalysis of DFH ad AJ-DFH uder disguised jammig. I. INTRODUCTION I wireless etworks, oe of the most commoly used techiques for limitig the effectiveess of a oppoet s commuicatio is referred to as jammig, i which the authorized user s sigal is deliberately iterfered by the adversary. Alog with the wide spread of various wireless devices, especially with the advet of user cofigurable itelliget devices such as cogitive radios, jammig attack is o loger limited to battlefield or military related evets, but has become a urget ad serious threat to civilia commuicatios as well [] [4]. As a widely used spread spectrum techique, frequecy hoppig FH was origially desiged for secure commuicatio uder hostile eviromets [5] [9]. I covetioal FH, each user hops idepedetly based o its ow PN sequece, a collisio occurs wheever there are two or more users trasmittig over the same frequecy bad. aily limited by the collisio effect, the spectral efficiecy of the covetioal FH is very low [], []. To improve the spectral efficiecy, FH systems that exploit high-dimesioal modulatio scheme have bee studied i the literature [] [6]. However, the performace of these systems are still limited by the collisio effect, also kow as self-jammig. Lei Zhag is with arvell Semicoductor Ic., 5488 arvell L, Sata Clara, CA 9554, USA. email: lei@marvell.com. Huahui Wag is with AT &T Shao Laboratories, Florham Park, NJ 793, USA. email: huahui@research.att.com Togtog Li is with the Departmet of ECE, ichiga State Uiversity, East Lasig, I 4884, USA. email:togli@egr.msu.edu Recetly, a three-dimesioal modulatio scheme, kow as message-drive frequecy hoppig DFH, was proposed i []. The basic idea of DFH is that part of the message acts as the PN sequece for carrier frequecy selectio at the trasmitter. ore specifically, selectio of carrier frequecies is directly cotrolled by the ecrypted iformatio stream rather tha by a pre-selected pseudo-radom sequece as i covetioal FH. The most sigificat property of DFH is that: by embeddig a large portio of iformatio ito the hoppig frequecy selectio process, additioal iformatio trasmissio is achieved with o extra cost o either badwidth or power. I fact, trasmissio through hoppig frequecy cotrol adds aother dimesio to the sigal space, ad the resulted codig gai ca icrease the system spectral efficiecy by multiple times []. I this paper, we aalyze the performace of DFH uder hostile jammig. It is observed that: DFH is particularly powerful uder strog jammig scearios, ad outperforms the covetioal FH by big margis. The uderlyig argumet is that for DFH, eve if the sigal is jammed, strog jammig ca ehace the power of the jammed sigal ad hece icreases the probability of carrier detectio. Whe the system experieces disguised jammig, where the jammig is highly correlated with the sigal, ad has a power level close or equal to the sigal power, it is the difficult for the DFH receiver to distiguish jammig from the true sigal, resultig i performace losses. As will be show i part II of the paper, this is essetially due to the existece of symmetricity betwee the jammig iterferece ad the authorized sigal. To overcome the drawback of DFH, i this paper, we propose a ati-jammig DFH AJ-DFH scheme. The mai idea is to isert some sigal idetificatio ID iformatio durig the trasmissio process. This secure ID iformatio is geerated through the Advaced Ecryptio Stadard AES [7] usig the shared secret betwee the trasmitter ad the receiver. The ID iformatio ca be exploited by the receiver to locate the true carrier frequecy. oreover, protected by AES, it is computatioally ifeasible for malicious users to recover the ID sequece. The major differece with DFH is that: i AJ-DFH, we add shared radomess betwee the trasmitter ad the receiver to break the symmetricity betwee the jammig iterferece ad authorized sigal. ore specifically, i AJ- DFH, secure ID sigals are itroduced to distiguish the true iformatio chael from the disguised chaels ivoked by jammig iterferece. Our aalysis idicates that: comparig with DFH, AJ-DFH ca effectively reduce the performace degradatio caused by disguised jammig. At the same

time, its spectral efficiecy is very close to that of DFH, which is several times higher tha that of covetioal FH. Sigle carrier AJ-DFH ca be further exteded to multicarrier AJ-DFH C-AJ-DFH. It is observed that by exploitig secure group geeratio, C-AJ-DFH ca icrease the system efficiecy ad jammig resistace sigificatly through jammig radomizatio ad eriched frequecy diversity. By assigig differet carrier groups to differet users, C-AJ-DFH ca also be used as a collisio-free multiple access system. We further ivestigate ID costellatio desig ad its impact o the performace of AJ-DFH uder both oise jammig ad disguised jammig. Noise jammig, where the jammig is modeled as Gaussia oise, has bee widely adopted i literature [8] []. However, disguised jammig ca be much more harmful for most commuicatio systems. For AJ- DFH, the worst case disguised jammig is ID jammig, for which the jammer tries to mimic the ID sigal, ad seds symbols from the same costellatio as that of the ID sigal. We show that uder oise jammig, the detectio error probability is maily determied by the sigal to jammig ad oise ratio. I this case, for a give power costrait, costat modulus costellatio delivers the best results i terms of detectio error probability. Uder ID jammig, the situatio is more complex. I the ideal case whe the system is oisefree, icreasig the ID costellatio size ca icrease the ID ucertaity, hece reduce the probability of error. I this case, the ideal costellatio size is =. However, whe oise is preset, we prove that the detectio error probability coverges as goes to ifiity. I other words, there exists a threshold t, icreasig the costellatio size over t will result i little improvemet i error probability. This result justifies the use of practical, fiite size costellatios i AJ-DFH. This paper is orgaized as follows. I Sectio II, we itroduce the cocept of disguised jammig ad evaluate DFH uder hostile jammig. I Sectio III, the proposed AJ-DFH scheme is itroduced, followig by the extesio to the multicarrier case. ID costellatio desig is ivestigated i Sectio IV. Spectral efficiecy is aalyzed i Sectio V. Simulatio examples are provided i Sectio VI ad we coclude i Sectio VII. Part II of the paper [] focuses o the capacity aalysis of DFH ad AJ-DFH uder disguised jammig. II. ESSAGE-DRIVEN FREQUENCY HOPPING A BRIEF REVIEW I this sectio, we briefly review the message-drive frequecy hoppig DFH system, ad evaluate its performace uder differet jammig scearios. A. System Descriptio The basic idea of DFH is that a major part of the iformatio is trasmitted through carrier frequecy selectio i the hoppig process. I other words, the hoppig patter is determied by the ecrypted message iformatio itself. Let N c be the total umber of available chaels, with {f, f,, f Nc } beig the set of all available carrier frequecies. The umber of bits used to specify a idividual chael here is B c = log N c, where x deotes the largest iteger less tha or equal to x. Without loss of geerality, we assume that N c = Bc. Let Ω be the selected costellatio that cotais symbols, each symbol i the costellatio represets B s = log bits. Let T s ad T h deote the symbol period ad the hop duratio, respectively, the the umber of hops per symbol period is give by N h = Ts T h. We assume that N h is a iteger larger tha or equal to oe. X Ecrypted Iformatio Sequece Fig.. Carrier Bit Vectors X, X, X, Nh S/P Partitioig X a Iformatio block structure X,, X,, N h Carrier Bits Y Ordiary Bits Carrier Frequecy Selectio Basebad Sigal Geeratio b Trasmitter Trasmitter structure of DFH. f,, f,, N h mt Ordiary Bit Vector Y odulatio The trasmitter structure of DFH is show i Fig.. We start by dividig the ecrypted iformatio stream ito blocks of legth L N h B c + B s. Each block is parsed ito N h B c carrier bits ad B s ordiary bits. The carrier bits are used to determie the hoppig frequecies, ad the ordiary bits are mapped to a symbol which is trasmitted through the selected chaels successively. Deote the th block by X, as illustrated i Fig. a. Note that i DFH, the whole block X is trasmitted withi oe symbol period. At the receiver, the trasmittig frequecy is captured usig a filter bak as i the FSK receiver rather tha usig the frequecy sythesizer. To detect the active frequecy bad, a bak of N c badpass filters BPF, each cetered at f i i =,,, N c, is deployed at the receiver frot ed, followed by a demodulator equipped with matched filter ad sampler. At each hoppig period, the ordiary bits ad the correspodig carrier frequecy ca be determied usig the miimum distace criterio. Jammig detectio algorithm ca be icorporated at the receiver to improve the system performace []. B. Performace of DFH uder Hostile Jammig First, we itroduce the cocept of disguised jammig. Disguised jammig deotes the case where the jammig is highly correlated with the sigal, ad has a power level close or equal to the sigal power. ore specifically, let st ad Jt be the user s sigal ad jammig iterferece, respectively. Defie ρ = T Ps P J t st t stj tdt as the ormalized cross-correlatio coefficiet of st ad Jt over the time period [t, t ], where T = t t, P s = t T t st dt ad P J = t T t Jt dt. We say that Jt is a disguised jammig to sigal st over [t, t ] if

3 Jt ad st are highly correlated. ore specifically, ρ > ρ, where ρ is a applicatio-orieted, predefied correlatio threshold. The jammig to sigal ratio JSR is close to db. ore specifically, P J P s < ϵ P, where ϵ P is a applicatioorieted, predefied jammig-to-sigal ratio threshold. I this paper, we cosider disguised jammig over each hoppig period, that is, [t, t ] = [mt h, m + T h ] for some iteger m. I the worst case, the costellatio Ω ad the pulse shapig filter of the iformatio sigal are kow to the jammer, the jammer ca the disguise itself by trasmittig symbols from Ω over a fake chael usig the same power level. That is, Jt = e iθ st for some phase θ. 3 covetioal FH: Disguised Jammig covetioal FH: Noise Jammig DFH: Disguised Jammig DFH: Noise Jammig this paper, we itroduce the ati-jammig DFH AJ-DFH system. Remark: It is iterestig to ote that i the primary user emulatio PUE attack i cogitive radio etworks, the malicious user mimics the primary user s sigal i fallow bads to preempt spectrum resources that could have bee used by legitimate secodary users [3] [5]. This implies that disguised jammig ca be used to iterfere the authorized user s sigal directly as discussed i this paper, ad ca also be used to distract other users such as i PUE attacks. Similar techiques ca be used to distract/prevet eavesdroppig as well. III. ANTI-JAING DFH AJ-DFH A. Trasmitter Desig The mai idea here is to isert some sigal idetificatio ID iformatio durig the trasmissio process. This secure ID iformatio is geerated through a cryptographic algorithm usig the shared secret betwee the trasmitter ad the receiver, ad ca be used by the receiver to locate the true carrier frequecy. Our desig goal is to reiforce jammig resistace without sacrificig too much o spectral efficiecy. 4 5 5 5 5 JSRdB Ecrypted Iformatio Iitial Vector, Key Chael Codig PN Sequece Geeratio Ecryptio Secure ID Geeratio Iterleavig Y X Symbol apper Carrier Frequecy Selectio s Basebad Sigal Geeratio f X odulatio st Fig.. Performace compariso uder sigle bad jammig, /N = db, N c = 64, N h = 3. DFH uses QPSK modulatio ad covetioal FH uses 4-FSK modulatio. I this case, the spectral efficiecy of DFH is roughly 3.3 times that of covetioal FH. We compare the performace of DFH with that of covetioal FH i AWGN chaels, uder both oise jammig ad disguised jammig. The result with o chael codig is show i Fig.. The jammig-to-sigal ratio is defied as JSR = P J P s, where P J ad P s deote the jammig power ad sigal power per hop, respectively. As ca be see, DFH delivers excellet performace uder strog jammig scearios i.e., JSR, ad outperforms covetioal FH by big margis. Note that i this case, the spectral efficiecy of DFH is 3.3 times that of covetioal FH. The uderlyig argumet is that: whe the jammig power is much stroger tha the sigal power, jammig ca be easily distiguished from the true sigal whe they are i differet bads; eve if jammig collides with the sigal, the true carrier frequecy ca still be detected as jammig ca eve ehace the power of the jammed chael ad hece icreases the probability of carrier detectio. For covetioal FH, o the other had, oce the jammig power reaches a certai level, the system performace is maily limited by the probability that the sigal is jammed. However, we also otice that uder disguised jammig, the system experieces cosiderable performace losses, sice it is difficult for the DFH receiver to distiguish disguised jammig from the true sigal. The sesitivity of DFH to disguised jammig is iflueced by the SNR. To ehace the jammig resistace of DFH uder disguised jammig, i Fig. 3. AJ-DFH trasmitter structure. The trasmitter structure of AJ-DFH is illustrated i Fig. 3. Each user is assiged a secure ID sequece. We propose to replace the ordiary bits i DFH with the ID bits. I order to prevet impersoate attack, each user s ID sequece eeds to be kept secret from the malicious jammer. The ID sequece ca be geerated usig two steps as i [6]: i Geerate a pseudo-radom biary sequece usig a liear feedback shift registerlfsr; ii Take the output of LFSR as the plaitext, ad feed it ito the Advaced Ecryptio Stadard AES [7] ecrypter. The AES output is the used as our ID sequece. Recall that B c = log N c ad B s = log, where N c is the umber of chaels, ad is the costellatio size. We divide the source iformatio ito blocks of size B c ad divide the ID sequece ito blocks of size B s. Deote the th source iformatio block ad ID block as X ad Y, respectively. Let f X be the carrier frequecy correspodig to X, ad s the symbol correspodig to ID bit-vector Y. It should be oted that the ID symbol is refreshed at each hoppig period. The trasmitted sigal ca the be represeted as st = { } Re s gt T h e jπf X t = Re = N c { = i= α i, s gt T h e jπfit }, where T h is the hop duratio, gt is the pulse shapig filter, { if fx = f i, α i, = otherwise.

4 B. Receiver Desig The receiver structure for AJ-DFH is show i Fig. 4. For each hop, the received sigal is first fed ito the badpass filter bak. The output of the filter bak is demodulated, ad the used for carrier bits i.e., the iformatio bits detectio. Fig. 4. r t BPF, f t BPF, f t BPF, fnct Iitial Vector, Key Demodulatio Secure ID Geeratio AJ-DFH receiver structure. Y Sigal Detectio & Extractio s Symbol apper Recovered Iformatio Demodulatio: Let st, Jt ad t deote the ID sigal, the jammig ad the oise, respectively. For AWGN chaels, the received sigal ca be represeted as rt = st + Jt + t. 3 For i =,,, N c, the output of the ith ideal badpass filter f i t is r i t = f i t rt. For demodulatio, r i t is first shifted back to the basebad, ad the passed through a matched filter. At the th hoppig period, for i =,, N c, the sampled matched filter output correspods to chael i ca be expressed as r i, = α i, s + β i, J i, + i,, 4 where s, J i, ad i, correspod to the ID symbol, the jammig iterferece ad the oise, respectively; α i,, β i, {, } are biary idicators for the presece of ID sigal ad jammig, respectively. Note that the true iformatio is carried i α i,. Sigal Detectio ad Extractio: Sigal detectio ad extractio is performed for each hoppig period. For otatio simplicity, without loss of geerality, we omit the subscript i 4. That is, for a particular hoppig period, 4 is reduced to: r i = α i s + β i J i + i, for i =,, N c. 5 Defie r = r,..., r Nc, α = α,..., α Nc, β = β,..., β Nc, J = J,..., J Nc ad =,..., Nc, the 5 ca be rewritte i vector form as: r = sα + β J +. 6 For sigle-carrier AJ-DFH, at each hoppig period, oe ad oly oe item i α is ozero. I this case, there are N c possible iformatio vectors: α =,,...,, α =,,...,,, α Nc =,,...,. If α k is selected, ad the biary expressio of k is b b b Bc, with B c = log N c, the the estimated iformatio sequece is b b b Bc. At each hoppig period, the iformatio symbol α, or equivaletly, the hoppig frequecy idex k, eeds to be estimated based o the received sigal ad the secure ID iformatio which ca be regeerated at the receiver through the shared secret. Whe the iput iformatio vectors are equiprobable, that is, P α i = N c for i =,,..., N c, the AP maximum a posteriori probability detector is reduced to the L maximum likelihood detector. For the L detector, the hoppig frequecy idex ˆk ca be estimated as: ˆk = arg max i N c P r{r α i }. 7 Whe,..., Nc, J,..., J Nc are all statistically idepedet, r,..., r Nc are also idepedet. I this case, the joit L detector i 7 ca be decomposed as: ˆk = arg max i N c j= = arg max = arg max N c P r{r j α i } N c i N c j=,j i N c i N c j= P r{r j α j = } P r{r i α i = } P r{r j α j = } P r{r i α i = } P r{r i α i = }. 8 Sice N c j= P r{r j α j = } is idepedet of i, 8 ca be further reduced to the likelihood ratio test ˆk = arg max i N c P r{r i α i = } P r{r i α i = }, 9 where P r{r i α i = } = β i P r{r i α i =, β i }P β i ad P r{r i α i = } = β i P r{r i α i =, β i }P β i, with β i {, }. I the ideal case whe β i is kow for i =,, N c, the L detector above ca be further simplified. If we assume that,, Nc are i.i.d. circularly symmetric Gaussia radom variables of zero-mea ad variace, ad J,, J Nc are i.i.d. circularly symmetric Gaussia radom variables of zero-mea ad variace J i, the it follows from 5 ad 9 that r i ˆk r i s = arg max, i N c where i = β ij i +. Note that i is geerally ukow. If we replace the overall iterferece power i with the istataeous power of the received sigal r i, the it follows from that: i r i s ˆk = arg mi i N c r i. For more tractable theoretical aalysis, we ca replace r i with the average sigal power observed i chael i, P i = E{ r i }. Defie Z i r i s Pi, the we have ˆk = arg mi Z i. i N c Discussios: I the covetioal FH, a frequecy sythesizer is used at the receiver to capture the trasmitted sigal. The strict requiremet o frequecy sychroizatio turs out to be a sigificat challege i FH system desig, especially for fast hoppig systems. I DFH ad AJ-DFH, a badpass filter bak is used to capture the hoppig frequecy. The complexity is similar to usig multiple FSK receivers i parallel. It is observed that: Comparig with covetioal FH, DFH based systems relax the frequecy sychroizatio

5 problem. At the same time, we show that if a badpass filter bak is used by a adversary, the the covetioal FH sigal as well as the PN sequece ca be easily captured, makig it fragile to follower jammig ad resultig i total loss of the trasmissio. I AJ-DFH, the ecrypted iformatio is trasmitted through hoppig frequecy cotrol, this is like usig the oe-time key pad, makig it impossible for the adversary to lauch follower jammig. Comparig with FSK, which has zero capacity uder disguised jammig, AJ- DFH is much more efficiet ad robust uder disguised jammig. C. Extesio to ulti-carrier AJ-DFH For more efficiet spectrum usage ad robust jammig resistace, i this sectio, we exted the cocept of DFH to multi-carrier AJ-DFH C-AJ-DFH. The idea is to split all the N c chaels ito N g o-overlappig groups, ad each subcarrier hops withi the assiged group based o the AJ-DFH scheme. To esure hoppig radomess of all the subcarriers, the groups eed to be reorgaized or regeerated securely after a pre-specified period, amed group period. A secure subgroup assigmet algorithm ca be developed as what we did i [7], to esure that: i Each subcarrier hops over a ew group of chaels durig each group period, so that it evetually hops over all the available chaels i a pseudoradom maer; ii Oly the legitimate receiver ca recover the trasmitted iformatio correctly. ulti-carrier AJ-DFH without Diversity: I this case, each subcarrier trasmits a idepedet bit stream. The spectral efficiecy of the AJ-DFH system ca be icreased sigificatly. Let B c = log N c ad B g = log N g, the the umber of bits trasmitted by the C-AJ-DFH withi each hoppig period is B C = B c B g N g = B c log N g N g. B C is maximized whe B g = B c or B g = B c, which results i B C = Bc. Note that the umber of bits trasmitted by the AJ-DFH withi each hoppig period is B c, it ca be see that B C > B c as log as B c >. Take N c = 56 for example, the the trasmissio efficiecy of AJ-DFH ca be icreased by B C B c = B c B c = 6 times. ulti-carrier AJ-DFH with Diversity: Uder multibad jammig, diversity eeds to be itroduced to the AJ- DFH system for robust jammig resistace. A atural solutio to achieve frequecy diversity is to trasmit the same or correlated iformatio through multiple subcarriers. The umber of subcarriers eeded to covey the same iformatio varies i differet jammig scearios. Geerally, the umber of correlated sigal subcarriers should ot be less tha the umber of jammed bads. At the receiver, the received sigals from differet diversity braches ca be combied for joit sigal detectio [8] [3]. As will be show i Sectio VI, C-AJ-DFH ca icrease the system efficiecy ad jammig resistace sigificatly through jammig radomizatio ad frequecy diversity. oreover, by assigig differet carrier groups to differet users, C-AJ-DFH ca also be used as a collisiofree multiple access system. IV. ID CONSTELLATION DESIGN AND ITS IPACT ON SYSTE PERFORANCE For AJ-DFH, ID sigals are itroduced to distiguish the true iformatio chael from disguised chaels ivoked by jammig iterferece. I this sectio, we ivestigate ID costellatio desig ad its impact o the performace of AJ- DFH uder various jammig scearios. A. Desig Criterio ad Jammig Classificatio The geeral desig criterio of the ID costellatio is to miimize the probability of error uder a give sigal power. Uder this criterio, the followig questios eed to be aswered: How does the size of the costellatio impact the system performace? How does the type or shape of the costellatio ifluece the detectio error? Which type should we use for optimal performace? I this sectio, we will try to address these questios uder differet jammig scearios. I literature, jammig has geerally bee modeled as Gaussia oise [8] [], referred as oise jammig. Recall that disguised jammig deotes the jammig iterferece which has similar power ad spectral characteristics as that of the true sigal. For AJ-DFH, whe the ID costellatio is kow to, or ca be guessed by the jammer, the jammer ca the disguise itself by sedig symbols take from the same costellatio over a differet or fake chael. I this case, it could be difficult for the receiver to distiguish the true chael from the disguised chael, leadig to high detectio error probability. We refer to this kid of jammig as ID jammig or ID attack. ID jammig is the worst case disguised jammig for AJ-DFH. B. Costellatio Desig uder Noise Jammig Without loss of geerality, we cosider the case where the ID symbol is trasmitted through chael, i.e., α = α,, α Nc =,,,. 3 Recall that for i =,, N c, r i = α i s + β i J i + i. Let ñ i = β i J i + i, ad deote its variace as i = β i J i +, which may vary from chael to chael. Followig the defiitio i, we have Z = ñ, ad Z i = ñ i s s + i for i N c. It ca be see that Z is a Rayleigh radom variable with probability desity fuctio PDF p Z z = z where = radom variable with PDF z e, z >, 4 s +. For i N c, Z i is a Ricia p Zi z i = z z i e i +ν zi ν I, z i >, 5 where ν = s i, = ad I x is the modified Bessel fuctio of the first kid with order zero. Accordig to, the carrier ca be correctly detected if ad oly if Z < Z i for all i N c. Assumig that

6 the symbols i costellatio Ω are equally probable, the the carrier detectio error probability is give by P e = P r{z < Z,..., Z < Z Nc s}p S s = N c P r{z i > z s, Z = z }p Z z dz. i= Note that Z,, Z Nc are i.i.d. Ricia radom variables, the it follows from 4 ad 5 that P e = N c Q s, z i s + z e s z dz, 6 where Q is the arcum Q-fuctio [3]. We have the followig result: Propositio : Assumig the true chael idex is k. Uder oise jammig, a upper boud of the carrier detectio error probability P e ca be obtaied as: P U e = i= + k s + k e s s + k where m = arg max{ l } for l N c, l k. Proof: See Appedix A. Assumig the true chael idex is k, let x = s x +x x+ e ζ k ad ax = x+ N c. The Pe U ca be writte as Pe U = ax with ζ = k / m. Note that whe x, ax N c x +x x+ e ζ x+ ãx. It ca be show that whe x, ãx is a covex fuctio. By Jese s iequality [3], we have Pe U s ã ã k k s Ps = ã k 8 The equality is achieved if ad oly if s = P s for all s Ω. This implies that: uder the coditio that the sigal. to jammig ad oise ratio over chael k satisfies s k, Pe U is approximately miimized whe the costellatio is costat modulus, that is, s = P s for all s Ω. A ituitive explaatio for this result is that the sigal power i costat modulus costellatios always equals to the maximal sigal power available. oreover, it ca be see that Pe U is idepedet of the costellatio size, but is oly a fuctio of P s /k. Next, we will ivestigate how the costellatio size affects the system performace uder ID attacks. C. Costellatio Desig uder ID Jammig Clearly, uder ID attacks, the ucertaity of the ID symbol eeds to be maximized. Uder the assumptio that all the symbols i a costellatio Ω of size are all equally probable, the the average symbol etropy Hs = log = log bits. 9 I the ideal case whe the chael is oise-free, the optimal costellatio size would be =. However, whe oise is preset, a larger also implies there is a larger probability for a ID symbol to be mistake for its eighborig symbols. ore specifically, we have the followig result: Theorem : For a give SNR ad assumig PSK costellatio is utilized, uder ID jammig, the carrier detectio error probability P e is a fuctio of costellatio size ad lim P e = P e. I other words, for ay give ϵ >, there always exists a t such that for all > t, P e P e < ϵ. The expressio of P e ad the proof of the theorem ca be foud i Appedix B. This theorem essetially says that: for a give SNR, due to the oise effect, icreasig the costellatio size over a threshold t will result i little improvemet i detectio error probability. This result justifies the use of fiite costellatio i AJ-DFH. V. SPECTRAL EFFICIENCY ANALYSIS Nc The spectral efficiecy ν is defied as the ratio of the m s + k, iformatio bit rate R b to the trasmissio badwidth W t, i.e., ν = R b W 7 t. I this sectio, we will aalyze ad compare the spectral efficiecy of the existig ad proposed frequecy hoppig schemes, icludig covetioal FH, DFH ad C-AJ-DFH. We start with the sigle-user case. Recall that T s ad T h deotes the symbol period ad the hoppig duratio, respectively; N h = T s /T h is the umber of hops per symbol period. For fair compariso, we assume that all systems have: i The same umber of available chaels N c ; ii The same hoppig period T h to esure the hoppig chaels have the same badwidth W c = /T h ; iii The same frequecy spacig f betwee two adjacet subcarriers, where f /T h is chose to avoid iter-carrier iterferece. Note that uder these assumptios, all systems have the same total badwidth W t = N c f + W c. For covetioal FH usig FSK modulatio, log bits are trasmitted durig each symbol period. The bit rate of covetioal FH ca be calculated as R b = log T h N h, ad the correspodig spectral efficiecy ca be obtaied as ν = R b W t = log T h N h W t. The bit rate ad spectral efficiecy of other frequecy hoppig schemes ca be obtaied similarly. The results are listed i Table I. = log T s TABLE I THE BIT RATE R b AND SPECTRAL EFFICIENCY ν COPARISON IN THE SINGLE-USER CASE covetioal FH DFH AJ-DFH C-AJ-DFH R b b/s log T h N h N h B c +B s N h T h B c T h T h ν b/s/hz log T h N h W t N h B c+b s T h N h W t B c T h W t B c log N g N g B c log N g N g T h W t Next, we cosider the more geeral multiuser case. The multiple access scheme for covetioal FH, amely FHA, was proposed i [6]. The multiple access extesio of DFH, deoted as E-DFH, has bee aalyzed i []. Due to the variability i multiple access system desig, a closed-form expressio of the spectral efficiecy is hard to obtai. Here we

7 compare the total iformatio bits allowed to be trasmitted by each system uder the same ad badwidth requiremets, ad illustrate the spectral efficiecy compariso through the followig example. Let N u deote the umber of users ad we choose the umber of chaels be N c = 64 i.e., B c = 6. For C- AJ-DFH, we choose N g = N u = 4; For E-DFH, we choose 8-PSK to modulate B s = 3 ordiary bits ad N g = N u = 4, N h = 3. For FHA, we choose 64-FSK modulatio, N h = 3, ad cosider N u =, 3, 4, respectively. The required is 4. Fig. 5 depicts the performace of these multiuser systems. From Fig. 5a, it ca be see that both C-AJ-DFH ad E-DFH achieve the desired at N 6.5dB. From Fig. 5b, it ca be observed that: due to severe collisio effect amog differet users, FHA ca oly N accommodate up to users at 6.5dB for the desired. I this particular example, the spectral efficiecy of both C-AJ-DFH ad E-DFH are 4 times ad 5 times that of FHA. For example, for /N = 5dB, we ca choose t = 3. This example demostrates the theoretical result i Theorem. Based o the result of Example, i the followig examples, we choose to use 3-PSK to modulate the ID sigal of AJ- DFH ad C-AJ-DFH. 3 /N =5dB /N =db /N =5dB Noise free 3 4 5 6 7 8 Costellatio size Fig. 6. Example : The performace of AJ-DFH with differet costellatio size, uder sigle-bad ID jammig. 3 4 C AJ DFH E DFH 5 6 7 8 9 /N db a E-DFH ad C-AJ-DFH, N u = 4 3 4 N u = N u =3 N u =4 5 6 7 8 9 /N db b FHA Fig. 5. Performace compariso of C-AJ-DFH, E-DFH ad FHA i the multiuser case. Here N u deotes the umber of users i the system. Based o our aalysis above, as well as the performace aalysis of AJ-DFH uder various jammig attacks i Sectio VI, it ca be show that: while AJ-DFH is much more robust tha DFH uder various jammig attacks, its spectral efficiecy is very close to that of DFH, which is several times higher tha that of covetioal FH. A comprehesive capacity aalysis for DFH ad AJ-DFH uder disguised jammig is provided i Part II of the paper []. VI. SIULATION RESULTS I this sectio, simulatio examples are provided to illustrate the performaces of the proposed AJ-DFH ad C- AJ-DFH schemes uder various jammig scearios. For all the systems cosidered i the followig examples, we assume the total umber of available chaels is N c = 64, that is, B c = 6. Example : Impact of the ID costellatio size I this example, we cosider the impact of the ID costellatio size o the performace of AJ-DFH uder sigle-bad ID jammig. From Fig. 6, it ca be see that i the ideal case where the system is oise-free, the performace of AJ-DFH improves cotiuously as the costellatio size icreases. However, whe oise is preset, the coverges oce the costellatio size reaches a certai threshold t. Example : Performace compariso uder sigle bad jammig I this example, we cosider both oise jammig ad disguised jammig. SNR is take as /N = db ad Jammig-to-Sigal Ratio JSR is defied as the ratio of the jammig power to sigal powers durig oe hoppig period. For covetioal FH, 4-FSK modulatio scheme is used ad we assume that the adjacet frequecy toes i 4-FSK correspod to the ceter frequecies of the adjacet DFH chaels. For DFH, QPSK is used to modulate ordiary bits ad the umber of hops per symbol period is N h = 3. From Fig. 7, it ca be observed that AJ-DFH ca effectively reduce the performace degradatio caused by disguised jammig, while remaiig robust uder oise jammig. Note that JSR=dB uder disguised jammig correspods to the ID jammig for AJ-DFH. It ca also be see that the performace of AJ-DFH improves sigificatly whe the jammig power differs from the sigal power. This implies that ucertaity i the sigal power is aother dimesio i combatig ID jammig. Fig. 7. 3 4 covetioal FH: Disguised Jammig covetioal FH: Noise Jammig DFH: Disguised Jammig DFH: Noise Jammig AJ DFH: Disguised Jammig AJ DFH: Noise Jammig 5 5 5 5 JSRdB Example : Performace compariso uder sigle bad jammig. Example 3: Performace compariso uder multi-bad oise jammig ad disguised jammig I this example,

8 /N = db. For disguised jammig, the jammer takes symbols radomly from the same costellatio as the ID sigal. The jammed bads are selected idepedetly ad radomly. For C-AJ-DFH without diversity, the chaels are divided ito 3 groups to maximize the spectral efficiecy; for C-AJ-DFH with diversity, each symbol is trasmitted simultaeously over 4 subcarriers to achieve frequecy diversity. The equal gai combiatio scheme is adopted for the joit detectio metric at the receiver. From Fig. 8 ad Fig. 9, it ca be see that C-AJ-DFH delivers much better performace tha sigle carrier AJ-DFH uder multi-bad jammig. Fig. 8. Fig. 9. jammig. 3 4 5 DFH covetioal FH AJ DFH C AJ DFH: without diversity C AJ DFH: with diversity 6 5 5 5 5 JSRdB Example 3: Performace compariso uder -bad oise jammig. 4 DFH covetioal FH AJ DFH C AJ DFH: without diversity C AJ DFH: with diversity 6 5 5 5 5 JSRdB Example 3: Performace compariso uder -bad disguised whe oise is preset, the detectio error probability coverges as the costellatio size goes to ifiity. oreover, AJ-DFH ca be exteded to C-AJ-DFH by allowig simultaeous multi-carrier trasmissio. With jammig radomizatio ad eriched frequecy diversity, C-AJ-DFH ca icrease the system efficiecy ad jammig resistace sigificatly, ad ca readily be used as a collisio-free multiple access scheme. ACKNOWLEDGEENT This work was supported i part by the Natioal Sciece Foudatio uder grats CNS-7468, CNS-783, CNS- 76, ad CNS-39. It follows from 6 that P e = APPENDIX A PROOF OF PROPOSITION N c Q s, z p Z z dz l l= s, z p Z z dz m Q Nc where m = arg max l Nc {l }. The iequality follows from the fact that for fixed s ad z, Q l s, z is a mootoically decreasig fuctio with respect to l. The equality ca be achieved whe = = Nc. Assume N c. For N c =, it is easy to show that P e = Pe U with m =. Note that for fixed s ad m, Q m s, z = P r{z m > z s, z = Z } is a fuctio of z. For N c >, fx = x Nc is covex whe x >. By Jese s iequality, we obtai Q N c s, z p Z z dz m [ ] Nc P r{z m > z s, Z = z }p Z z dz = [P r{z < Z m s}] N c. 3 Accordig to [33], P r{z < Z m s} ca be calculated as VII. CONCLUSIONS I this paper, we proposed a efficiet ati-jammig scheme, AJ-DFH, based o the message-drive frequecy hoppig techique. It was show that by isertig a secure ID sequece i trasmissio, AJ-DFH ca effectively reduce the performace degradatio caused by disguised jammig. The impact of ID costellatio desig o system performace was ivestigated uder both oise jammig ad ID jammig. It was proved that for a give power costrait, costat modulus costellatio delivers the best results uder oise jammig i terms of detectio error probability; While uder ID jammig, P r{z < Z m s} = s + Followig -4, P e s + e s s + m s +. 4 e s s + Nc m s + = P U e for N c >. 5 Overall, P e P U e for N c..

9 APPENDIX B PROOF OF THEORE Note that the system is uder ID attack. If the power of the -ary PSK costellatio is P s, the the sigal ad jammig symbol ca be writte as s = P s e j πm s, J j = P s e j πm J respectively, where = ad m s, m J. Without loss of geerality, we assume that: i Both the ID ad jammig take the symbols i Ω with equal probability /; ii The sigal is trasmitted i chael α = ad chael j is jammed β j =. Whe j =, jammig collides with the ID sigal. I this case, r = s + J + ad r l = l for l =,..., N c. J We have Z = + ad Z l = l s s+j +, where is the oise variace. The detectio error probability i this case ca be calculated as P e = [P r{z < Z s, J, z }] N c J Ω p Z z dz. 6 Note that Z is a Ricia radom variable with PDF p Z z = z Ps s+j +, = e z +ν s+j + I zν, where ν = ad Z l s are i.i.d. Ricia radom variables with ν = P s, =. The 6 ca be writte as i 7, where κ m s m J mod is uiformly distributed over [, ]. Whe j =,..., N c, jammig does ot collide with ID sigal. I this case, r = s +, r j = J j + j ad r l = l for l =,..., N c, l j. We have Z = Z j = Jj s+j Ps + probability i this case ca be calculated as P e = Ps + ad Z l = l s. The detectio error J j Ω P r{z j > z s, J j, z } [P r{z l > z s, z }] Nc p Z z dz. 8 Note that Z is a Rayleigh radom variable with PDF p Z z = z e z, where = Ps + P s+, Z j is a Ricia radom variable with ν = J j s, =, P s + ad Z l s are i.i.d. Ricia radom variables with ν = P s, =. The 8 ca be writte as i 9. The overall detectio error probability i oisy eviromet is give as P e = P r{j = }P e + P r{ j N c }P e = N c P e + N c N c P e. 3 Whe P s is fixed, it follows from 7 ad 9 that P e is a fuctio of give as P e = κ= b πκ, 3 where bx is give i 3. As approaches ifiity, P e coverges. I fact, we have, P e = lim P b πκ π e = lim π = π = π b lim κ= π κ= πκ π bxdx. 33 Note that bx is the detectio error probability whe the agle betwee the sigal symbol ad the jammig symbol is x, hece bx ad we have P e = π π bxdx. 34 That is: ϵ >, there always exists a iteger t such that > t, P e P e < ϵ. REFERENCES [] A. pitziopoulos, D. Gavalas, C. Kostatopoulos, ad G. Patziou, A survey o jammig attacks ad coutermeasures i WSNs, IEEE Commu. Surveys Tuts., vol., o. 4, pp. 4 56, 9. [] H. Yag, F. Ricciato, S. Lu, ad L. Zhag, Securig a wireless world, Proc. IEEE, vol. 94, o., pp. 44 454, Feb. 6. [3] W. Xu, W. Trappe, Y. Zhag, ad T. Wood, The feasibility of lauchig ad detectig jammig attacks i wireless etworks, i Proc. AC It. Symp. obile Ad Hoc Networkig Comput. AC, 5, p. 57. [4]. K. Simo, J. K. Omura, R. A. Scholtz, ad B. K. Levitt, Spread Spectrum Commuicatios Hadbook. cgraw-hill, 994. [5] G. Cooper ad R. Nettleto, A spread spectrum techique for high capacity mobile commuuzatio, IEEE Tras. Veh. Techol., vol. 7, pp. 64 75, Nov. 978. [6] A. Viterbi, A processig satellite traspoder for multlple access by low rate mobile users, i Proc. Digital Satellite Commu. Cof., otreal, Caada, Oct 978. [7]. Simo ad A. Polydoros, Coheret detectio of frequecy-hopped quadrature modulatios i the presece of jammig part I: QPSK ad QASK modulatios, IEEE Tras. Commu., vol. 9, pp. 644 66, Nov 98. [8] R. Pickholtz, D. Schillig, ad L. ilstei, Theory of spread-spectrum commuicatios: A tutorial, IEEE Tras. Commu., vol. 3, o. 5, pp. 855 884, ay 98. [9]. Pursley ad W. Stark, Performace of reed-solomo coded frequecy-hop spread-spectrum commuicatios i partial-bad iterferece, IEEE Tras. Commu., vol. 33, pp. 767 774, Aug 985. [] K. Choi ad K. Cheu, Performace of asychroous slow frequecyhop multiple-access etworks with FSK modulatio, IEEE Tras. Commu., vol. 48, o., pp. 98 37, Feb. [] Q. Lig ad T. Li, essage-drive frequecy hoppig: Desig ad aalysis, IEEE Tras. Wireless Commu., vol. 8, o. 4, pp. 773 78, April 9. [] L.-L. Yag ad L. Hazo, Overlappig -ary frequecy shift keyig spread-spectrum multiple-access systems usig radom sigal sequeces, IEEE Tras. Veh. Techol., vol. 48, o. 6, pp. 984 995, Nov 999. [3] S. Glisic, Z. Nikolic, N. ilosevic, ad A. Pouttu, Advaced frequecy hoppig modulatio for spread spectrum WLAN, IEEE J. Sel. Areas Commu., vol. 8, o., pp. 6 9, Ja. [4] J. Cho, Y. Kim, ad K. Cheu, A ovel frequecy-hoppig spreadspectrum multiple-access etwork usig -ary orthogoal Walsh sequece keyig, IEEE Tras. Commu., vol. 5, o., pp. 885 896, Nov. 3. [5] K. Choi ad K. Cheu, aximum throughput of FHSS multiple-access etworks usig FSK modulatio, IEEE Tras. Commu., vol. 5, pp. 46 434, ar 4. [6] Y. Kim, K. Cheu, ad K. Yag, A badwidth-power efficiet modulatio scheme based o quaterary quasi-orthogoal sequeces, IEEE Commu. Lett., vol. 7, o. 7, July 3.

P e = J Ω e s+j + z +P s = [ e κ= Q N c Ps, s + J + z z z I Ps s + J + dz Ps πκ +cos + Q N c Ps ], z z P s I z z [ Ps P s + cos πκ + cos πκ + P s ] + dz, 7 P e = Ps + Q J j s, z J Ω Q Nc Ps, Ps + z z e Ps+ z dz = Q P s cos πκ, Ps + κ= z Q N c Ps, Ps + z z e Ps+ z dz. 9 bx = N c [ ] e Ps +cos x+ N c N c Q N c Q N c Ps z P s Ps, z, z I z [ ] Ps z + cos x + P s + cos x + P s dz Q Ps cos x, Ps + z Ps + z e Ps+ z dz. 3 [7] Advaced Ecryptio Stadard, FIPS-97, Natioal Istitute of Stadards ad Techology Std., Nov.. [8] J. Lee ad L. iller, Error performace aalyses of differetial phaseshift-keyed/frequecy-hoppig spread-spectrum commuicatio system i the partial-bad jammig eviromets, IEEE Tras. Commu., vol. 3, o. 5, pp. 943 95, ay 98. [9] J. Kag ad K. Teh, Performace of coheret fast frequecy-hopped spread-spectrum receivers with partial-bad oise jammig ad AWGN, IEE Proc. Commu., vol. 5, o. 5, pp. 679 685, Oct. 5. [] C. Esli ad H. Delic, Atijammig performace of space-frequecy codig i partial-bad oise, IEEE Tras. Veh. Techol., vol. 55, o., pp. 466 476, arch 6. [] L. Zhag ad T. Li, Ati-jammig message-drive frequecy hoppig: Part II capacity aalysis uder disguised jammig, IEEE Tras. Wireless Commu.,, to appear. [] L. Zhag, J. Re, ad T. Li, Spectrally efficiet ati-jammig system desig usig message-drive frequecy hoppig, i Proceedigs of IEEE Iteratioal Coferece o Commuicatios, Ju. 9. [3] R. Che, J.-. Park, ad J. Reed, Defese agaist primary user emulatio attacks i cogitive radio etworks, IEEE Joural o Selected Areas i Commuicatios, vol. 6, o., pp. 5 37, 8. [4] H. Li ad Z. Ha, Dogfight i spectrum: Combatig primary user emulatio attacks i cogitive radio systems, part i: Kow chael statistics, IEEE Trasactios o Wireless Commuicatios, vol. 9, o., pp. 3566 3577,. [5], Dogfight i spectrum: Combatig primary user emulatio attacks i cogitive radio systems, part ii: Ukow chael statistics, IEEE Trasactios o Wireless Commuicatios, vol., o., pp. 74 83,. [6] T. Li, Q. Lig, ad J. Re, Physical layer built-i security aalysis ad ehacemet algorithms for CDA systems, EURASIP J. Wireless Commu. Networkig, vol. 7, pp. Article ID 83 589, 7 pages, 7. [7] L. Lightfoot, L. Zhag, ad T. Li, Secure collisio-free frequecy hoppig for OFDA based wireless etworks, EURASIP J. Advaces Sigal Process., vol. 9, 9. [8] R. Viswaatha ad K. Taghizadeh, Diversity combiig i FH/BFSK systems to combat partial bad jammig, IEEE Tras. Commu., vol. 36, o. 9, pp. 6 69, Sep 988. [9] J. Lee, L. iller, ad Y. Kim, Probability of error aalyses of a BFSK frequecy-hoppig system with diversity uder partial-bad jammig iterferece part II: Performace of square-law oliear combiig soft decisio receivers, IEEE Tras. Commu., vol. 3, o., pp. 43 5, Dec 984. [3] L. iller, J. Lee, ad A. Kadrichu, Probability of error aalyses of a BFSK frequecy-hoppig system with diversity uder partialbad jammig iterferece part III: Performace of a square-law selformalizig soft decisio receiver, IEEE Tras. Commu., vol. 34, o. 7, pp. 669 675, Jul 986. [3] J. G. Proakis ad. Salehi, Digital Commuicatios, 5th ed. New York: cgraw-hill, 8. [3]. Kuczma, A Itroductio to the Theory of Fuctioal Equatios ad Iequalities: Cauchy s Equatio ad Jese s Iequality, d ed. Spriger, 9. [33] S. Stei, Uified aalysis of certai coheret ad ocoheret biary

commuicatios systems, IEEE Tras. If. Theory, vol., o., pp. 43 5, Ja 964. Lei Zhag received the B.S. ad.s. degrees i commuicatio egieerig i 5 ad 7, respectively, both from Xidia Uiversity, Xi a Chia. He received the Ph.D. degree i electrical ad computer egieerig i, from ichiga State Uiversity, East Lasig I. Dr. Zhag joied arvell Semicoductor i, ad is curretly workig i the area of mobile SOC desig ad verificatio. Huahui Wag received his B.S. degree i Electroics from Pekig Uiversity, Chia i, ad the.eg ad Ph.D degrees i Electrical Egieerig from Natioal Uiversity of Sigapore ad ichiga State Uiversity i 3 ad 6, respectively. Dr. Wag s research iterests ivolved various desig aspects of wireless commuicatio ad etworkig, icludig PHY layer system desig ad AC/higher lay protocol aalysis. From 7 to 8, he worked as a research egieer i LG/Zeith Electroics ad cotributed to the stadardizatio of the ew mobile DTV systems for the Uited States. From 9 to, he worked as a Research Associate at ichiga State Uiversity workig o cogitive radios ad ati-jammig systems. I, he joied AT&T Labs at Florham Park, NJ, where he is curretly a Seior ember of Techical Staff. His mai research activities are i the area of etwork plaig ad optimizatio for both UTS ad LTE systems. Togtog Li received her Ph.D. degree i Electrical Egieerig i from Aubur Uiversity. From to, she was with Bell Labs, ad had bee workig o the desig ad implemetatio of 3G ad 4G systems. Sice, she has bee with ichiga State Uiversity, where she is ow a Associate Professor. Dr. Li s research iterests fall ito the areas of wireless ad wired commuicatios, wireless security, iformatio theory ad statistical sigal processig. She is a recipiet of the Natioal Sciece Foudatio NSF CAREER Award 8 for her research o efficiet ad reliable wireless commuicatios. She served as a Associate Editor for IEEE Sigal Processig Letters from 7-9, ad a Editorial Board ember for EURASIP Joural Wireless Commuicatios ad Networkig from 4-. She is curretly servig as the Associate Editor for IEEE Trasactios o Sigal processig.