Frequency hopping does not increase anti-jamming resilience of wireless channels

Transcription

1 Frequeny hopping does not inrease anti-jamming resiliene of wireless hannels Moritz Wiese and Panos Papadimitratos Networed Systems Seurity Group KTH Royal Institute of Tehnology, Stoholm, Sweden {moritzw, Abstrat The effetiveness of frequeny hopping for antijamming protetion of wireless hannels is analyzed from an information-theoreti perspetive. The sender an input its symbols into one of several frequeny subbands at a time. Eah subband hannel is modeled as an additive noise hannel. No ommon randomness between sender and reeiver is assumed. It is shown that apaity is positive, and then equals the ommon randomness assisted CR) apaity, if and only if the sender power stritly exeeds the jammer power. Thus ompared to transmission over any fixed frequeny subband, frequeny hopping is not more resilient towards jamming, but it does inrease the apaity. Upper and lower bounds on the CR apaity are provided. I. INTRODUCTION A wireless hannel is open to inputs from anybody operating on the same frequeny. Therefore ommuniation has to be proteted against deliberate jamming. This means that ommuniation protools have to be devised whose appliation enables reliable data transmission even if attaed by a jammer. If a suffiiently broad frequeny band is available, and if the jammer does not have simultaneous aess to the omplete band, a method whih suggests itself is frequeny hopping FH). The frequeny spetrum is divided into subbands. In eah time slot, the sender hooses a subband in a random way and uses only that frequeny to transmit data in that time slot. In some models [4], [6], the reeiver hops over frequenies, too, and only listens to one subband at a time. The idea is that in this way, the hannel will not be jammed all the time with positive probability, and some information will go through. To sueed, the basi FH idea requires ommon randomness nown to sender and reeiver, but unnown to the jammer. A areful analysis of that situation has been performed in [4]. It is learly neessary that the ommon randomness realization be nown before transmission starts. As the hannel annot be used to distribute this nowledge, this leads to a irle alled anti-jamming/ey-establishment dependeny in [6]. In [6] it has been investigated for the first time whether FH an be used for data transmission without the availability of ommon randomness. Moreover, the jammer is allowed to distribute its power arbitrarily over all frequeny subbands and use these simultaneously. It is assumed that whether the jammer inserts, modifies or jams messages only depends on the relation of its own and the sender s power. A protool is found whih ahieves a positive throughput depending on the jammer s strategies, e.g. whether or not it an listen to the sender s signals. We tae a different perspetive in this wor. The entral figure of merit for our ommuniation system is the message transmission error inurred under a jamming atta. A good FH protool should mae this error small. We assume that the jammer annot listen to symbols sent through the hannel this in partiular differs from [6]), that it nows the hannel and the ode, but not the speifi message sent, and that it nows when the transmission of a new odeword begins. It an input symbols into any frequeny subset of a given size. We also assume that the reeiver listens to all frequenies simultaneously. Within these boundaries, any jammer strategy is allowed. The jammer is suessful if no oding strategy an be found maing the transmission error vanish with inreasing oding blolength for any jamming strategy. This is an operational approah to measure the suess of jamming, in ontrast to the approah of [6] desribed above. Using the information-theoreti model of an additive Arbitrarily Varying Channel AVC) and the analysis in [2], we find that the suess of a jammer indeed depends on the relation between its own and the sender s power. In fat, if the sender power is stritly larger than the jammer power, the same, positive apaity is ahieved as in the ase where sender and reeiver have aess to ommon randomness whih is unnown to the jammer. If the onverse relation between sender and jammer power holds, then no data transmission at all is possible. This is independent of the number J of hannels the jammer an influene at the same time. On the other hand, it is nown that for eah frequeny subband the same holds: If the jammer has more power than the sender, no ommuniation is possible over this band, whereas the ommon randomness assisted apaity is ahieved in ase the sender power exeeds the jammer power. Thus in the ase that no single frequeny subband has a positive apaity without ommon randomness, then no FH sheme ahieves a positive apaity either. Seen from this perspetive, FH does not provide any additional protetion against jamming ompared to shemes whih sti to one single frequeny. However, FH does in general inrease the ommon randomness assisted apaity ompared to the use of one single

2 subhannel, and hene also the apaity without ommon randomness if positive the FH sequene may depend on the message and thus reveal additional information. In [8], [7] this is alled message-driven frequeny hopping.) The ommon randomness assisted apaity will in general depend on the number J of subhannels the jammer an simultaneously influene. Thus the apaity ahievable without ommon randomness, if positive, also depends on J. We give a lower bound for the ommon randomness assisted apaity. If the noise is Gaussian and J is suffiiently large, we also provide an upper bound whih differs from the lower bound by the logarithm of the number of frequeny bands. The bounds involve a waterfilling strategy for the distribution of the jammer s power over the frequenies. Organization of the paper: Setion II presents the hannel model and the main results. Setions III-VI ontain the proofs of these results. A disussion onludes the paper in Setion VII. II. SYSTEM MODEL AND MAIN RESULTS The total frequeny band available for ommuniation is divided into K frequeny subbands. These are modeled as parallel hannels with additive noise. The reeiver listens to all frequenies simultaneously. Frequeny hopping FH) means that the sender at eah time instant hooses one of the K subhannels into whih it inputs a signal. For a fixed number J with 1 J K, the jammer an at eah time instant hoose a subset I of the K subhannels with I = J and input its own signals in subhannels belonging to this subset. The overall hannel, alled FH hannel in the following, an be desribed as an additive Arbitrarily Varying Channel AVC) with additive noise. For any K = {1,..., K}, we set e 1,..., e K ) = e to be the vetor with e = 1 and e l = 0 for l. Further for any I with I = J, we set e I,1,..., e I,K ) = e I to be the vetor satisfying e I,l = 1 if l I and e I,l = 0 else. If the sender hooses symbol x R to transmit over subhannel, it inputs xe into the hannel. We denote the set R K by X. The jammer hoooses a subset I K of subhannels for possible jamming I = J) and a vetor s 1,..., s K ) = s R K of real numbers satisfying s l = 0 if l / I. Then it inputs s e I into the hannel, where the symbol denotes omponent-wise multipliation. We denote the set of possible jammer hoies by S := {I} {s R K : l I s l = 0} I K: I =J The noise on different frequenies is assumed to be independent. Thus the noise probability distribution of the overall hannel is determined by the noise distributions on the subhannels. For subhannel, let N be the noise random variable. Its mean is assumed to be zero and its variane is denoted by σ 2. The random vetor N 1,..., N K ) is denoted by N. Given sender input xe and jammer input s e I, the reeiver obtains a real K-dimensional output vetor y 1,..., y K ) = y through the FH hannel whih satisfies y = xe + s e I + N. In partiular, on frequenies without sender or jammer inputs, the output is pure noise. The hannel is memoryless over time, i.e. outputs at different time instants are independent onditional on the sender and jammer inputs. Note that this is an additive AVC, but as its input alphabet is a strit subset of R K, the speial results of [2] on additive-noise AVCs do not apply here. The general theory developed in [2] is appliable, though: All alphabets involved are omplete, separable metri spaes 1, the hannel output distribution ontinuously depends on the sender and jammer inputs, and the onstraints on sender and jammer inputs to be defined below are ontinuous. Hene the entral hypotheses H.1)-H.4) of [2] are satisfied. The protools used for data transmission are blo odes. A blolength-n ode is defined as follows. We assume without loss of generality that the set of messages M n is the set {1,..., M n }. An enoder is a mapping f n from M n into the set of sequenes of sender hannel inputs of length n, {x 1 e 1,..., x n e n ) : x i, i ) X 1 i n)}. Note that this means that the sequene of frequeny bands used by the sender may depend on the message to be sent. Every odeword an be onsidered as a K n-matrix whose i-th olumn is the i-th hannel input vetor. The deoder at blolength n is a mapping ϕ n : R K n M n. Additionally, for some Γ > 0, the sender has the power onstraint n i=1 f nm) i 2 nγ for all m M n, where f n m) i denotes the i-th olumn of the K n-matrix f n m) and denotes the Eulidean norm on R K. A ode f n, ϕ n ) with blolength n whih satisfies the power onstraint for Γ is alled an n, Γ)-ode. We are interested in the transmission error inurred by a ode f n, ϕ n ). This error should be small for all possible jammer input sequenes. Thus we first define the transmission error for a given length-n jamming sequene I 1, s 1 ),..., I n, s n )). This sequene an be given matrix form as well. We denote by S the K n-matrix whose i- th olumn equals s i. By Ẽ RK n, we denote the matrix with olumns e I1,..., e In. Of ourse, S Ẽ = S. We eep Ẽ expliit beause S itself does not in general uniquely determine the sequene I 1,..., I n ), as some omponents of s i ould be zero 1 i n). Just lie the sender, the jammer also has a power onstraint. We require that n i=1 s i 2 nλ for some Λ > 0 and denote the set of S Ẽ satisfying this power onstraint by J Λ. It is lear that a realisti jammer annot transmit at arbitrarily large powers, so this is a reasonable assumption. Note that the jammer is free to distribute its power over the subhannel subset it has hosen for jamming. In partiular, the power an be onentrated on one single frequeny no matter what J is. 1 Giving a disrete set K the metri ρ, l) = 1 if l and ρ, ) = 0 for all, l K maes K a omplete metri spae whose Borel algebra is its omplete power set.

3 Now let f n, ϕ n ) be a blolength-n ode and S Ẽ R K n a jammer input. Then the average error inurred by f n, ϕ n ) under this jamming sequene is defined to equal ēf n, ϕ n, S Ẽ) = 1 P[ϕ n f n m) + M n S m M n Ẽ + Ñ) m], where Ñ is a matrix whose olumns are n independent opies of the noise random vetor N. The overall transmission error for f n, ϕ n ) under jammer power onstraint Λ is given by ēf n, ϕ n, Λ) = sup ēf n, ϕ n, S Ẽ). S G J Λ This error riterion maes the FH hannel an AVC. A nonnegative real number is said to be an ahievable rate under sender power onstraint Γ and jammer power onstraint Λ if there exists a sequene of odes f n, ϕ n )) n=1, where f n, ϕ n ) is an n, Γ)-ode, satisfying lim inf n 1 n log M n R, lim ēf n, ϕ n, Λ) = 0. n The supremum CΓ, Λ) of the set of ahievable rates under power onstraints Γ and Λ is alled the Γ, Λ)-apaity of the hannel. Now we as under whih onditions the Γ, Λ)-apaity of the FH hannel is positive, and in ase it is positive, how large it is. A preise statement an be made upon introdution of the ommon randomness assisted apaity C r Γ, Λ). This is the maximal rate ahievable if sender and reeiver have a ommon seret ey unnown to the jammer. The ey size is not restrited. As noted in the introdution, the presene of a ertain amount of ommon randomness is a frequent assumption in the literature on frequeny hopping. For given power onstraint Γ > 0, we desribe a ommon randomness assisted n, Γ)-ode as a random variable F n, Φ n ) on the set of n, Γ)-odes whih is independent of the hannel noise. The error it inurs under jamming sequene S Ẽ is defined to equal the mean E[ēF n, Φ n, S Ẽ)] over all possible realizations of F n, Φ n ), and the overall transmission error under jammer power onstraint Λ > 0 is set to equal sup E[ēF n, Φ n, S Ẽ)]. S Ẽ JΛ The definition of ommon randomness assisted ahievable rate under power onstraints Γ and Λ is now a straightforward extension of the orresponding notion for the deterministi ase. The supremum of all ommon randomness assisted rates under power onstraints Γ and Λ is alled the ommon randomness assisted Γ, Λ)-apaity and denoted by C r Γ, Λ). Theorem 1. CΓ, Λ) is positive if and only if Γ > Λ. If it is positive, it equals C r Γ, Λ). Corollary. 1) If CΓ, Λ) > 0, then every fixed-frequeny subhannel also has a positive apaity. In this sense FH is not neessary to ahieve a positive rate. 2) If CΓ, Λ) > 0, then ommon randomness does not inrease the maximal transmission rate. For Γ > Λ, it is thus desirable to have bounds on C r Γ, Λ). These an be provided for all pairs Γ, Λ). Note that the hoie of Λ 1,..., Λ K is a waterfilling strategy. Theorem 2. 1) Let Λ 1,..., Λ K be nonnegative numbers satisfying { σ 2 + Λ = if σ 2 <, Λ = 0 if σ 2 with suh that Λ Λ K = Λ. Then C r Γ, Λ) 1. 1) In partiular, C r Γ, Λ) > 0. 2) If the noise is Gaussian and J { K : σ 2 < }, then C r Γ, Λ) 1 + log K. 2) Remar. 1) Set K := { K : σ 2 < }. As omparison with 2) shows, 1) is a good bound if J K and the noise is Gaussian. The la of a similar bound for the ase J < K an be explained by the fat that the jammer in this ase has to leave some of the highest-throughput subhannels unjammed. C r Γ, Λ) in general depends on J, and should inrease for dereasing J. 2) The proof of Theorem 2 shows that the logarithmi terms in 1), 2) are ahievable without frequeny hopping, whereas frequeny hopping ontributes at most log K bits to apaity. Aording to the lower bound, the ommon randomness assisted apaity grows to infinity as Λ is ept fixed and Γ tends to infinity. Thus asymptotially for large Γ, the relative ontribution to C r Γ, Λ) of information transmitted through the FH sequene vanishes. 3) Non-trivial frequeny hopping will in general be neessary both to ahieve C r Γ, Λ) and CΓ, Λ). Although we will not prove this, this is implied by the mutual information haraterization of C r Γ, Λ) see the proof of Theorem 2). III. PROOF OF THEOREM 2 Although Theorem 1 and its orollary are our main results, we first prove Theorem 2, whih is needed for the proof of Theorem 1. From [2, Theorem 4] it follows that C r Γ, Λ) = sup min X,κ): ι,s): E[X 2 ] Γ E[ S 2 ] Λ = min ι,s): sup X,κ): E[ S 2 ] Λ E[X 2 ] Γ IXe κ ; Xe κ + S e ι + N) IXe κ ; Xe κ + S e ι + N). Here Xeκ is a random variable on the possible sender inputs determined by an X -valued random pair X, κ). Similarly, S e ι is the jammer s random hannel input determined by a random S-valued pair ι, S) independent of X, κ).

4 Define Y = Xe κ + S e ι + N. The expression IXe κ ; Y) is onave in the distribution P κ of κ and onvex in the distribution P ι of ι. Therefore the sender will in general have to use frequeny hopping to approah apaity and liewise, the jammer will not sti to one onstant frequeny subset I for jamming. The mutual information term appearing in the above formula for C r Γ, Λ) an be written as IXe κ ; Y) = IXe κ, κ; Y) Iκ; Y Xe κ ) = IX; Y κ) + Iκ; Y), 3) upon appliation of the hain rule in eah of the equalities and observing that the sequene κ Xe κ Y is a Marov hain. The seond term in 3) is between 0 and log K. Thus to bound C r Γ, Λ), it remains to bound min sup ι,s): X,κ): E[ S 2 ] Λ E[X 2 ] Γ = min ι,s): sup κ,γ) E[ S 2 =1 ] Λ IX; Y κ) 4) K P κ ) sup IX; Y κ = ), X:E[X 2 κ=] Γ where the supremum over κ, Γ) is over κ and nonnegative vetors Γ = Γ 1,..., Γ K ) satisfying P κ )Γ Γ. We ontinue with the proof of the lower bound. For any K, IX; Y κ = ) IX; Y κ = ). 5) Fix any ι, S) with E[ S 2 ] Λ. Let S I, be distributed aording to the projetion onto the -th oordinate of P S ι [ ι = I] and denote the seond moment of S I, by Λ I,. Note that Λ I, = 0 if / I. The -th oordinate output onditional on the event κ = has the form y = x + Z, 6) where Z is a real-valued random variable whose distribution equals P Z = P ι {I : / I})P N + P ι I)P N +S I, I: I if ξ is a random variable, then by P ξ we mean its distribution). If we set Λ := I P ιi)λ I,, then Z has the variane σ 2 + Λ. Observe that Λ Λ K Λ. As 6) is an additive hannel with the real numbers as input and output alphabet, it is a well-nown fat [5, Theorem 7.4.3] that sup IX; Y κ = ) 1 X:E[X 2 κ=] Γ σ 2 + Λ. Hene the right-hand side of 4) an be lower-bounded by min max 1 K P κ ) log 1 + Γ ) Λ κ,γ) 2 σ 2 + Λ, 7) =1 where the minimum is over vetors Λ = Λ 1,..., Λ K ) with nonnegative omponents satisfying Λ Λ K Λ. By hoosing κ to be onstantly equal to the orresponding to the maximal log-term in 7) and by putting all power Γ onto this, 7) is lower-bounded by min max 1 Λ 1+ +Λ K Λ 2 log 1 + Γ σ 2 + Λ ). 8) By this hoie of κ, 8) is obtained without frequeny hopping. It is now straightforward to show by omparison that waterfilling for the jammer is the optimal hoie of Λ 1,..., Λ K in 8). This bound on 4) together with 3) proves 1). Next we prove the upper bound 2). Assume that all noise random variables are Gaussian. It is suffiient to upper-bound 4). We are now free to hoose any ι, S) obeying the onditions. Let Λ 1,..., Λ K satisfy the waterfilling sheme. Further, let I be a set ontaining K := { : σ 2 < } and hoose ι to be onstant equal to this set reall that J K by assumption). Define random variables S 1,..., S K, independent of eah other and of the noise, by setting S = 0 if / K and, for K, by letting S be Gaussian distributed with mean 0 and variane Λ. The independene of S 1,..., S K maes 5) an equality. Conditional on the event κ =, the -th oordinate output random variable is given by the formula y = x + S + N, whih is an additive Gaussian noise hannel with noise variane σ 2 + Λ. Applying [5, Theorem 7.4.2], we thus obtain sup IX; Y κ = ) = 1 X:E[X 2 κ=] Γ σ 2 + Λ. So altogether, realling the hoie of Λ 1,..., Λ K, the righthand side of 4) an be at most { sup P κ ) 1 κ,γ) K + P κ ) 1 } σ 2. 9) / K By replaing all σ 2 by and exploiting the onavity of the logarithm, one thus obtains that 4) is upper-bounded by 1, 10) as laimed. Note that 9) is equal to 10) if κ is onentrated on one fixed K and the sender uses maximal power on this, so 10) is also valid without frequeny hopping. Note also that in the ase of Gaussian noise and J K, together with the lower bound proved before, we have thus obtained a losed-form haraterization of 4). This ompletes the proof of Theorem 2. IV. PROOF OF DIRECT PART OF THEOREM 1 The proof of Theorem 1 bases on the suffiient riterion for CΓ, Λ) = C r Γ, Λ) provided by the orollary to [2, Theorem 4]. To formulate this riterion, we first have to say what it

5 means for the FH hannel to be symmetrized by a stohasti ernel. A stohasti ernel U with inputs from X and outputs in S gives, for every x, ) X, a probability measure U x, ) on the Borel algebra of S suh that for every Borel-measurable A X, the mapping x, ) UA x, ) is measurable. U x, ) is speified by its values on all pairs I, B), where I = J and B is a Borel set on R K suh that for all b B, it holds that l / I implies b l = 0. One an thus write UI, B x, ) = U 1 I x, )U 2 B x,, I). U 1 x, ) determines a random variable ι U x, ) on the set of subsets of K with ardinality J. U x, ) then determines a random variable S U x, ) whih, onditional on the event ι U x, ) = I, has the distribution U 2 x,, I). These random variables give rise to a random jammer input, Zx, U := SU x, ) e ι U x,). Thus any pair x, ) X together with U defines the following hannel: y = xe + Z U x, + N, where x, ) X is the sender input, the output set is R K, and the noise is Z U x, + N. By definition, the FH hannel is symmetrized by U if all sender input pairs x, ) and x, ) satisfy xe + Z U x, + N D = x e + Z U x, + N, where D = means that the left-hand and the right-hand side have the same distribution. In partiular, this implies xe + E [ Z U x, + N] = x e + E [ Z U x, + N ] or equivalently, as the noise is mean-zero, xe + E [ Z U x, ] = x e + E [ Z U x,]. 11) To state the riterion for the equality of the Γ, Λ)-apaities with and without ommon randomness, some more definitions are neessary. Let U 0 be the lass of stohasti ernels U that symmetrize the FH hannel and for whih Z U x, has finite variane for all x, ). Let X X be finite and X, κ) be onentrated on X. Assume that for every x, ) X, the onditional distribution of the random variable ZX,κ U given {X = x, κ = } equals that of Zx, U. Then define τ X X, κ, Λ) = 1 Λ inf U U 0 E [ Z U X,κ 2]. We also write C r, X Γ, Λ) for the ommon randomness assisted apaity of the FH hannel with the same power onstraints, but whose inputs are restrited to the finite subset X of X. By the orollary of [2, Theorem 4], CΓ, Λ) = C r Γ, Λ) if there exists a family F of finite subsets of X satisfying that every finite subset of X is ontained in some member of F and that for every X F, there is an X, κ) onentrated on X and satisfying E[X 2 ] Γ with IXe κ ; Y) = C r, X Γ, Λ) and τ X X, κ, Λ) > 1. We will now losely follow the proof of [2, Theorem 5] to prove that the above riterion is satisfied for the FH hannel if Γ > Λ. Fix Γ, Λ > 0. Let X0 be a finite set satisfying C r, Γ, Λ) > C X0 r Γ, Λ) for some Γ > Γ. Suh a set exists by the fat [2, Theorem 4]) that for all Γ, Λ, C r Γ, Λ) = sup C r, X Γ, Λ) X X finite and the lower bound on C r Γ, Λ) of Theorem 2 showing that C r Γ, Λ) tends to infinity as Λ is fixed and Γ tends to infinity. We hoose F as the family of finite subsets X of X satisfying X 0 X and K X = X {}, =1 where X is symmetri about the origin. Obviously, every finite subset of X is ontained in some X F. We first need to show that for every finite input set X F there exist C r, X Γ, Λ)-ahieving hannel input distributions whih exhaust all the power and are symmetri on every frequeny subband. Lemma. Let X F. Then there exists a pair X, κ) of random variables with values in X satisfying and min ι,s): IXe κ ; Xe κ + S e ι + N) = C r, X Γ, Λ) 12) E[ S 2 ] Λ E [ X 2] = Γ, 13) P X κ ) = P X κ ) 1 K) 14) Here P X κ denotes the onditional probability of X given κ, and P X κ is defined analogously. Proof. Fix X F. By definition of X0 and Γ we have C r, X Γ, Λ) C r, X0 Γ, Λ) > C r, X Γ, Λ). Let X, κ) and X, κ ) assume values in X suh that X, κ) ahieves C r, X Γ, Λ) and X, κ ) ahieves C r, X Γ, Λ). Then any X, κ) distributed aording to a nontrivial onvex ombination of P X,κ) and P X,κ ) ahieves a rate larger than C r, X Γ, Λ) beause the left-hand side of 12) is onave in P X,κ) by [2, Lemma 5]. Moreover X, κ ) uses stritly more power than X, κ), so the seond moment of X must equal Γ. This proves 13). If we replae X, κ) by X, κ), then the left-hand side of 12) remains unhanged. This is due to the symmetry of the jammer input onstraints. Hene a random input X, κ) distributed aording to 1 2 P X,κ) + P X,κ) ) satisfies 12)- 14). Let X F and X, κ) as in the Lemma. We now show that τ X X, κ, Λ) > 1 if Γ > Λ. To do so, hoose any U U 0.

6 Then for any x, ) X, using Jensen s inequality, E [ Z U X,κ 2] = P X,κ) x, )E [ Z U x, 2] x,) X x,) X P X,κ) x, ) E [ Z U x, ] 2. 15) As U symmetrizes the FH hannel, we an apply 11) and lower-bound 15) by x,) X P X,κ) x, ) xe x e + E[Z U x, ] 2 P κ ) x X P X κ x ) x x e + E[Z U x, )] 2, 16) where we denote by Z U x, ) the -th omponent of ZU x,. By 14), P X κ ) is symmetri for every, so its mean equals 0 and min a x X P X κ x ) x a 2 = x X P X κ x ) x 2. Using this in 16) and applying 13) yields the lower bound x,) X P x, ) x 2 = E[X 2 ] = Γ for E [ Z U X,κ 2]. We onlude that τ X X, κ, Λ) > 1 for all X F and the orresponding X, κ) if Γ > Λ, implying that CΓ, Λ) = C r Γ, Λ). As the ommon randomness assisted Γ, Λ)-apaity is positive for positive Γ, this further implies that CΓ, Λ) > 0 if Γ > Λ, and the proof of the diret part of Theorem 1 is omplete. V. PROOF OF CONVERSE FOR THEOREM 1 The onverse follows the lines of the proof of the onverse of [3, Theorem 1]. Let Γ Λ. Let f n, ϕ n ) be any n, Γ)- ode with M n 2 messages. We will prove that there exists a jammer input sequene S Ẽ suh that ēf n, ϕ n, S Ẽ) 1/4. This sequene will be found among the following inputs. Assume that f n m) = x 1 e 1,..., x n e n ). Then let Ẽm) be the matrix whose i-th olumn is e i, and let the i-th olumn of the matrix Sm) equal x i e i. This gives a set { Sm) Ẽm) : m M n } of jammer input sequenes. Note that the power of any of these is at most Λ. Observe that for m, m M n with m m, P[ϕ n f n m) + Sm ) Ẽm ) + Ñ) m] = P[ϕ n f n m ) + Sm) Ẽm) + Ñ) m] 1 P [ϕ n f n m ) + Sm) Ẽm) + Ñ) m ]. Thus 1 ēf n, ϕ n, M n Sm) Ẽm)) m M n 1 M n 2 P[ϕ n f n m) + Sm ) Ẽm ) + Ñ) m] m,m M n 1 M n 2 M n M n 1) Therefore one of the jammer inputs Sm) Ẽm) maes the average error inurred by the ode f n, ϕ n ) at least one quarter. This proves the onverse of Theorem 1. VI. PROOF OF THE COROLLARY TO THEOREM 1 The seond laim of the orollary is obvious from Theorem 1. The first statement follows from [2, Theorem 5], whih says that an additive-noise hannel with R as sender, jammer and output alphabet has positive apaity then equal to the random oding assisted apaity) if and only if the sender power exeeds the jammer power. So if both the sender and the jammer in the FH hannel onentrate their power on any frequeny band K and Γ > Λ, already a positive apaity equal to max X:E[X 2 ] Γ min IX; X + S + N ) S:E[S 2 ] Λ and lower-bounded by 1 σ 2 + Λ > 0 will be ahievable. In partiular, this rate an be obtained without frequeny hopping. On the other hand, if no transmission is possible over the subhannels, then Γ Λ, and the FH hannel also has zero apaity. VII. DISCUSSION For non-disrete AVCs, there is no general statement that apaity without ommon randomness always equals 0 or the ommon randomness assisted apaity lie the Ahlswede dihotomy in [1] for disrete AVCs. Thus it is not possible to justify Theorem 1 just by observing that the apaity of every subhannel is positive if Γ > Λ. Lie [8], [7] we assume here that the reeiver simultaneously listens on all frequenies. A different approah is taen in [6], [4], where the reeiver listens randomly on only one frequeny band at a time. The above analysis an be performed in a similar way for this situation and leads to analogous results: The apaity without ommon randomness shared between sender and reeiver is positive if and only if the sender power exeeds the jammer power. Of ourse, the apaity will in general be smaller than if the reeiver listens on all frequenies. The onverse shows that in order to find a good jamming sequene, the jammer needs nowledge of the hannel and the transmission protool. Further, it should now when the transmission of a odeword starts, so it has to be synhronized with the sender. If this is given, then the suessful jamming

7 strategy in the ase Γ Λ is to onfuse the reeiver: There exists a legitimate odeword suh that if the jammer inputs this into the FH hannel, the reeiver annot distinguish the sender s messages. The ase of a jammer listening to the sender s input into the hannel lie in [6], [4] was not treated here beause there exist few results on AVCs in this diretion. REFERENCES [1] R. Ahlswede. Elimination of orrelation in random odes for arbitrarily varying hannels. Z. Wahrsheinliheitstheorie verw. Gebiete, 44: , [2] I. Csiszar. Arbitrarily varying hannels with general alphabets and states. Information Theory, IEEE Transations on, 386): , Nov [3] I. Csiszar and P. Narayan. Capaity of the gaussian arbitrarily varying hannel. Information Theory, IEEE Transations on, 371):18 26, Jan [4] Y. Eme and R. Wattenhofer. Frequeny hopping against a powerful adversary. In Y. Afe, editor, Distributed Computing, volume 8205 of LNCS, pages Springer, [5] R. G. Gallager. Information theory and reliable ommuniation. Wiley, New Yor, [6] M. Strasser, C. Popper, S. Capun, and M. Cagalj. Jamming-resistant ey establishment using unoordinated frequeny hopping. In Seurity and Privay, SP IEEE Symposium on, pages 64 78, May [7] L. Zhang and T. Li. Anti-jamming message-driven frequeny hopping part ii: Capaity analysis under disguised jamming. Wireless Communiations, IEEE Transations on, 121):80 88, January [8] L. Zhang, H. Wang, and T. Li. Anti-jamming message-driven frequeny hopping part i: System design. Wireless Communiations, IEEE Transations on, 121):70 79, January 2013.