Channel Model. The channel model has three terminals — a sender, a The equivalence can be established by noting that the modified channel preserves the same knowledge of the side information sequences as the original problem, the rate and equivocation terms only depend on the joint distribution p(y¯n, y¯n, xn, sn) receiver and an eavesdropper. The sender communicates with and for any input distribution n n r e t the other two terminals over a discrete-memoryless-channel controlled by a random state parameter. The transition proba- bility of the channel is pyr,ye|x,s (·) where x denotes the channel satisfies p(y¯n, y¯n|xn, sn) = p(x |st ), the extended channel Yn p(y¯ri, y¯ei|xi, sti), (6) input symbol, whereas yr and ye denote the channel output r e t
i=1 symbols at the receiver and the eavesdropper respectively. The symbol s denotes a state variable that controls the channel transition probability. We assume that it is independent and · identically distributed (i.i.d.) from a distribution ps ( ) in each channel use. Further, the entire sequence sn is known to the sender before the communication begins. As explained in section II-C the model generalizes easily to take into account correlated side information sequence at each of the receivers.
Channel Model. We assume a simple system of two transmitter-receiver (Tx-Rx) pairs in which each Tx has N antennas and each Rx has only one antenna. This results in a two-user MISO IC, which is illustrated in Figure 4.1 as an example with N = 3. We assume linear pre-coders and the Txs use the same Gaussian codebooks and therefore the Rxs, if the channel qualities allow, can decode the interference and subtract it from the received signal. Also, we assume that the interference is successfully decoded if the rate of the interference signal is smaller than the Xxxxxxx capacity of the interference channel. Denote the transmit beamforming vector of Tx i by wi and the channel from Tx i to Rx ¯i, where i,¯i ∈ {1, 2} ,¯i i, hi¯i ∈ CN×1. Note that the channel gains are i.i.d complex Gaussian coefficients with zero mean and unit variance. The received signal at Rx i is therefore √ √
ii i ¯i i i i The noise ni is a complex Gaussian random variable with zero mean and unit variance. Pi is the transmit power at each Tx and we assume the same power constraint for both Txs, Pi ≤ P ∗. The symbol xi is the transmit symbol at Tx i with unit power. The transmit beamformer has unit norm ǁwiǁ = 1. Denote
Channel Model. In the channel model, the m legitimate terminals have access to two resources: an authenticated but public communication channel, and a discrete memoryless broadcast channel (DMBC), described by the conditional law q(x2, x3, ..., xm, z|x1). Any message sent on the public channel will be heard by all terminals including the eavesdropper. The eavesdropper is assumed to be passive and cannot tamper with the messages sent on the public channel. The input of the broadcast channel X1 is controlled by the first legitimate terminal. The DMBC has outputs X2, X3, ..., Xm at the remaining m − 1 legitimate terminals, and output Z at the eavesdropper. Before providing a formal definition, we begin with an intuitive description of a secret key generation scheme in the channel model: The secret key generation scheme begins by the first terminal inserting random variable X1(1) at the input of q(x2, x3, ..., xm, z|x1). The other legitimate terminals and the eavesdropper receive X2(1),
Channel Model. ≥ We consider the channel type model from [5] with a wiretapper. Xxxxx and Xxx communicate over a two way block- fading (MIMO) channel. In addition they can also use a public discussion channel with unlimited capacity. We pose the constraint that a node cannot transmit and receive at the same time. We assume that Alice, Bob, and Eve have nA, nB, and nE antennas, respectively, with nA nB without loss of generality. If Xxxxx uses the channel, the received signals at Xxx and Eve at time t are given by YB(t) = H(t)XA(t)+ VB(t), YE(t) = GA(t)XA(t)+ VE(t), and if Xxx uses the channel, the received signals at Xxxxx and Eve are given by YA(t) = H†(t)XB(t)+ VA(t), YE(t) = GB(t)XB(t)+ VE(t). ∈ ∈ ∈ ∼ ∈ ∈ Here XA(t) CnA , or XB(t) CnB is the transmitted signal, YA(t) CnA , YB(t) CnB , and YE(t) CnE are Alice’s, Bob’s, and Eve’s received signals, respectively. H(t) represents the channel matrix between Xxxxx and Xxx, and GA(t) and GB(t) are the channel matrices between Xxxxx and Eve, and Xxx and Eve, respectively. The noise terms VA(t) CN ∼ CN ∼ CN (0, InA ), VB(t) (0, InB ), and VE(t) (0, InE ) CN are i.i.d. and independent of each other and all other variables. We assume that the entries of H(t), GA(t), and GB(t) are distributed as (0, 1), independent of each other, and stay fixed for T channel uses, but are independent between different fading blocks. We further assume a short-term average power constraint on the input symbols
(1) Denote the message sent over the public channel by Xxxxx at time k by Φk and the message sent by Xxx at time k by Ψk. We consider key agreement over several coherence blocks N , and denote the symbols transmitted and received by Xxxxx and Xxx in these N blocks by (XN, YN) and where H1 CMB ×MA , H2 CMB ×(nA−MA), H3 ∈ ∈ ∈ ∈ C(nB −MB )×MA , and H4 C(nB −MB )×(nA−MA). Bob’s re- ceived signal at time j at antenna i is yB,i(j) = Hi,j√P + vB,i(j). (2) B (XN, YN), respectively. The message transmitted over the The XXXX xxxxxxxx XˆX,x,x xx Xx,x is a circularly symmetric public channel at time k can depend on previous observed Gaussian random variable with variance P/(P + 1) given by symbols: Φ = X (Xx−0, Xx−0, Xx−0, Xx−1), and Ψ = √ Ψ (Xk−1 k−1 A A x x−1 ˆ k B , YB , Ψk−1, Φ ). After N coherence blocks, HB,i,j = yB,i(j). (3) and K uses of the public channel Xxxxx generates a key A A KA = K1(XK, YK, ΨK, ΦK) ∈ K, and Xxx generates a The estimation error eB,i,j = Hi,j − HˆB,i,j is distributed A A key KB = KB(XK, YK, ΨK, ΦK) ∈ K. A key rate R is as CN ...
