We give an example to show that the secrecy capacities of the two cases are in general unequal, and we show that in both cases the computation of the secrecy capacity is NP-complete. The presence of bottleneck link 3 to 4 requires coding on this link to achieve multicast rate 2.
Wireless Broadcast: Double-Edged Sword
This thesis examines new techniques to exploit wireless broadcasting to improve the throughput and efficiency of wireless networks and use network coding to enable secure communications. Wireless broadcast, on the other hand, can also be leveraged to increase throughput and improve the reliability of wireless networks.
Network Coding: A New Communication Paradigm
1.3, where node 1 and node 2 want to send a packet to each other with the help of relay node 3. Using network coding at the network layer, the relay node broadcasts the XOR of the packets received from two terminals, which the number of broadcasts.
Outline and Contributions
In fact, the abs based scheme resembles network coding where the relay performs an XOR on the decoded data from the terminals [48]. However, in an abs-based scheme, the relay receives the sum of the real value of the data from the two terminals plus noise at the physical layer, while in network coding, the addition is performed over a finite field at the network layer.
System Model
The amount N represents the additive white Gaussian noise (AWGN) on the relay with mean zero and variance σr2. It is also assumed that the terminals and the relay know h1 and h2, which can be obtained by using channel estimation at the relay or the feedback channel of the two terminals, see e.g. [23].
Relay Strategies for the BPSK Case
Non-Abs-Based Strategies
- Amplify-and-Forward
- Detect-and-Forward
- Estimate-and-Forward
- Optimized Relay Function
The terminal fault probability is optimized over the relay threshold w, the relay transfer values a and b, and the terminal detection thresholds v1 and v2, taking into account the average power limit on the relay. We can then derive the optimal a and b given the power limit on the relay by substituting (2.10) into (2.7).
Abs-Based Strategies
- Abs-Based Amplify-and-Forward (AAF)
- Abs-Based Detect-and-Forward (ADF)
- Abs-Based Estimate-and-Forward (AEF)
- Optimized Relay Strategy
For caseh1 =h2 = 1, the average failure probability at each terminal (2.20) can be written as. 2.24) has the good property that the optimization with respect to w and v is separate. Therefore, (2.39) is the optimal solution when v = 0, which can be achieved in the high SNR regime as shown below.
Comparison Between Two Classes of Strategies
These results suggest that when the channel is highly asymmetrical or the relay has more power than the terminals, we should use DF. Therefore, when the channel is symmetrical and the relay has greater power than the terminals, we must use DF with (2.16).
Higher Order Constellations
The minimum possible constellation size of the relay function is equal to the chromatic number of G. AAF can be easily generalized by setting the relay function to a piecewise linear function based on h(u), such as.
Peak Power Constraint
For the no-abs strategy, is the sum of average failure probabilities at both terminals. We find that under a peak power constraint, the ADF strategy is the optimal abs strategy, while the optimal non-abs strategy is similar to the DF strategy (2.12) except for the current constraint.
Simulation Results
It can also be seen that in this scenario non-abs-based strategies perform better than abs-based strategies at low SNR and worse than abs-based strategies at high SNR. The reason for this is that non-abstract strategies do not exploit a priori information about the signal available at each offering terminal; This a priori information produces additional redundancy, which is particularly useful at low SNR.
Conclusion
At high SNR, it is clear that energy savings dominate the ADF performance. At low SNR, we find that the performance degrades as M decreases, which means that M = 7 achieves the best performance.
Appendix
In this chapter, we discuss inter-layer optimization in wireless broadcast networks, focusing on the problem of distributed scheduling of broadcast links. We also provide a generalization of existing cross-layer optimization results for multicast network coding in wireless networks.
Related Work
System Model
By the power sharing property of network coding [7] and the rate limitation, we have the following two limitations.
Cross-Layer Design with Broadcast Advantage and Network Coding . 48
- Problem Formulation
- Local Optimal Algorithms
- Randomized Algorithm
- Hybrid Algorithm
Let WM W and WLO be the values of the second term in (a) with maximum weighted hypergraph matching and Algorithm 3.1, respectively. Each while loop requires 2 time slots, and the expected execution time of Algorithm 3.3 is also O(log|E|).
Simulation Results
Wireless Butterfly Network
Maximum Weighted Hypergraph Matching Hypergraph Matching Algorithm 3.1 Hypergraph Matching Algorithm 3.2 Maximum Weighted Graph Matching Local greedy graph matching. 3.1, where our cross-layer design with maximally weighted hypergraph matching, algorithm 3.3, algorithm 3.4 and algorithm 3.5, and the algorithm in [18] with maximally weighted graph matching.
Random Networks
Our results show that it is more advantageous to use hypergraph matching when the multicast group is large.
Conclusion
Unlike schedule-based medium access that requires a central authority, multiple nodes share the medium using random access in contention-based MAC. In both works [31, 32] it is shown that the achievable throughput using MPR is higher than that using Aloha.
A Motivating Example
Without rate splitting, each user's packet can always be decoded if its transmission rate is less than R2. Therefore, the proposed strategy is optimal for the case of decentralized control and without using rate sharing.
Multiple Access MAC in Aloha Type Networks
MAC on AWGN Multiple Access Channels
We can use pseudo-random variables with random seeds to choose the transmission rate at each node, so that each receiver with the random seed knows the transmission rate of each user at each time window and thus the order in which the users must be decoded. In the absence of random seeds or other information about the transmitters' transmission rates, the receiver may first try to decode the lowest packets for each source.
Achievable Results
Let ηj,k denote the probability that j given users transmit at a rate less than or equal to Rk and that their packets can be correctly decoded by the receiver. Assuming that N −j other users transmit at a rate greater than Rk , all packets from users j can be decoded if and only if N −j +l ≤k and the remaining j−l packets from users can be decoded.
Fixed Number of Transmission Rates
Although the optimal pk that maximizes the average throughput may depend on Rk and N in a complicated way that does not lead to simple practical protocol design, the following theorem shows that the optimal pk asymptotically has a simple form as N. Compared to conventional Aloha, which achieves a throughput of only 0.0137 bits/s/Hz (not shown in Fig. 4.3), the proposed scheme achieves a much higher throughput.
Multiple Access MAC in WLAN
PsTt+ (1−Ps−pNK)Tc+pNKTSLOCK. 4.30) where sd is the average number of data bits successfully transmitted in a time slot by one node, Ps is probability of at least one node with successful transmission, and R¯ is the average throughput of each node in a half-duplex slotted network, η1 = TTt. As we use variable rate transmission, we assume that Td or Tt is the same for all transmission rates.
Game-Theoretic Model of Contention Control
Random Access Game
A random access game is a rather general model for contention control, since the payoff function can be constructed by reverse engineering from (4.35). In a random access game, one of the most important questions is whether a Nash equilibrium exists or not.
Utility Function Design
- Reverse Engineering from Existing Protocols
- Reverse Engineering from Desired Operating Points . 95
When N is large, we get the utility function as 4.46) Similarly, when α > 1, the random access game with utility function (4.46) has a unique non-trivial Nash equilibrium. Most algorithms can be reverse engineered to be a random access game with specific utility function.
Dynamics of Random Access Game
- Basic Dynamic Algorithms
- Best Response
- Gradient Play
- Asynchronous Dynamic Algorithms
- Best Response
- Gradient Play
- Dynamic Algorithms under Estimation Uncertainties
- Equilibrium Selection
Using Theorem 4.15, it is easy to obtain the conditions for ai and bi in (4.48) such that the best response strategy converges to a nontrivial equilibrium of the corresponding game. From (4.65), we can see that a larger α indicates a smaller convergence rate of the outer loop, while a larger α results in a higher convergence rate of the inner loop, as suggested in Theorem 4.16.
Rate Splitting
The proposed approach can be easily generalized to the example of choosing M virtual users in each node. According to Theorem 4.1, the achievable throughput of users at layer m can be approximated as.
Simulation Results
Aloha Type Networks
The throughput of Aloha decreases as N increases, while the throughput of the proposed protocol increases as N increases. 4.7 we find that the slope ρ of the proposed local search protocol is the same as that of Aloha, which is 1e.
Half Duplex WLAN
- Throughput Comparison with Capacity Formula
- Throughput Comparison with Convolutional Codes . 116
It can be seen that the throughput of the protocol without SIC and IEEE 802.11 DCF decreases as N increases, while the throughput of the protocol with SIC increases as N increases. The throughput of both the protocol without SIC and the protocol with SIC converges to a constant value, while the throughput
Appendix
- Proof of Theorem 4.3
- Proof of Theorem 4.16
- Proof of Theorem 4.18
- Proof of Theorem 4.19
- Proof of Theorem 4.20
- Proof of Theorem 4.21
In this chapter, we consider secure communication over networks with deletions and networks with unequal link capacities in the presence of an eavesdropper that can eavesdrop on any subset of k links.
Introduction
With equal (unit) link capacities, the secrecy capacity is given by the cut set bound, regardless of whether the location of the eavesdropped links is known or not. In contrast, we show that for unequal connection capacities, the secrecy capacity is generally not the same when the location of the eavesdropped connections is known or unknown.
Network Model and Problem Formulation
The network interdiction problem [79] is to minimize the maximum flow of the network when k links are removed from the entire network. We consider an optimization problem from the point of view of the communicating users who seek to maximize their communication rates subject to the requirement that the message be secret, regardless of the choice of intercepted links.
Secrecy Capacity Region When the Location of the Wiretapped Links
Assuming that the destination has full knowledge of the deletion locations on each network link and the locations of the intercepted links, the secrecy capacity is given by . Talk: Let V be a cut of the network and A ⊆[Vs,Vsc],|A| ≤k be the set of intercept edges.
Achievable Strategies when the Location of the Wiretapped Links is
To achieve secrecy, we must have w ≥ 25x, where the minimum cut condition on the first layer requires r+w ≤ x. Since the source injects all random keys, the minimum cut condition on the first layer requires r+w≤4.
Unachievability of Cut Set Bound
Restricted Wiretap Set (Scenario 3)
The constraints required are that the source information is a function of the signals on the sink's incoming links, and that there is no mutual information between the source information and the signals on the links in each conflicting subset. From Cases 1, 2a and 2b, we conclude that the secrecy rate without knowledge of the eavesdropping set using any non-linear or linear coding strategy is smaller than two obtained for the case where such knowledge is present at the source.
Unequal Link Capacities (Scenario 4)
Therefore, 5/3 is an upper bound on the secrecy rate when the location of the wiretap is unknown, which is less than the secrecy rate 2 achievable when such information is known.
NP-Completeness
Now we consider the case where the eavesdropping set is unknown and show that the secrecy capacity of ˆGH when the eavesdropper accesses any unknown subset of k =|Eh| − r2. We now show that the latter condition is equivalent to the condition that the secrecy capacity of GH when the eavesdropper accesses any unknown subset of k links from A1 is r.
Conclusion
In this thesis, we have considered wireless broadcasting at different layers of wireless networks and studied network coding for secure communication. Our results have demonstrated the usefulness of wireless broadcasting and network coding for network design.
