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Dudbridge demonstrates that the gray-level sum, noted gk1 , of the sub-elements included in the element Ak1 can be decomposed as follows [DUD 95b, MON 95a]: gk1 = ak1 x dx dy + bk1 y dx dy

.

and that, consequently, the sum of the gray-levels of the N elements Ak1 reads:

1 (1035)

Page 218 Project success (Continued) infrastructure, 17, 29, 163 project management, 17, 29, 163 quality standard, 17, 29, 163 supply-chain management, 17, 29, 163 short-term vs long-term assessment of success/failure, 69, 74 75, 134 over time, 68 73, 134 unique setting of project management, 61 64 value chains and, 140 141 Project supply chain management (PSCM), 17, 29, 30, 49, 103 126, 147, 150 151, 163 analysis of strengths/weaknesses, 117 approved vendors, 120 benchmarking, 122 123 buyer-vendor coordination, 114 challenges in effective, 119 125 components of supply chain, 106 112 control, 112, 115 116, 118, 123 124 coordination/planning, 112, 113 115 designing/setting up project supply chain, 112 119 diagram, 105 distribution, 106, 110 112, 118 environmental analysis, 112, 113 fabrication, 106, 109 110 goals, 112 113, 118 modeling, 118 operational planning/control, 123 124 proactivity, 119 procurement, 106, 107 109 production-distribution coordination, 114 115 project execution plan, 49 stakeholders and, 121 122, 147, 150 151 steps in developing a project supply chain, 112 top down approach, 117 tracking work in process, 120 121 vs traditional supply chain management, 104, 105 vendor analysis, 118 what-if analysis, 124 125 Project team(s), 34, 42 46, 81, 82, 83, 146, 147, 153 154, 156, 196, 198 199

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Similarly, gk1 k2 stands for the gray-level sum of the sub-elements of Ak1 k2 and reads: gk1 k2 = ak1 x dx dy + bk2 y dx dy

The decoding procedure decomposes as follows: the sum k=1 gk is directly calculated from the coef cients of G (formula (1035)) The result is equal to the gray-level sum of the pixels in the original image; according to (1034), gk1 (k1 = 1 N ) is a function of the variables ak1 , bk1 , N tk1 , sk1 and of the value k=1 gk previously calculated; according to (1036), gk1 k2 is a function of the variables ak2 , bk2 , tk2 , sk2 and of the previously calculated gk2 ; etc This way, each element of the invariant function g at resolution m can be recursively calculated The reconstruction procedure is not iterative, in contrast to most algorithms of fractal decoding The method presented in this section, approximates the luminance function f in each square block of an original image partition Each block is approximated by an invariant function g, independently of the rest of the image At a given resolution m, the approximation is performed in the least squares sense, using an IFS associated

We next show how this mechanism can be used to detect wormholes Assume that nodes D and E are malicious and they intend to form a wormhole between nodes A and B Consider hello messages being transmitted by node A in order to determine the set of real neighbors In this case, node D receives the hello message of A and tunnels it to node E, which then retransmits the Hello message in its neighborhood Node B receives this message and reports back using a reply message that the Hello message of A was received in zone 4 of node B If this reply of node B is replayed back by node D, then node A receives this reply in its zone 4 Thus, node A can infer that there is an inconsistency and reject this reply from node B Note, though, that the same mechanism does not protect a wormhole from being established between nodes A and C since they are in the opposite zones as compared with the wormhole endpoints Thus, we see that the simple mechanism given earlier is not suf cient to detect all wormholes, although it does detect many of them In order to address this shortcoming of the simple mechanism, the authors enhance the basic scheme The enhancement requires that nodes cooperate with their neighbors by having other nodes called veri ers validate the legitimacy of the various packets being transmitted Restrictions are placed on the nodes that can act as veri ers This prevents the adversary from leveraging the weakness earlier whereby nodes that are in opposite directions to the wormhole endpoints are tricked into accepting each other as neighbors In this case, after the messages of the basic protocol, the veri er is requested to corroborate the zones from which the messages were received by the veri er As pointed out earlier, however, this approach requires perfectly aligned directional antennae In [73], the authors present a graph theoretic framework for modeling the wormhole attack They provide a necessary and suf cient condition that any solution to the wormhole problem needs to satisfy In addition, the authors also propose the use of local broadcast keys whereby the keys in different geographic regions are different As a result, an encrypted message replayed via the wormhole in a different location cannot be decrypted by the receivers in that region.

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