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(ak1 x + bk1 y + tk1 ) dx dy + sk1 v(p) f (p)

(1033)

p k1 (Pm 1 )

The N summations are performed on the sub-block k1 of the original image, noted k1 (Pm 1 ) (recall that this is one of the four quadrants of the original image) The sum v(p) depends on the resolution m of approximation of the luminance of element p The minimization of function (1033) amounts to solving the following system of equations [DUD 95b]: 2 x x y x 1 v(p) x p p p p p p p p p p 2 x y y y 1 v(p) y ak1 p p p p p bk 1 p p p p p 2 tk 1 x 1 y 1 1 v(p) 1 sk1 p p p p p p p p p p 2 v(p) v(p) x v(p) y v(p) 1

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Geographic leashes require loose time synchronization among the nodes in the network, as opposed to temporal leashes, where the nodes in the network need to have tightly synchronized clocks with nanosecond precision In addition, temporal leashes might not be practical in networks (such as those based on the use of 80211) that make use of a contention-based MAC protocol Temporal leashes will also be effective only against the rst and easier mode of tunneling, where a packet is encapsulated Of course, nodes in systems dependent on geographic leashes need to be able to determine their locations securely (see 7) Another approach for detecting wormhole attacks is given in [72] In this case the authors assume the presence of directional antennae The approach here is based on the use of packet arrival direction to detect that packets are arriving from the proper neighbors Such information is possible due to the use of directional antennae This information about the direction of packet arrival is expected to lead to accurate information about the set of neighbors of a node As a result, wormhole attacks can be detected since such attacks emanate from false neighbors To illustrate this idea consider Figure 431 Here we show two nodes, A and B We show the directional antenna with six zones explicitly for both nodes Each node is assumed to have knowledge of the zone from where a packet is received Given this, the basic idea that is used to determine the set of authentic neighbors is that, if a node is in a given direction of another node, then the latter node is in the opposite direction of the former node For example, in Figure 431, node B is in zone 6 of node A while node A is in the opposite zone, which is zone 3 of node B An implicit assumption here is that the directional antennae on the various nodes are perfectly aligned Now consider nodes A, B, C, D, and E as shown in Figure 432 Assume that B transmits a message (call it a Hello message for simplicity) in its neighborhood which includes C Then node C replies back to B informing node B of the zone from which node B s hello message was received If node B receives this reply in the opposite zone to what zone C reports, then node C can possibly be an authentic neighbor Thus, in this case node C replies back to node B that the hello message was received in zone 1 of node C This reply is received by node B in its zone 4, which is the zone opposite to zone 1 Thus, B can infer that node C is an authentic neighbor Every node can repeat this and form the list of authentic neighbors Any message that does not emanate from an authentic neighbor is rejected.

= f (p)

f (p) y p f (p) 1 p f (p)v(p)

An image (a square block of the original image partition) is then coded by a sequence of 4 4 real coef cients Decoding The decoding algorithm allows for a fast and non-iterative reconstruction of the invariant function g associated with the operator G It is only necessary to know the coef cients ak1 , bk1 , tk1 and sk1 associated with each of the N spatial transformations k1

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