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on receiving this route update rst validates the HMAC value of the route update If the HMAC value does not match, then the message is dropped If the HMAC value matches, then the neighbor node classi es the entries into authoritative and nonauthoritative entries 1 The authoritative entries are accepted without any further check 2 The nonauthoritative entries are accepted only after checking for their consistency To explain the consistency check, consider three nodes A, B, and C Consider a routing update received by node C from node B, where node B advertises a route to the destination A with a metric of 1 Node C considers this as a nonauthoritative entry from node B In order to check the consistency of such a nonauthoritative entry, node C needs to know the node from which node B received such a nonauthoritative entry In this case, the node from which node B received the nonauthoritative entry is node A Then node C sends a request to node A asking for the metric that node A has to the destination A and also information about the metric related to the path between nodes A and B In addition, node A is also expected to send the sequence number corresponding to the destination of the nonauthoritative entry for which information is requested Node A sends all this information to node C while ensuring the integrity of the message using the HMAC key shared between nodes A and C When node C receives this message, it checks to see if the sequence numbers have been changed in the message received from node A and the routing update received from node B Additionally, node C will also check if the value of the metric as received from the routing update from node B is the sum of the metric as received in the message from node A and the cost associated for routing between nodes A and B The metric is accepted as reliable if it passes these tests Thus, S-DSDV ensures that the metrics associated with DSDV cannot be manipulated by malicious nodes acting along A problem with S-DSDV, though, is the tremendous routing overhead associated with the consistency checks on the unauthoritative routes.

The following theorem, proven in [DAO 96], enables us to calculate the multifractal spectrum d( ) of WSA functions The corresponding theorem for ordinary self-similar functions can be found in [JAF 02a] (see also 3 in this volume) Let us de ne, for any j 1, the d-tuple (uj , , uj ) by: 0 d 1 (uj , , uj ) = ( j0 , , jd 1 ) 0 i i d 1 where (i0 , , id 1 ) is the permutation of (0, , d 1) which gives: j0 jd 1 i i In other words, for any j, (uj , , uj ) is the d-tuple ( j , , j ) rearranged 0 0 d 1 d 1 in ascending order Then, the following theorem holds THEOREM 92 Let us suppose that there exist two scalars a > 0 and b > 0 such that, for any i {0, , d 1} and j 1, we have: 0 < a uj b < 1 i

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exists for any (x0 , , xd 1 ) [a; b]d Finally, let us suppose that g is uniformly more regular than f We note by d( ) the (Hausdorff) multifractal spectrum of f , ie, d( ) = dimH {x : (x) = }, where (x) is the H lder exponent of f at point x Then: d( ) = if [ min ; max ] where: logd (u1 ) + + logd (un ) d 1 d 1 min = lim n n 1 n max = lim logd (u0 ) + + logd (u0 ) n n if [ min ; max ], then: d( ) = inf q (q)

This theorem shows that the multifractal formalism is valid for the WSA functions Therefore, the Hausdorff multifractal spectrum of large deviations and Legendre transform (see 1 or 4 for de nitions) coincide, are concave and can be easily calculated 97 Signal representation by WSA functions The method we use to represent a given signal is based on its approximation by one or many WSA functions In this section, we develop a practical technique to approximate, in the L2 sense, a signal through a WSA function In practice, only discrete data are available; thus, in what follows, we will suppose that we have a signal {f (m), m = 0, , 2J 1} Our purpose is to nd the parameters (d, g, (Si )i , ( j )k,j , ( j )i,j ) of the WSA i k function which provide the best L2 -approximation of f In its general form, this problem is dif cult to solve However, it is possible to consider a simpli ed and less general form, for which a solution can be found by using a fast algorithm based on the wavelet decomposition of f Let us describe this sub-optimal solution Let

n denote the scaling function, the corresponding wavelet and wk the resulting wavelet coef cients of the signal f : J 1 j wk (2j x k) j=0 0 k<2j

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