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and of gray-level images The concept of IFS is also discussed, from a broader perspective, in 9 for the generation of the continuous function graphs and the characterization of H lder functions 1021 Contracting transformations and iterated transformation systems 10211 Lipschitzian transformation Let : R2 R2 be a transformation de ned on the metric space (R2 , d) The symbol d indicates the distance between two points of R2 The transformation is called Lipschitzian with strictly real positive Lipschitz factor s, if: d (x), (y) 10212 Contracting transformation Let : R2 R2 be a transformation de ned on the metric space (R2 , d) The transformation is said to be contracting with real contraction factor s, 0 < s < 1, if: d (x), (y) 10213 Fixed point A contracting transformation possesses a single xed point xf R2 , such that (xf ) = xf Let us note by on ( ) the application of reiterated n times For any point x element of R2 , the sequence { on (x) : n = 0, 1, 2, } converges towards xf :

the radius R of its zone in the RLSU eld It then selects a random number x and applies a hash function on the number h(x) h(x) is put in the hops_traversed eld and h R(x) in the zone_radius eld Subsequent nodes that forward the LSU apply the hash function on the hops_traversed eld and replace that eld with the new value Thus we have hops_traversed h(hops_traversed) The SLSP_LUS_SEQ contains a 32-bit sequence number that is increased monotonically with new LSUs generated by the node The node also appends the signature into the LSU_signature eld The TTL value of the IP packet is set to R 2 1 so that routing advertisements stay within the zone It is also possible that the certi ed key is attached by the node to the LSU message itself (rather than in the periodic broadcast of the key) That ensures that the node receiving the LSU will be able to verify the message A node that is within the zone of the originator of an LSU most likely has the public key of the originator When such a node receives the LSU message it can verify the authenticity of the message Authenticity of the message is veri ed as follows The node checks the hops_traversed eld The node knows the number of hops the message has already traveled which is the zone radius R minus the TTL value in the current packet (the TTL value is decreased by one at each hop) Therefore, the node applies the hash function TTL times on the hops_traversed eld and compares it with the zone_radius eld [which equals h R(x)] The two values should be equal: hTTL (hops traversed) hTTL hR TTL (x) hR (x) zone radius If the LSU is validated, the node decrements the TTL, applies the hash function to the hops_traversed eld and then rebroadcasts the LSU The LSU is only kept until an LSU from the other side of the link is also received When the LSU is con rmed it is used for updating routes on the node, otherwise the LSU is discarded There is still a possibility that two nodes may collude to advertise a nonexisting link SLSP can not prevent such attacks.

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This result makes it possible to de ne the local exponent by using any intervals family containing x0

DEFINITION 13 Let f be a function de ned in a neighborhood of x0 Let {In }n N be a decreasing sequence of open intervals converging to x0 By de nition, the local H lder exponent of f at x0 , noted l (x0 ), is: l (x0 ) = sup l (In ) = lim l (In )

(121)

Let us brie y note that the local exponent is related to a notion of critical exponent of fractional derivation [KOL 01] We may understand the difference between p and l as follows: let us suppose that there exists a single couple (y, z) such that (f, B(x, )) = f (y) f (z) Then p results from the comparison between (f, B(x, )) and , whereas for l , we compare (f, B(x, )) to |y z| This is particularly clear in the case of the chirp, where the distance between the points (y, z) realizing the oscillation tends to zero much faster than the size of the ball around 0 Accordingly, it is easy to demonstrate that l (0) = /(1 + ) for the chirp The exponent l thus sees oscillations around 0: for xed , the chirp is more irregular (in the sense of l ) when is larger The local exponent possesses an advantage over p : it is stable under the action of pseudo-differential operators However, as well as p , l cannot by itself completely characterize the irregularity around a point Moreover, l is, in a certain sense, less precise than p The results presented below support this assertion PROPOSITION 12 For any continuous f and for all x:

The routing protocols that we have discussed so far in this chapter focus on ensuring that the information propagated by the routing protocol is authentic A facet of routing that has not been considered by these routing protocols is traf c analysis Traf c analysis is a

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