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SEAD can forward a routing update entry without increasing the metric or hashing the authentication value, as discussed earlier SuperSEAD protects against such same-distance fraud It does this by making use of the concept of hash tree chains The approach here is to tie the authenticator (the hash value) to the address of the node sending the route update An adversary therefore cannot replay the same message as the one sent by the neighbor unless the adversary takes the identity of the neighbor In order to achieve this, a special one-way chain called the hash tree chain is constructed A hash tree chain is a hybrid between a hash tree and a hash chain The hash chain is used in the same way as in SEAD to ensure that nodes cannot decrease the distance metric while the hash tree property is used to authenticate the identity of a node We next describe the construction of the hash tree chain Consider the hash chain as constructed earlier and as shown in Figure 411 A hash tree is constructed between each pair of values in this hash chain The hash tree is built as follows Given each value in the chain, a set of values denoted b0, b1, , bn is derived using a one-way hash function The value bj is given as bj h(v k j) where v is the value of the hash chain and j represents the identity of a node in the system A hash tree is then built above these values as shown in Figure 413 In this gure we assume that there are four nodes in the system and hence the four vertices ( j 0, , 3) derived from the hash chain value These vertices correspond to the leaf of the hash tree The root of the tree becomes the next value in terms of the generation of the hash chain This is repeated for the entire hash chain giving a hash tree chain, as shown in Figure 414 The generation of the hash tree chain is from left to right while the usage as described earlier for SEAD is from right to left in groups of km each Thus, we see that the identity of each node in the network is encoded into each element of the chain Now consider a node with the identity 1 Such a node corresponds to the node b1 in Figure 413 In SEAD this node sends the value of the hash chain as an authenticator, but with SuperSEAD, this node uses the elements in the hash tree to authenticate the routing updates To do this, node with identity 1 forwards the values b1, b00 and b23 (the shaded nodes in Figure 413) along with the routing updates to its neighboring nodes Each neighboring node can then use these values to verify the authenticity of the hash chain (since each neighboring node has the authentic hash chain anchor corresponding to the destination as in case of SEAD) Now consider a neighboring node with identity 3 Such a node cannot replay the hash values received from node with identity 1 since the node with identity 3 will have to forward the values b3, b20 and b01 along with the routing updates In other words, each node has to specify a different path up the hash tree Therefore a node cannot advertise the same metric as its neighbor and needs to follow the hash tree structure to the next level Thus, the neighboring node is forced to increase the distance metric A malicious node that does not do so can be easily detected Packet leashes (see Section 471) can be used here to ensure that the node that sent it the packet is really its neighbor Thus, packet leashes provide hop-by-hop authentication, thereby preventing an adversary from impersonating another node The hash chain as well as the hash-tree chain though have two problems namely:.

Let: l (x0 ) = sup{ l (Oi )}

.

(120)

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physics applications [MANT 89, VRS 91b], image compression [BAR 93a, BAR 93b, CAB 92, FIS 93, JAC 93a, JAC 93b, VRS 91a] or signal processing [MAZ 92] The inverse problem is de ned as follows: given a signal f , nd a (generalized) iterated function system whose attractor approximates at best f , in the sense of a xed norm This problem is extremely complex in the case of the IFS, yet becomes much easier for that of GIFS In particular, it is shown in [DAO 98] that, for a function f C 0 ([0; 1]), we can nd a GIFS whose attractor is as close to f as wished, in the |||| sense This is obviously different when IFS have a nite number of functions However, the consequence of simplifying and improving the approximation, is that we move from a nite to an in nite number of parameters in the modeling A practical solution to the inverse problem for GIFS is detailed in the next section For now, let us note that in the particular case where: m = 2 Lk (t) = i t + i i m mk if we write: 1 1 sj = f (k + )2 j f (k2 j ) + f (k + 1)2 j k 2 2 and: cj = k sj k j 1 s[ k ] 2 k = 0, , 2j 1 (94) k {0, , 2j 1}

then, since m and Lk are xed, the inverse problem is solved if cj satisfy Conditions i k (c) and (c ) Indeed, in this case, the attractor of the GIFS de ned by the interpolation points {(0, 0); ( 1 , f ( 1 )); (1, 0)} and the coef cients cj is the graph of f To calculate i 2 2 the H lder function of f , we note: 1 S = f C 0 ([0; 1]) : |cj | [ + ; 1 ] k 2 where cj are determined by (94) and where is a strictly positive xed scalar (as k small as we wish) If we have f S, we can apply Proposition 94 and the pointwise H lder exponents of f S are obtained by applying this proposition NOTE 92 If we have f (0) = f (1) = 0, then: f (x) =

An adversarial node with fresh (higher) sequence number information can use this to fabricate lower metric authenticators for old (lower) sequence numbers An attacker can ood a node in the network with route updates containing recent or very large sequence numbers with false authenticators

where: j,k (x) = (2j x k)

and: (x) = 1 |2x 1| if x [0; 1] 0 if x [0; 1]

which is simply the decomposition of f in the Schauder basis 952 Algorithm Let f C 0 ([0; 1]) To simplify notations, let us suppose that f (0) = f (1) = 0 The function f is then written as: f (x) =

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