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Inspired by the formal analysis of SRP and Ariadne, the authors of [55] proposed a routing protocol that can be proven to be statistically secure They call this protocol endairA, which is the reverse of Ariadne They propose that each intermediate node should sign the route reply instead of signing the route request as in Ariadne The operation of the protocol is shown in Figure 410 The initiator of the route request generates a message that contains the identi ers of the source and the destination as well as a randomly generated request identi er Each intermediate node that receives this route request message appends its identi er to the route accumulated so far and rebroadcasts it Route request messages received more than once are discarded When the route request message arrives at the destination, it generates a route reply message The route reply message contains the identi ers of the source and destination nodes, the accumulated route as obtained from the route request message and a digital signature of the target on the entire message The route reply is then sent back on the reverse route as given by the accumulated route in the route request message Each intermediate node on the reverse route veri es that its identi er is present in the accumulated route In addition, it also veri es if the predecessor and successor nodes to it in the accumulated route are indeed its neighbors If both these tests are valid, the intermediate node signs the route reply message and passes it to the next node in the path As a result, the route reply reaches the source node This node veri es whether it received the message from its neighbor and if this neighbor is the rst node on the path If so, it veri es the signatures of all the nodes in the reply The path is then accepted to be valid if all the signatures are veri ed We show the various steps assuming that source node S desires to determine a route to destination node D in Figure 410 The various steps are self-explanatory The authors have shown that the endairA is provably secure in a network that contains a single adversary and a single compromised node The protocol is also less resource-intensive since nodes need to sign only the route reply messages This is much.

C + (t0 )

This is also equivalent to: = sup{ > 0 : f C (t0 )} 93 Iterated function systems (IFS) Let K be a complete metric space, with distance d Given m continuous functions Sn of K in K, we call an iterated functions system (IFS) the family {K, Sn : n = 1, 2, , m} Let H be the set of all non-empty closed parts of K Then the set H is a compact metric space for the Hausdorff distance h [HUT 81] de ned, for any A, B in H, by: h(A, B) = max sup inf d(x, y), sup inf d(x, y)

Let us consider the operator W : H H de ned by:

.

W (G) =

.

Sn (G)

.

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We call any set A H which is a xed point of W an attractor of the IFS {K, Sn : n = 1, 2, , m}, ie, it veri es: W (A) = A An IFS always possesses at least one attractor Indeed, given any set G H, the closure of the accumulation set of points {W (m) (G)} , with W (m) (G) = m=1 W (W (m 1) (G)), is a xed point of W If all Sn functions are contractions, then the IFS is said to be hyperbolic In this case, W is also a contraction for the Hausdorff metric; thus, it possesses a single xed point which is the single attractor of the IFS When the IFS is hyperbolic, the attractor can be obtained in the following manner [BAR 85a]: let p = (p1 , , pm ) be a probability vector with pn > 0 and n pn = 1 From the xed point x0 of S1 , let us de ne the sequence xi by successively choosing xi {S1 (xi 1 ), , Sm (xi 1 )}, where the probability pn is linked to the occurrence xi = Sn (xi 1 ) Then, the attractor is the closure of the trajectory {xi }i N In this chapter, we focus on IFS which make it possible to generate continuous function graphs [BAR 85a] Given a set of points {(xn , yn ) [0; 1] [u; v], n = 0, 1, , m}, with (u, v) 2 , let us consider the IFS given by m contractions Sn (n = 1, , m) which are de ned on [0; 1] [u; v] by: Sn (x, y) = Ln (x); Fn (x, y) where Ln is a contraction which transforms [0; 1] into [xn 1 ; xn ] and where Fn : [0; 1] [u; v] [u; v] is a contraction with respect to the second variable, which satis es: Fn (x0 , y0 ) = yn 1 ; Fn (xm , ym ) = yn (91).

Then, the attractor of this IFS is the continuous function graph which interpolates the points (xn , yn ) In general, this type of function is called a fractal interpolation function [BAR 85a] The most studied class of IFS is that of af ne iterated function systems, ie, IFS for which Ln and Fn are af ne functions We will study this class later We also assume that the interpolation points are equally spaced Then, Sn (0 n < m) can be written in a matrix form as: Sn t x = 1/m 0 an cn t n/m + x bn

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