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for m = 0, 1 and n = 0, 1, 2 with > H2 The symbol C denotes a generic constant HYPOTHESIS 62 In the preceding de nition, we have a, b C 1 ([0, 1]) and, for every t [0, 1], we have a(t)b(t) = 0 Limiting ourselves to an expansion of order 2 is a convention adopted with the aim of facilitating the presentation of the identi cation algorithms For the same reason, we suppose that ltered white noises are the processes indexed by t For a better understanding of the relationship between ltered white noises and FBM, it is enough to suppose that R(t, ) is identically zero We nd Xt = CH1 a(t)BH1 (t) + CH2 b(t)BH2 (t), for BH1 , BH2 two fractional Brownian motions and the CH constant de ned in (68) It should, however, be noted that, even in this simpli ed example, BH1 and BH2 are not independent and therefore

the law of X is not trivially deduced from FBM This last example illustrates an additional virtue of ltered white noises: their de nition authorizes not only the functional parameters a(t) and b(t) to vary according to the position, but also the superposition of self-similar phenomena of orders H1 and H2 In addition, we will nd an interpretation of H2 in terms of long dependence in 5 On the other hand, starting from formula (627), it is dif cult to nd a convenient expression of the reproducing kernel Hilbert space of a ltered white noise Moreover, the ltered white noises do not maintain the overall properties of FBMs, that is to say, self-similarity and stationarity of the increments, which is an advantage when modeling certain phenomena Since the lter of a white ltered noise is asymptotically equivalent in high frequency to that of a FBM, only the local properties remain For example, the ltered white noises verify a property of local self-similarity of the following type PROPOSITION 61 A ltered second order white noise X associated with a lter of form (626) is locally self-similar of constant multifractional function equal to H1 : lim + X(t + u) X(t)

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= CH1 a(t) BH1 (u)

The level of security afforded could be related to cost A higher level of security could result in the system being secure even against well-funded adversaries Note, though, that the level of security itself is a qualitative measure The functionality would be related to the attributes that can be achieved using the proposed primitive For example, nonrepudiation would need the use of concepts from asymmetric key cryptography

(628)

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A proof of this result can be found in [BEN 98a], regarding the multifractional processes 633 Elliptic Gaussian random elds (EGRP) Another manner of generalizing FBM, which constitutes the approach adopted in [BEN 97b], consists of starting from the reproducing kernel Hilbert space By returning to De nition 63, we can already represent the reproducing kernel Hilbert space norm by means of the operator J and the formula: J ( 1 ), J ( 2 )

= 1 , 2

L2 (

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