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As seen above, a function f (t) is H lderian with exponent (0, 1) at t if there exists a constant c such that for all t in a neighborhood of t, |f (t) f (t )| c|t t | In terms of oscillations, this condition may be written as: c, osc (t) c where osc (t) = sup|t t | f (t ) inf |t t | f (t ) = supt ,t [t ,t+ ] |f (t ) f (t )| The estimator is then simply de ned as the slope in the least-square linear regression of the logarithms of the oscillations versus the logarithms of the size of balls used to calculate the oscillations 1132 Wavelet coef cient regression (W CR) A method using wavelet coef cients is described in this section It relies on a theorem by S Jaffard This theorem shows how we can estimate the regularity at the point t0 using the wavelet coef cients (provided the wavelets verify some regularity properties [JAF 04]) THEOREM 111 (S Jaffard) Let f be a uniformly H lderian function and the pointwise H lder exponent of f at t0 Then there exists a constant c such that the wavelet coef cients verify: |cj,k | c2 j( + 2 ) (1 + |2j t0 k|)

Conversely, if j, k Z2 we obtain |cj,k | c2 j( + 2 ) (1 + |2j t0 k|) for a < then the H lder exponent of f at t0 is From this theorem, a traditional local regularity estimator is obtained if we consider only the indexes j, k such that |k 2j t0 | < cste This amounts to making the assumption that the local and pointwise H lder exponent of f at t0 coincide [LEV 04b] Under this hypothesis, an estimator is obtained through the regression slope p of log2 |cj,k | versus j More precisely, at each point t0 of a signal decomposed on n scales we estimate the regularity by: 1 sj log2 |cj,k | (n, t0 ) = Kn 2 j=1

Page 25 packages are broken down, individual assignments are created, and the process is delineated

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encryption is not at a layer at which it operates For example, when traf c is encrypted at the application layer, IDS modules can detect attacks at transport and lower layers, for example, ping-of-death, TCP SYN ood, smurf, or bubonic [79] If encryption is used at all layers, which is sometimes the case in battle eld networks, then schemes have to be designed to distribute the keys securely to the IDS active nodes so that the IDS can decrypt the traf c and analyze it for detecting attacks The NIDS approach in a wireless ad hoc network has another limitation It may be possible to devise attacks based on packet fragments and multiple routes that send fragments of the attacks through different paths NIDS sensors do not get the chance to observe all the fragments of the attack and therefore may not be able to detect the attack Such attacks are more effective in a wireless ad hoc network since nodes move often and routes change frequently This makes it more likely that a single NIDS entity may not observe all relevant traf c for detecting the attack It is clear from the earlier discussion that both the HIDS and the NIDS approaches have their own advantages and disadvantages Therefore, both approaches are often used together in complementary roles In a typical enterprise environment today, the activity that could be monitored by an IDS is incredibly large and it is often impractical to monitor all of it Enterprises usually select the key points in the enterprise to monitor depending on what is important to protect and where potential attacks may be originating from Hence, it is typical for enterprises to monitor activities on critical servers and applications through HIDS It is also typical for enterprises to position NIDSs right behind rewalls protecting the enterprise from external traf c since most attacks are expected to originate outside the enterprise and enter through the rewall A few well-placed NIDSs in an enterprise network can monitor most of the traf c in the network and detect malicious activities This architectural approach is shown in Figure 51 In the remainder of this chapter we will focus mostly on NIDS because they are impacted more by the unique features of wireless ad hoc networks 5112 Data Analysis Components Once the data is collected by IDS sensors, it has to be analyzed so that malicious activity can be detected IDSs typically incorporate analysis engines that automatically analyze the data collected by the various sensors to detect.

1133 Wavelet leaders regression (W L) This method is very similar to the previous one, but often provides better results For more information on wavelet leaders see [JAF 04]

DEFINITION 113 The wavelet leaders are d = sup |c | j (t0 ) is the dyadic cube of size 2 j at scale j containing the point t0 Let dj (t0 ) = sup adj( j (t0 )) |c | with adj( j (t0 )) the set of dyadic cubes adjacents to j (t0 ) PROPOSITION 111 (S Jaffard) If f C (t0 ), then c > 0, j 0, dj (t0 ) c2 ( + 2 )j

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