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[56] Y-C Hu, A Perrig, and Davic B Johnson Ariadne: a Secure On-demand Routing Protocol for Ad Hoc Networks, in Proceedings of the ACM Conference on Mobile Computing and Networking (Mobicom), 2002, pp 12 23 [57] C E Perkins and P Bhagwat, Highly Dynamic Destination-Sequenced Distance-vector Routing (DSDV) for Mobile Computers, SIGCOMM, UK, 1994 [58] J Broch, D Maltz, D Johnson, Y Hu, and J Jetcheva, A Performance Comparison of Multihop Wireless Ad Hoc Network Routing Protocols, Proceedings of the Fourth Annual ACM/ IEEE International Conference on Mobile Computing and Networking (Mobicom 98), October 1998 [59] Y-C Hu, D B Johnson, and A Perrig SEAD: Secure Ef cient Distance Vector Routing for Mobile Wireless Ad Hoc Networks, Ad Hoc Networks, 1(1), 175 192 (2003) [60] T Wan, E Kranakis, and P Van Oorschot, Securing the Destination Sequenced Distance Vector Routing Protocol (S-DSDV), in Proceedings of 6th International Conference on Information and Communications Security (ICICS 04), October 2004, Malaga Lecture Notes in Computer Science, Vol 3269, J Lopez, S Qing, and E Okamoto, eds Berlin, Springer, 2004 [61] T Clausen and P Jacquet, eds, Optimized Link State Routing Protocol (OLSR), IETF RFC 3626, October 2003; //wwwietforg/rfc/rfc3626txt [62] A Halfslund, A Tonnesen, R B Rotvik, J Andersson, and O Kure, Secure Extension to the OLSR Protocol, OLSR Interop and Workshop, 2004 [63] D Raffo, T Clausen, C Adjih, and P Muhlethaler, An Advanced Signature System for OLSR, SASN 04, October 2004 [64] C Adjih, T Clausen, P Jacquet, A Laouiti, P Muhlethaler, and D Raffo, Securing the OLSR Protocol, Proceedings of Med-Hoc-Net, June 2003 [65] P Papadimitratos and Z J Haas, Secure Link State Routing for Mobile Ad Hoc Networks, in Proceedings of the 2003 Symposium on Applications and the Internet Workshops (SAINT 03 Workshops), 2003, p 379 [66] J Kong and X Hong, ANODR: Anonymous On Demand Routing with Untraceable Routes for Mobile Ad-hoc Networks, ACM MobiHoc, June 2003 [67] Y Zhang, W Liu, W Lou, and Y Fang, MASK: an Anonymous Routing Protocol for Mobile Ad Hoc Networks, IEEE Transaction on Wireless Communications, in press [68] Y Zhang, W Liu, and W Lou, Anonymous Communications in Mobile Ad hoc Networks, IEEE INFOCOM 2005, Miami, FL, March 2005 [69] D Boneh and M Franklin, Identify-based Encryption from the Weil Pairing, in Proc CRYPTO 01, Springer, Berlin, 2001 [70] P S L M Barreto, H Y Kim, B Bynn, and M Scott, Ef cient Algorithms for PairingBased Cryptosystems, in Proc CRYPTO 02, Springer Berlin, August 2002 [71] Y Hu, A Perrig, and D Johnson, Packet Leashes: a Defense Against Wormhole Attacks in Wireless Ad Hoc Networks, in Proceedings of INFOCOM, San Francisco, CA, April 2003, pp 1976 1986 [72] L Hu and D Evans, Using Directional Antennas to Prevent Wormhole Attacks, Network and Distributed System Security Symposium, San Diego, CA, 5 6 February 2004 [73] R Poovendran and L Lazos, A Graph Theoretic Framework for Preventing the Wormhole Attack in Wireless Ad Hoc Networks, ACM Journal on Wireless Networks (WINET), in press [74] Y-C Hu, A Perrig, and D B Johnson, Rushing Attacks and Defense in Wireless Ad Hoc Network Routing Protocols, Proceedings of the 2003 ACM Workshop on Wireless Security (WiSe 2003), ACM, San Diego, CA, September 2003, pp 30 40.

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Requiring that | (t) b (dt)| resp | (t)Mb (t)dt| should be bounded away from zero in order to insure (478), though satis ed in some simple cases, seems unrealistically restrictive to be of practical use A few comments are in order here First, this condition can be weakened to arbitrarily allow small values of these integrals, as long as all their negative moments exist This can be shown by an argument using the Borel-Cantelli lemma Second, the condition may simplify in two ways For iid multipliers we know that these integrals are equal in distribution to C0,0 , thus only n = k = 0 has to be checked Further, for the Haar wavelet and symmetric multipliers, it becomes simply the condition that M0 be uniformly bounded away from zero (see (460)), or at least that E[|M0 1/2|q ] < for all negative q Third, if we drop iii) and allow the multiplier distributions to depend on scale (see (n,k) (n,k) [RIE 99]), then | (t) b (dt)| resp | (t)Mb (t)dt| has to be bounded away from zero only for large n In applications such as network traf c modeling we nd n+1 n+1 that on ne scales M2k M2k+1 is best modeled by discrete distributions on [0, 1] with large variance, ie without mass around 1/2

Fourth, another way out is to avoid small wavelet coef cients entirely in a multifractal analysis More precisely, we would follow [BAC 93, JAF 97] and n replace Cn,kn in the de nition of wkn (412) by the maximum over certain wavelet coef cients close to t Of course, the multifractal formalism of section 44 still holds [JAF 97] gives conditions under which the spectrum w, b (a) based on this n modi ed wk agrees with the H lder spectrum dim(Ea ) based on hn (Mb ) k 462 Multifractal properties of the derivative Corollary 45 establishes for the binomial what intuition suggests in general, ie that the multifractal spectra of processes and their derivative should be related in a simple fashion at least for certain classes of processes As we will show, increasing processes have this property, at least for the wavelet based multifractal spectra

h However, the order of H lder regularity in the sense of the spaces Ct (see Lemma 41) might decrease under differentiation by an amount different from 1 This is particularly true in the presence of highly oscillatory behavior such as chirps , h as the example ta sin(1/t2 ) demonstrates In order to assess the proper space Ct a 2-microlocalization has to be employed For good surveys see [JAF 95, JAF 91]

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