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[16] W Du, J Deng, Y S Han, P Varshney, J Katz, and A Khalili, A Pairwise Key Predistribution Scheme for Wireless Sensor Networks, ACM Transactions on Information and System Security (TISSEC), 2005, pp 228 258 [17] L Eschenauer and V Gligor, A Key-management Scheme for Distributed Sensor Networks In Proceedings of the 9th ACM Conference on Computer and Communications Security, November 2002, pp 41 47 [18] D Liu, P Ning, and R Li, Establishing Pairwise Keys in Distributed Sensor Networks, ACM Transactions on Information Systems Security, 8(1), 41 77 (2005) [19] D Balfanz, D Smetters, P Stewart, and H Wong, Talking to Strangers: Authentication in Ad Hoc Wireless Networks, in Proceedings of 9th Annual Network and Distributed System Security Symposium, San Diego, CA, 2002 [20] H Deng and D P Agrawal, TIDS: threshold and identity based security scheme for wireless ad hoc networks, Ad Hoc Networks, 2(3), 291 307 (2004) [21] A Khalili, J Katz, and W A Arbaugh, Toward Secure Key Distribution in Truly Ad-hoc Networks, in Proceedings of IEEE Workshop on Security and Assurance in Ad-Hoc Networks, 2003 [22] N Arora and R K Shyamasundar UGSP: Secure Key Establishment Protocol for Ad-hoc Networks In R K Ghosh and H Mohanty, eds, Distributed Computing and Internet Technology: First International Conference, ICDCIT 2004, Bhubaneswar, India, December 22 24, 2004 Proceedings, Volume 3347 of Lecture Notes in Computer Science, Springer, Berlin, 2004, pp 391 399 [23] D J Malan, M Welsh, and M D Smith, A Public-key Infrastructure for Key Distribution in TinyOS Based on Elliptic Curve Cryptography, IEEE Secon 2004 [24] L Zhou and Z Haas, Securing Ad Hoc Networks, IEEE Network, 13(6), 24 30 (1999) [25] H Luo, J Kong, P Zerfos, S Lu, and L Zhang, URSA: Ubiquitous and Robust Access Control for Mobile Ad Hoc Networks, IEEE/ACM Transactions on Networking, December 2004, pp 1049 1063 [26] J Kong, P Zerfos, H Luo, S Lu, and L Zhang, Providing Robust and Ubiquitous Security Support for Mobile Ad Hoc Networks, in Proceedings of Ninth International Conference on Network Protocols (ICNP), November 2001 [27] S Capkun, L Buttyan, and J-P Hubaux, Self-Organized Public-Key Management for Mobile Ad Hoc Networks, IEEE Transactions on Mobile Computing, 2(1), 52 64 (2003) [28] J-P Hubaux, L Buttyan, and S Capkun, The Quest for Security in Mobile Ad hoc Networks, in MobiHoc ACM, New York, 2001, pp 146 155 [29] G O Shea and M Roe, Child-proof Authentication for MIPv6 (CAM), ACM Computer Communications Review, 31(2), 4 8 (2001) [30] G Montenegro and C Castelluccia, Statistically Unique and Cryptographically Veri able (SUCV) Identi ers and Addresses, in Proceedings of NDSS, 2002 [31] M Cagalj, S Capkun, and J P Hubaux, Key Agreement in Peer-to-Peer Wireless Networks, in Proceedings of the IEEE (Special Issue on Cryptography and Security), February 2006, pp 467 478 [32] C Ellison and S Dohrmann, Public-key Support for Group Collaboration, ACM Transactions on Information Systems Security, 6(4), 547 565 (2003) [33] C Gehrmann, C Mitchell, and K Nyberg, Manual Authentication for Wireless Devices, RSA Cryptobytes, 7(1), 29 37 (2004) [34] C Gehrmann and K Nyberg, Enhancements to Bluetooth Baseband Security, in Proceedings of 6th Nordic Workshop on Secure IT Systems, Copenhagen, November 2001, pp 39 53 [35] J-H Hoepman, The Ephemeral Pairing Problem, in Intl Conference on Financial Cryptography, Key West, FL, February 2004, Vol 3110, pp 212 226.

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The log-normal framework also allows us to calculate F (a) explicitly, demonstrating which rescaling properties of the marginal distributions of the increment processes of Mb are captured in the multifractal spectra Indeed, if all n n ln(Mk ) are N (m, 2 ) then ln(2) k is N (m, 2 /n) The mean value theorem of integration gives

n P [| k a| < ] =

ln(2)( a+ ) ln(2)( a )

with xa,n [a , a + ] for all n Keeping only the exponential term in n and substituting m = 2 ln(2) we nd 1 n log2 (2n P [| k a| < ]) n 1 ln(2) 2 2 xa,n 1 2 2 ln(2)

(474)

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Comparing with (473) we see that T (a) = F (a), as stated in Theorem 42 The above computation shows impressively how well adapted a multiplicative iteration with log-normal multipliers is to multifractal analysis (or vice versa): F extracts, basically, the exponent of the Gaussian kernel Since the multifractal formalism holds for Mb these features can be measured or estimated using the re-normalized histogram, ie the grain based multifractal spectrum f (a) This is a property which could be labeled with the term ergodicity Note, however, that classical ergodic theory deals with observations along an orbit of increasing length, while f (a) concerns a sequence of orbits 455 Beyond dyadic structure We elaborate here generalizations of the binomial cascade Statistically self-similar measures: a natural generalization of the random n binomial, denoted here by Mc , is obtained by splitting intervals Jk iteratively n+1 n+1 n+1 n+1 n into c subintervals Jck , , Jck+c 1 with length |Jck+i | = Lck+i |Jk | and n+1 n+1 n mass c (Jck+i ) = Mck+i c (Jck ) In the most simple case, we will require mass n+1 n+1 conservation, ie Mck + + Mck+c 1 = 1, but also Ln+1 + + Ln+1 ck ck+c 1 = 1 which guarantees that c lives everywhere Assuming the analogous properties of ii) and iii) to hold for both the length- as well as the mass-multipliers we nd that T Mc (q) is the unique solution of E (M0 )q (L0 ) T (q) + + (Mc 1 )q (Lc 1 ) T (q) = 1 (475)

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This formula of T (q) can be derived rigorously by taking expectations where appropriate in the proof of [RIE 95a, Prop 14] Doing so shows, moreover, that T (q) assumes a limit in these examples Multifractal formalism: it is notable that the multifractal formalism holds for the class of statistically self-similar measures described above in Theorem 44 (see [ARB 96])

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