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Indeed, for H > 1/2 this expression decays much slower than 1/m As is shown in [COX 84] var(X (m) ) m2H 2 is equivalent to rX (k) k 2H 2 and so, G(k) is indeed LRD for H > 1/2 Let us demonstrate with FGN how to relate LRD with multifractal analysis based only on the fact that it is a zero-mean processes, not (41) To this end let

After a certain duration, the sender discloses the key K to all the receivers At this point every receiver can check if the disclosed key K is indeed valid This can be done by checking if the hash of K corresponds to the hashed value sent authentically by the sender earlier If this is true, then every receiver can verify the authenticity of every packet Note that the receiver must not accept any packet the HMAC of which has been calculated using key K after the disclosure of the key K This is achieved by the sender disclosing the timeout schedule to the receivers and by having loose time synchronization between the receivers and the sender The approach given above can be easily extended (and is in fact more ef cient) when considering multiple time intervals In this case, each sender splits time into multiple intervals It then chooses a secret key and generates a one-way key chain through repeated use of the one-way hash function property The last value of the hash chain is then transferred to the receivers authentically Each generated key that is one of the values of the hash chain will be used in one time interval The keys are used in the reverse order of generation Thus, the message authentication keys used for packets in the previous interval are derived (via hashing) from the message authentication keys used in the current interval During each interval the sender calculates the HMAC of each packet using the key corresponding to that interval The transmitted packet contains the original contents of the packet, the calculated HMAC value over the original packet and the most recent one-way chain value that can be disclosed Thus, the sender discloses the keys used after the time interval of their use The receivers must know the key disclosure schedule, so when the receiver receives a packet, it checks that the key used to compute the HMAC on the received packet is still secret It can do so using its knowledge of the key disclosure schedule since the receivers and the sender are assumed to be loosely synchronized in time As long as this key is still secret, the receiver buffers the packet If the key is not secret then the receiver has to drop the packets When the key is disclosed, the receiver checks the correctness of the key (using the one way property of hash functions) and then authenticates the buffered packets This operation of TESLA is illustrated in Figure 212 Thus, TESLA achieves asymmetry needed for broadcast authentication through clock synchronization and a delayed key disclosure Note also that TESLA is robust to packet loss A drawback, though, is the need for receiver to buffer packets transmitted during an.

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= 2 n denote the nest resolution we will consider, and let 1 be the largest For m = 2i (0 i n) the process mG(m) (k) becomes simply BH ((k + 1)m ) BH (km ) = BH ((k + 1)2i n ) BH (k2i n ) However, the second moment of this expression which is also the variance is exactly what determines T (2) More precisely, using stationarity of G and substituting m = 2i , we obtain 2 (n i)T (2)

E S n i (2) =

E |mG(m) (k)|2 (488)

= 2n i 22i var G(2

.

This should be compared with the de nition of the LRD parameter H using var(G(m) ) m2H 2 or var(G(2 ) )

2i(2H 2)

Time Did the project come in on schedule Cost Did the project come in on or under its budget Performance Does the project perform according to the technical specifications we originally developed for it

(489)

At this point a conceptual dif culty arises Multifractal analysis is formulated in the limit of small scales (i ) while LRD is a property for large scales (i ) Thus, the two exponents H and T (2) can in theory only be related when assuming that the scaling they represent is actually exact at all scales, and not only asymptotically In any real world application, however, we will determine both H and T (2) by nding a scaling region i i i in which (488) and (489) hold up to satisfactory precision Comparing the two scaling laws in i yields T (2) + 1 2 = 2H 2, or H= T (2) + 1 2 (490)

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