4.4 HELLMAN S TIME-MEMORY TRADEOFF Table 4.3: Algorithm to Find the Key from S P

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// Is key K at position t findKey(i,l,j)

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1 in chain t of table i

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Y = Fz(E(P,Y ) )

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return(K) e l s e / / false alarm r e t u r n (not found) end i f end findKey block as necessary. In contrast, the issue of merging and cycling chains is of fundamental importance in this TMTO attack.

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c = E ( P , K ) then

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4.4.3 Success Probability

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What is Trudy s probability of success when she uses Hellman s TMTO attack The fundamental problem is that keys can appear within more than one chain. Therefore, estimating the probability of success is equivalent estimating the probability of such duplication. Perhaps the easiest2 way to estimate the success probability for Hellman s TMTO attack is to use the classic occupancy problem, which is described nicely by Feller [49]. The details of the derivation are left as a homework problem, but the result is that Trudy s probability of successfully finding a key is approximately P(success) = 1 - e--mt+ . (4.3)

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The probabilities given by (4.3) for various choices of mtr are given in Table 4.4. Hellman suggests choosing m = t = r = 2 13 (4.4)

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and, as can be seen in Table 4.4, the estimated probability of success for this choice of parameters is about 0.63. In general, the cryptanalytic TMTO pre-computation requires mtr encryptions. The necessary storage is proportional to rm; the number of chains.

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Table 4.4: Approximate TMTO Success Probabilities P(success) 0 2k-5 0.03 21;-4 0.06 21;-3 0.12 2k-2 0.22 21;-1 0.39 21; 0.63 2krl 0.86 2k+2 0.98 2k+3 0.99 0 0 1.00 If key K lies on one of the pre-computed chains then the time required when the attack is executed is about t (that is, t / 2 steps, on average, are needed to find the matching E P and then another t / 2 steps are required. on average, to find K ) . For the parameters in equation (4.4), this gives a pre-computation of 2k encryptions, a memory requirement of 2 2 k / 3 , and a time requirement of 22k/ . For example for DES-the cipher for which Hellman originally developed his attack --this yields a costly pre-computation of 256, but then the resulting time and memory requirements for each instance of the attack phase are both less than 238, with a high probability of success. Although the attack is only probabilistic, the probability of success is high, provided that the necessary pre-computation is feasible.

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4.4.4 Distributed TMTO

Hellman s TMTO is easily adapted to a distributed attack. This version of the attack employs distinguished points \20]. The crucial insight is that we need not use fixed-length chains, but, instead, we can simply construct it chain until some easily distinguished point is found. For example, we can construct each chain until we obtain an output of the form

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. . , Zs-1, 0 , 0 , . . . , 0 ) .

Then each chain will, on average, be of length 271-s. In practice we would want to set a limit, on the maximum length of a chain and reject, any chain that exceeds the limit. Using distinguished points, the pre-computation is similar to the case

4.4 HELLMA N S TIME-M E M O R Y TRADE-OFF

described above, except that we now retain triples

where l j is the length of chain j (that is, the number of elements computed before a distinguished point was found). We must also keep track of the maximum length of any chain within a table; for table i, denote this as Mi. Now suppose that r computers are available. Then each computer can search one of the T tables of chains. Computer i only needs to know the function Fi along with the ciphertext C and Mi, as well as the definition of a distinguished point. In particular, the triples in equation (4.5) do not need to be transmitted to any of the r computers, saving significant bandwidth and reducing the storage requirement on the individual computers. Each computer then proceeds with the attack as described above, with the exception that instead of looking for a matching EPj at each step, a distinguished point is sought. If computer i finds such a point within Mi iterations, the distinguished point is returned. Then secondary testing is necessary to determine whether the putative solution is an actual endpoint from table i or a false alarm. This secondary testing requires access t o all (SPj,E P j , l j ) triples in (4.5). Note that the overall work for secondary testing can be adjusted by selecting the definition of a distinguished point appropriately. If an endpoint is found, the process of attempting to recover K from the corresponding starting point proceeds exactly as in the non-distinguished point case discussed above.