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Wang's multi-step modifications (also known as multi-message modifications) make it possible to satisfy some of the conditions in steps beyond 15. It is critical that when we satisfy conditions by this approach, we do not violate the out,put conditions from previous steps. This makes the multi-step modification more complex than the single-step modifications. There are actually several multi-step modification techniques, some of which are very convoluted, and some of which are not entirely deterniinistic, that is; the condition can fail with some small probabilit,y. Here, we describe the simplest example of a multi-step modification. The paper [16] discusses some other multi-step modifications, while Daum [34] provides a good description of several such techniques. 2 1 5 ) be the message block h f ~ after single-step Let, f = (XO, I , i " X p 16, where we want the output condition specified modifications. Consider by ( Q l 6 = 0)" to hold (see the Qle row of Table A-6). We have
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where I ~-~ I= .X i and f 1 5 = G(Q15, Q 1 4 , Q I : ~ ) . o and define D = -qoEo, where E is given i Let Q l ( j = (qO,ql, q : % l ) in (5.53). Then it is easy t,o verify that Q16 = Q l s + D will satisfy the required condition at step 16. As with the singlestep modification, we replacc with W1(; that so
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Since W ~ S = XI, we must ensure that all of the conditions in the first round involving X1 still hold. Since Qi, for i = 1,2,3,4,5, also depend on XI, we must carefully consider each of these steps. However, since no conditions were previously specified on Q1, the i = 1 case is not a concern. We have determined a new input at step 16, namely, W ~ S XI. From = the single-step modification, we have
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that is, 2 is the new Q1 that results from the modified X1 computed in step 16. Since no conditions were specified on Q1, we will not violate any previous conditions by letting Q1 = 2. Next, recall that
Q2 =
z+ ( f l ( Q 1 , Qo, Q-I) + Q-2 + 2;. + K2) << ~ <
z+ ( f l ( z ,Qo, Q-1) + Q-z + X2 + K2) << 5 2 , <
Using the same approach as the single-step modification, we choose that
Q2 =
which implies that
X = 2
2)>> s2) - f l ( z , Qo, Q-I) >
Observe that by selecting X2 in this way, the modification we made when selecting XI, as required for step 16, will not affect any of the output conditions from step 2. That is, all of the conditions on Q 2 that hold as a result of the single-step modifications still hold true. Similarly, we choose
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X = (64 4 (2
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and, finally,
X5 =
5 - f 4 ( Q 4 , Q3, Q2) )
2 - K5.
Since Z (the new Q1) is not used in the calculation of any other Q i , no other X i must be modified. The bottom line here is that we now have deterministically satisfied the conditions on step 16, while maintaining all of the conditions on steps 0
through 15 that resulted from the single-step modifications. The multi-step modification considered here is the sirnplest case. Several other methods have been developed in an effort t o slightly reduce the work factor of Wang s attack. The evolution of Wang s attack seems t o have reached the point wherc the attack is so efficient (about two minutes for Stevens [144] implementation) and the difficulty and complexity of finding improved multi-step modifications is now so high and many of the more advanced modification techniques only hold probabilistically-that it appears likely that further improvement along these lines will be incremental, a t best.