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ence is used. This is the part of the attack that is most shrouded in mystery, but we provide some hints on the methodology, below. In any case, finding useful difference patterns is obviously extremely challenging, since all MD5 attacks to date are based on Wang s lone differential patterns.
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3 . Given the differential patterns, derive a set of sufficient conditions on the outputs (along with a few necessary conditions on intermediate values). Provided all of these conditions are met, the differential path will hold and we will therefore obtain a collision.
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4. Finally, we must determine a pair of 1024-bit messages for which all of the sufficient conditions are satisfied. This is the computational part of the attack and Wang s approach for solving it consists of the following.
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Generate a random 512-bit message Mo. Use single-step modifications (as described below) to modify Mo so that all of the sufficient conditions in the early steps are forced to hold. This is accomplished via a direct modification of message words, and it can be done in a way that preserves the input differential conditions and any previously satisfied conditions. Use multi-step modifications (as outlined below) to force some of the sufficient conditions in the middle steps for Mo to hold. This a more complex modification technique than the single step modification. The difficulty arises since we must satisfy the differential conditions while maintaining all previously-satisfied sufficient conditions. Check the conditions for all of the remaining steps. If any of these conditions are not satisfied, goto 4b. These remaining sufficient conditions are satisfied probabilistically, that is, the attack is iterated until all of these probabilistic sufficient conditions hold. These iterations can be done efficiently and, since the input differential was chosen to behave nicely in the later steps, the probability of success is relatively high. Once Mo has been found, generate a random 512-bit message M i . Use single-step modifications to modify M1 so that all of the conditions for the early steps are satisfied. Note that the initial values for M I are not the MD5 initial values. Instead, the MD5 output from processing Mo must be used for the initial values. Use multi-step modifications to force the sufficient conditions in the middle steps for M I to hold. Check the conditions for all of the remaining steps. If any of these conditions are not satisfied, goto 4f.
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(i) Compute MA = MO AM, and M i = M I AM1. The precise values of AM0 and AM1 are specified by the input differential. For Wang s differential, these values are given in the next section.
(j) The MD5 hash of message M = ( M 0 , M l ) is equal to the MD5 hash of the message M = (MA,M i ) .
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See Problem 25 for one simple improvement to the computational phase of the attack as described here. For Wang s differential, the work factor for the computational part of the attack is dominated by finding Mo. The work factor for finding MO is on the order of 2n MD5 hashes, where n is the number of conditions for MO that are not satisfied by the modification techniques mentioned above (and described in more detail, below). As originally implemented by Wang, the computational phase had a work factor significantly greater than 240. Subsequent improvements have steadily lowered the work factor and, to date, the best claimed work factor is on the order of about 232.25 MD5 hashes [144]. It is possible that this will be reduced further by incremental improvements to Wang s attack. Below, we discuss each part of the attack, with the emphasis on the computational aspects. But before diving into the details, we mention an interesting insight due to Daum [34]. Suppose we have an MD-like hash function that only has three rounds--such as MD4 but not MD5. Then we would expect to find a collision using Wang s technique, since, roughly speaking, the input differential can be selected so that the third round conditions hold, singlestep modifications can then be used to ensures that the first round conditions hold, and, finally, multi-step modifications can ensure that the second round conditions hold. However, MD5 has four rounds, so some special property of MD5 must be exploited to make Wang s attack succeed. We briefly consider this special feature after discussing the attack outlined above.
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