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Figure 5.5: MD5 step. authors in subsequent papers [94, 127, 1611. To date, the most thorough and insightful description of the MD5 attack methodology was published by Daurri in his PhD dissertation [34]. Below, we describe the MD5 attack which originally appeared in [157]. As mentioned above, this attack can be used to efficiently find a pair of 1024-bit messages whose MD5 hashes are the same. But before giving details of the attack, we first present some background and motivation. This background material relies primarily on information in Wang s paper [157] and in Daum s PhD dissertation [34].

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A Note on Notation

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Unfortunately, everyone who writes about Wang s attack seems to have their own pet notation. Even more unfortunately, we are no exception. Our notation is closest to that in [16], with the only major difference being that we number bits from left-to-right-in our notation, the high-order (leftmost) bit is bit 0, while the low-ordw (rightmost) bit of a 32-bit word is bit number 31. As noted above, we denote the step outputs for a message block as QO through Q 6 3 , with the IV consisting of Q - 4 , Q-1, Q - 2 , Q-3. In contrast, in several papers, including [81, 1441, the IV is denoted Q - 3 , Qo,Q-1, Q - 2 and the computed outputs are Q1 t,hrough Q 6 4 (but, as if t o further confuse matters, these authors number thc message words 0 through 63). The papers of Wang [155, 156, 1571 and various other authors, use a completely different numbering of the outputs. Instead of Qj, the outputs arc denoted as a j , b j , cj, or d j , where j = 1 , 2 , .. . ,16 (or, in some cases, j = 0 , 1 , .. . , 15) depending on the round. While this is more consistent with

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Daurn s work was supervised by Hans Dobbertin, whose MD4 attack is discussed in Section 5.3.2.

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5.4 11.105

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the notation used in the original description of MD5 [122], it is awkward for analysis of the algorithm. Strangely, Wang numbers the bits of 32-bit words from 1 to 32 (right-to-left). Some authors use 6 for the modular difference and A for the XOR difference, while we use A for the modular difference. The bottom line is that considerable care must be taken when attempting to analyze results culled from a variety of papers, since it is not a trivial task to translate the results into a consistent notation.

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A Precise Differential

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Wang s MD5 attack is a differential attack. Recall that Dobbertin s MD4 attack [42] uses subtraction modulo 232 as the difference operator. Wang s attack uses this same modular difference for inputs. However, some parts of the MD5 attack require more detailed information than modular subtraction provides, so a kind of precise differential [157] is also employed. This differential combines modular subtractions with information on the precise location of the bit differences. In effect, this precise differential includes both a modular difference and an XOR difference, and also provides additional information beyond what these two standard differentials provide. To motivate this precise differential, consider the pair of bytes given by y = 00010101 and y = 00000101 and another pair of bytes z = 00100101 and z = 00010101. Then

= z/ -

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= O O O ~ O O O O= 24,

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which implies that with respect to the modular difference, these two pairs are indistinguishable. However, in the MD5 attack, we must distinguish between cases such as these, and to do so, we need more information than modular subtraction can provide. To this end, we employ a differential that includes modular subtraction along with an explicit specification of the bit positions that differ between the two terms. While an XOR difference specifies the bit positions that differ, we actually require even more information than modular subtraction and the XOR together can provide. Specifically, we need to know whether the difference in each bit position corresponds t o a +1 or -1 in the modular difference, and this level of detail is not provided by the XOR. Let y = (yo, y1,. . . , y7) and y = (yb, y;, . . . , yb), where each yi and yi is a bit. Using the same example as above, we have