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Insert code 128 barcode in .NET F ( A ,B ,C ) KO if 0 5 i 5 15 G ( A , B , C ) K1 if 16 5 i 5 31 H ( A ,B , C ) + K2 if 32 5 i 5 47.
F ( A ,B ,C ) KO if 0 5 i 5 15 G ( A , B , C ) K1 if 16 5 i 5 31 H ( A ,B , C ) + K2 if 32 5 i 5 47.
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H A S HFUNCTIONS
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Table 5.1: MD4 Algorithm
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iLf = (Yo, Y1,. . ,Y N - ~ ) , . message to hash, after padding Each Y , is a 32-bit word and N is a multiple of 16 MD4(111) / / initialize (A,B, C , D ) = IV ( A ,B , D ) = (0x67452301,Oxef cdab89,Ox98badcfe,0x10325476) for i = 0 t o N/16 - 1 // Copy block i into X X j = Y l ~ i +for j = 0 t o 15 j, // copy to w Wj = Xocj,, f o r j = 0 to 47 / / initialize Q
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Q-3, Q-2, Q-1) = (A, D , C , B ) Rounds 0, 1 and 2 RoundO(Q, X ) Roundl(Q, X ) Round2(Q, X ) // Each addition is modulo 232 (A,B , C ,D ) = (Q44 + Q-4, Q47 Q-I, next i return A, B !C, D end MD4
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(Q-4,
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+ Q-2,
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MD4 Attack
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The attack described in this section is due to Dobbertin [42]. Our description uses different notation than the original, and we have rearranged and
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Qi-2 Qi-3 Qi-4
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<<< s. si
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Qi- 1
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Qi-2
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Qi-3
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Figure 5.3: MD4 step.
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5.3 MD4
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Table 5.2: MD4 Rounds RoundO(Q, W ) // steps 0 through 15 for i = 0 t o 15
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Qi = (Qi-4
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next i end Round0
+ F(Qi-I, Qi-2,
Qi-3)
+ Wi+ K O )<< <
Roundl(Q, W ) // steps 16 through 31 for i = 16 to 31
next i end Round1
= (Qi-4
+ G(Qi-1, Qi-2,
Qi-3)
+ Wi+ ~
<< si < )
Round2(Q, W ) // steps 32 through 47 for i = 32 t o 47
next i end Round2
= (Qi-4
+ H(Qi-1,Qi-2,
Qi-3)
+ Wi+ K2) << si <
expanded on the exposition at several points. This attack finds two distinct 512-bit blocks that hash to the same value, thereby yielding a collision. As noted in Section 5.1, if any common bit string is appended to two colliding blocks, the resulting strings will also yield a collision. Dobbertin s attack includes a differential phase, where we require that the pair of 512-bit inputs satisfy a certain differential property at an intermediate stage of the algorithm. When this differential property holds, then with a
Table 5.3: MD4 Shifts Stepi 1 0 Shift s;I 3 Step i 16 Shifts, 3 Step i 32 Shift s7 3
7 17 5 33 9
2 3 4 11 19 3
18 9 35 11
19 13 35 15
20 3 36 3
5 7 21 5 37 9
6 11 22 9 38 11
7 19 23 13 39 15
8 3 24 3 40 3
9 7 25 5 41 9
10 11 12 13 14 15 11 19 3 7 11 19 26 9 42 11 27 13 43 15 28 3 44 3 29 5 45 9 30 9 46 11 31 13 47 15
HASH FUNCTIONS
Table 5.4: MD4 Input Word Order
0 0 ) 16 o(z) 0 Step i 32 ~ ( i )0
Stepi ~ ( i Step i
5 5 21 5 33 35 35 36 37 8 4 1 2 2 10
1 1 17 4
2 3 4 2 3 4 18 19 20 8 12 1
6 6 22 9 38 6
7 7 23 13 39 14
8 8 24 2 40 1
9 9 25 6 41 9
1011 10 11 26 27 10 14 42 43 5 13
12 12 28 3 44 3
131415 13 14 14 29 30 31 7 11 15 45 46 47 11 7 15
probability of about 1/222, we obtain a collision for the full hash. The trick then is to efficiently find a sufficient number of pairs of inputs that satisfy this differential property. After describing the differential phase of the attack, we then specify a systeni of nonlinear equations, any solution of which will satisfy the desired differential property. For the attack to be practical, we must be able to efficiently solve this system of equations, and we explain how to accomplish this. There is also a third phase of the attack that connects the second phase to the initial steps of the hash algorithm. This third phase is relatively simple. Before we dive into the computational details behind the attack, we try to motivate Dobbertin s approach to the problem. Here, we only provide a quick overview -for inorc details see Daum [34].