MD4 Algorithm

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The MD4 algorithm is described by Rivest in RFC 1320 [ l a l ] , where an efficient implementation (in C) is given. Here, we only provide enough details to implement the attack described below, and we use different notation than is found in [ l a l ] . MD4 operates on 32-bit words. The four bytes of each word are interpreted so that the leftmost byte is the low-order (least significant) byte. That is, a little-endian convention is followed. This is not a concern for the attack described here, but it does become an issue if we want to construct meaningful collisions. Let M be the message t o be hashed. The message M is padded so that its length (in bits) is equal to 448 modulo 512, that is, the padded message is 64 bits less than a multiple of 512. The padding consists of a single 1 bit, followed by enough zeros to pad the message to the required length. Padding is always used, even if the length of M happens to equal 448 mod 512. As a result, there is at least one bit of padding, and at most 512 bits of padding. Then the length (in bits) of the message (before padding) is appended as a

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5.3 MD4

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64-bit block. Padding is not a concern for the attack presented here; for the precise details, see [lal]. The padded message is a multiple of 512 bits and, therefore, it is also a multiple of 32 bits. Let M be the message and N the number of 32-bit words in the (padded) message. Denote the message words as Y,, so that M = (Yo,Yl,. ,Y N - ~Due to the padding, N is a multiple of 16. .. ). Define the three functions

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F ( A , B , C )= ( A A B ) v ( ~ A A C ) G ( A , B , C ) ( AA B ) v ( AA C ) v ( BA C) =

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(5.4) (5.5) (5.6)

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H(A,B,C) A @ B @ C =

where is the bitwise AND operation, V is the bitwise OR operation, @ is the XOR, and 1A is the complement of A. Each of these functions has a simple interpretation. The function F uses the bits of A to select between the corresponding bits of B and C , the function G is a majority vote in each bit position, while H can be viewed as a bitwise parity function. The MD4 hash algorithm appears in Table 5.1. In this algorithm, addition of 32-bit words are to be taken modulo 2 . In MD4, each 512-bit block is processed through three rounds, denoted as RoundO, Roundl, and Round2 in Table 5.1. Taken together, these three round functions and the final addition operation comprise the MD4 compression function, since they compress the 512-bit block and the 128-bit initial value into a 128-bit result. Table 5.2 shows how each of the three rounds is expanded into 16 steps, where the function F is used in round 0, the function G in round 1, and the function H in round 2, and <<< is a left rotation. Round i use the constant Ki, where

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KO= 0x00000000,

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K1 = Ox5a827999,

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and K2 = OxGed9ebal.

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Note that KO= 0, but we include it here to simplify some of the notation. The three rounds give a total of 48 steps, each involving one application of F , G or H . We number these steps consecutively, from 0 through 47, where in round 0, steps 0 through 15 occur, in round 1, steps 16 through 31 occur, and round 2 consists of steps 32 through 47. The shift for step i is denoted s i . The values of s i , for i = 0 , 1 , . . . ,47, are listed in Table 5.3. In Table 5.1, the permutation of the input words is denoted by 0, that is, Wi = X a ( z ) . The values of CJ(~), i = 0, 1 , 2 , . . . ,47, for are given in Table 5.10. At step i of MD4, only the 32-bit value Qi changes. A single MD4 step is illustrated in Figure 5.3, where

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