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Network Working Group R. Rivest
Request for Comments: 1186 MIT Laboratory for Computer Science
October 1990
The MD4 Message Digest Algorithm
Status of this Memo
This RFC is the specification of the MD4 Digest Algorithm. If you
are going to implement MD4, it is suggested you do it this way. This
memo is for informational use and does not constitute a standard.
Distribution of this memo is unlimited.
Table of Contents
1. Abstract .................................................... 1
2. Terminology and Notation .................................... 2
3. MD4 Algorithm Description ................................... 2
4. Extensions .................................................. 6
5. Summary ..................................................... 7
6. Acknowledgements ............................................ 7
APPENDIX - Reference Implementation ............................. 7
Security Considerations.......................................... 18
Author's Address................................................. 18
1. Abstract
This note describes the MD4 message digest algorithm. The algorithm
takes as input an input message of arbitrary length and produces as
output a 128-bit "fingerprint" or "message digest" of the input. It
is conjectured that it is computationally infeasible to produce two
messages having the same message digest, or to produce any message
having a given prespecified target message digest. The MD4 algorithm
is thus ideal for digital signature applications, where a large file
must be "compressed" in a secure manner before being signed with the
RSA public-key cryptosystem.
The MD4 algorithm is designed to be quite fast on 32-bit machines.
On a SUN Sparc station, MD4 runs at 1,450,000 bytes/second. On a DEC
MicroVax II, MD4 runs at approximately 70,000 bytes/second. On a
20MHz 80286, MD4 runs at approximately 32,000 bytes/second. In
addition, the MD4 algorithm does not require any large substitution
tables; the algorithm can be coded quite compactly.
The MD4 algorithm is being placed in the public domain for review and
possible adoption as a standard.
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RFC 1186 MD4 Message Digest Algorithm October 1990
(Note: The document supersedes an earlier draft. The algorithm
described here is a slight modification of the one described in the
draft.)
2. Terminology and Notation
In this note a "word" is a 32-bit quantity and a byte is an 8-bit
quantity. A sequence of bits can be interpreted in a natural manner
as a sequence of bytes, where each consecutive group of 8 bits is
interpreted as a byte with the high-order (most significant) bit of
each byte listed first. Similarly, a sequence of bytes can be
interpreted as a sequence of 32-bit words, where each consecutive
group of 4 bytes is interpreted as a word with the low-order (least
significant) byte given first.
Let x_i denote "x sub i". If the subscript is an expression, we
surround it in braces, as in x_{i+1}. Similarly, we use ^ for
superscripts (exponentiation), so that x^i denotes x to the i-th
power.
Let the symbol "+" denote addition of words (i.e., modulo- 2^32
addition). Let X <<< s denote the 32-bit value obtained by circularly
shifting (rotating) X left by s bit positions. Let not(X) denote the
bit-wise complement of X, and let X v Y denote the bit-wise OR of X
and Y. Let X xor Y denote the bit-wise XOR of X and Y, and let XY
denote the bit-wise AND of X and Y.
3. MD4 Algorithm Description
We begin by supposing that we have a b-bit message as input, and that
we wish to find its message digest. Here b is an arbitrary
nonnegative integer; b may be zero, it need not be a multiple of 8,
and it may be arbitrarily large. We imagine the bits of the message
written down as follows:
m_0 m_1 ... m_{b-1} .
The following five steps are performed to compute the message digest
of the message.
Step 1. Append padding bits
The message is "padded" (extended) so that its length (in bits)
is congruent to 448, modulo 512. That is, the message is
extended so that it is just 64 bits shy of being a multiple of
512 bits long. Padding is always performed, even if the length
of the message is already congruent to 448, modulo 512 (in
which case 512 bits of padding are added).
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RFC 1186 MD4 Message Digest Algorithm October 1990
Padding is performed as follows: a single "1" bit is appended
to the message, and then enough zero bits are appended so that
the length in bits of the padded message becomes congruent to
448, modulo 512.
Step 2. Append length
A 64-bit representation of b (the length of the message before
the padding bits were added) is appended to the result of the
previous step. In the unlikely event that b is greater than
2^64, then only the low-order 64 bits of b are used. (These
bits are appended as two 32-bit words and appended low-order
word first in accordance with the previous conventions.)
At this point the resulting message (after padding with bits
and with b) has a length that is an exact multiple of 512 bits.
Equivalently, this message has a length that is an exact
multiple of 16 (32-bit) words. Let M[0 ... N-1] denote the
words of the resulting message, where N is a multiple of 16.
Step 3. Initialize MD buffer
A 4-word buffer (A,B,C,D) is used to compute the message
digest. Here each of A,B,C,D are 32-bit registers. These
registers are initialized to the following values in
hexadecimal, low-order bytes first):
word A: 01 23 45 67
word B: 89 ab cd ef
word C: fe dc ba 98
word D: 76 54 32 10
Step 4. Process message in 16-word blocks
We first define three auxiliary functions that each take
as input three 32-bit words and produce as output one
32-bit word.
f(X,Y,Z) = XY v not(X)Z
g(X,Y,Z) = XY v XZ v YZ
h(X,Y,Z) = X xor Y xor Z
In each bit position f acts as a conditional: if x then y else
z. (The function f could have been defined using + instead of
v since XY and not(X)Z will never have 1's in the same bit
position.) In each bit position g acts as a majority function:
if at least two of x, y, z are on, then g has a one in that bit
position, else g has a zero. It is interesting to note that if
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RFC 1186 MD4 Message Digest Algorithm October 1990
the bits of X, Y, and Z are independent and unbiased, the each
bit of f(X,Y,Z) will be independent and unbiased, and similarly
each bit of g(X,Y,Z) will be independent and unbiased. The
function h is the bit-wise "xor" or "parity" function; it has
properties similar to those of f and g.
Do the following:
For i = 0 to N/16-1 do /* process each 16-word block */
For j = 0 to 15 do: /* copy block i into X */
Set X[j] to M[i*16+j].
end /* of loop on j */
Save A as AA, B as BB, C as CC, and D as DD.
[Round 1]
Let [A B C D i s] denote the operation
A = (A + f(B,C,D) + X[i]) <<< s .
Do the following 16 operations:
[A B C D 0 3]
[D A B C 1 7]
[C D A B 2 11]
[B C D A 3 19]
[A B C D 4 3]
[D A B C 5 7]
[C D A B 6 11]
[B C D A 7 19]
[A B C D 8 3]
[D A B C 9 7]
[C D A B 10 11]
[B C D A 11 19]
[A B C D 12 3]
[D A B C 13 7]
[C D A B 14 11]
[B C D A 15 19]
[Round 2]
Let [A B C D i s] denote the operation
A = (A + g(B,C,D) + X[i] + 5A827999) <<< s .
(The value 5A..99 is a hexadecimal 32-bit
constant, written with the high-order digit
first. This constant represents the square
root of 2. The octal value of this constant
is 013240474631. See Knuth, The Art of
Programming, Volume 2 (Seminumerical
Algorithms), Second Edition (1981),
Addison-Wesley. Table 2, page 660.)
Do the following 16 operations:
[A B C D 0 3]
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RFC 1186 MD4 Message Digest Algorithm October 1990
[D A B C 4 5]
[C D A B 8 9]
[B C D A 12 13]
[A B C D 1 3]
[D A B C 5 5]
[C D A B 9 9]
[B C D A 13 13]
[A B C D 2 3]
[D A B C 6 5]
[C D A B 10 9]
[B C D A 14 13]
[A B C D 3 3]
[D A B C 7 5]
[C D A B 11 9]
[B C D A 15 13]
[Round 3]
Let [A B C D i s] denote the operation
A = (A + h(B,C,D) + X[i] + 6ED9EBA1) <<< s .
(The value 6E..A1 is a hexadecimal 32-bit
constant, written with the high-order digit
first. This constant represents the square
root of 3. The octal value of this constant
is 015666365641. See Knuth, The Art of
Programming, Volume 2 (Seminumerical
Algorithms), Second Edition (1981),
Addison-Wesley. Table 2, page 660.)
Do the following 16 operations:
[A B C D 0 3]
[D A B C 8 9]
[C D A B 4 11]
[B C D A 12 15]
[A B C D 2 3]
[D A B C 10 9]
[C D A B 6 11]
[B C D A 14 15]
[A B C D 1 3]
[D A B C 9 9]
[C D A B 5 11]
[B C D A 13 15]
[A B C D 3 3]
[D A B C 11 9]
[C D A B 7 11]
[B C D A 15 15]
Then perform the following additions:
A = A + AA
B = B + BB
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RFC 1186 MD4 Message Digest Algorithm October 1990
C = C + CC
D = D + DD
(That is, each of the four registers is incremented by
the value it had before this block was started.)
end /* of loop on i */
Step 5. Output
The message digest produced as output is A,B,C,D. That is, we
begin with the low-order byte of A, and end with the high-order
byte of D.
This completes the description of MD4. A reference
implementation in C is given in the Appendix.
4. Extensions
If more than 128 bits of output are required, then the following
procedure is recommended to obtain a 256-bit output. (There is no
provision made for obtaining more than 256 bits.)
Two copies of MD4 are run in parallel over the input. The first copy
is standard as described above. The second copy is modified as
follows.
The initial state of the second copy is:
word A: 00 11 22 33
word B: 44 55 66 77
word C: 88 99 aa bb
word D: cc dd ee ff
The magic constants in rounds 2 and 3 for the second copy of MD4 are
changed from sqrt(2) and sqrt(3) to cuberoot(2) and cuberoot(3):
Octal Hex
Round 2 constant 012050505746 50a28be6
Round 3 constant 013423350444 5c4dd124
Finally, after every 16-word block is processed (including the last
block), the values of the A registers in the two copies are
exchanged.
The final message digest is obtaining by appending the result of the
second copy of MD4 to the end of the result of the first copy of MD4.
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