From: Andrew Roos <AndrewR@beetle.vironix.co.za>
Date: Fri, 22 Sep 95 03:01:07 PDT
To: "'rivest@theory.lcs.mit.edu>
Subject: Weak Keys in RC4
Message-ID: <3062899D@beetle.vironix.co.za>
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                  A CLASS OF WEAK KEYS IN THE RC4 STREAM CIPHER

                                PRELIMINARY DRAFT

                                   ANDREW ROOS
                          VIRONIX SOFTWARE LABORATORIES


1.  INTRODUCTION

This paper discusses a class of weak keys in RSA's RC4 stream cipher. It   
shows
that for at least 1 out of every 256 possible keys the initial byte of   
the
pseudo-random stream generated by RC4 is strongly correlated with only a   
few
bytes of the key, which effecitively reduces the work required to   
exhaustively
search RC4 key spaces.


2.  STATE TABLE INITIALIZATION IN RC4

Although the RC4 algorithm has not been published by RSA Data Security,   
source
code to implement the algorithm was anonymously posted to the Cypherpunks   

mailing list several months ago. The success of the Cypherpunks'   
brute-force
attack on SSL with a 40-bit key indicates that the source code published   
did
accurately implement RC4.

RC4 uses a variable length key from 1 to 256 bytes to initialize a   
256-byte
state table which is used for the subsequent generation of pseudo-random   
bytes.
The state table is first initialized to the sequence {0,1,2,...,255}.   
Then:

1    index1 = 0;
2    index2 = 0;
3
4    for(counter = 0; counter < 256; counter++)
5    {
6        index2 = (key_data_ptr[index1] + state[counter] + index2) % 256;   
                 

7        swap_byte(&state[counter], &state[index2]);
8        index1 = (index1 + 1) % key_data_len;
9    }

Note that the only line which directly affects the state table is line 7,   
when
two bytes in the table are exchanged. The first byte is indexed by   
"counter",
which is incremented for each iteration of the loop. The second byte is
indexed by "index2" which is a function of the key. Hence each element of   
the
state table will be swapped at least once (although possibly with   
itself),
when it is indexed by "counter". It may also be swapped zero, one or more   

times when it is indexed by "index2". If we assume for the moment that
"index2" is a uniformly distributed pseudo-random number, then the   
probability
that a particular single element of the state table will be indexed by
"index2" at some time during the initialization routine is:

     P = 1 - (255/256) ^ 255
       = 0.631

(The exponent is 255 because we can disregard the case when "index2" and
"counter" both index the same element, since this will not affect its   
value.)

Conversely, there is a 37% probability that a particular element will   
_not_ be
indexed by "index2" during initialization, so its final value in the   
state
table will only be affected by a single swap, when it is indexed by   
"counter".
Since key bytes are used sequentially (starting again at the beginning   
when the
key is exhausted), this implies:

A.   Given a key length of K bytes, and E < K, there is a 37% probability   
that
     element E of the state table depends only on elements 0..E   
(inclusive) of
     the key.

(This is approximate since "index2" is unlikely to be uniformly   
distributed.)

In order to make use of this, we need to determine the most likely values   
for
elements of the state table. Since each element is swapped at least once   
(when
it is indexed by "counter"), it is necessary to take into account the   
likely
effect of this swap. Swapping is a nasty non-linear process which is hard   
to
analyze. However, when dealing with the first few elements of the state   
table,
there is a high probability that the byte with which the element is   
swapped
has not itself been involved in any previous exchanges, and therefore   
retains
its initial value {0,1,2,...,255}. Similarly, when dealing with the first   
few
elements of the state table, there is also a significant probability that   
none
of the state elements added to index2 in line 6 of the algorithm has been   

swapped either.

This means that the most likely value of an element in the state table   
can be
estimated by assuming that state[x] == x in the algorithm above. In this   
case,
the algorithm becomes:

1    index1 = 0;
2    index2 = 0;
3
4    for(counter = 0; counter < 256; counter++)
5    {
6        index2 = (key_data_ptr[index1] + counter + index2) % 256;   
                 

7        state[counter] = index2;
8        index1 = (index1 + 1) % key_data_len;
9    }

Which can be reduced to:

B.   The most likely value for element E of the state table is:

     S[E] = X(E) + E(E+1)/2
     where X(E) is the sum of bytes 0..E (inclusive) of the key.

(when calculating the sum of key elements, the key is considered to "wrap   

around" on itself).

Given this analysis, we can calculate the probability for each element of   
the
state table that it's value is the "most likely value" of B above. The   
easiest
way to do this is to evaluate the state tables produced from a number of
pseudo-randomly generated RC4 keys. The following table shows the results   
for
the first 47 elements from a trial of 100 000 eighty-bit RC4 keys:

          Probability (%)

0-7       37.0  36.8  36.2  35.8  34.9  34.0  33.0  32.2
8-15      30.9  29.8  28.5  27.5  26.0  24.5  22.9  21.6
16-23     20.3  18.9  17.3  16.1  14.7  13.5  12.4  11.2
24-31     10.1   9.0   8.2   7.4   6.4   5.7   5.1   4.4
32-39      3.9   3.5   3.0   2.6   2.3   2.0   1.7   1.4
40-47      1.3   1.2   1.0   0.9   0.8   0.7   0.6   0.6

The table confirms that there is a significant correlation between the   
first
few values in the state table and the "likely value" predicted by B.


3.  WEAK KEYS

The RC4 state table is used to generate a pseudo-random stream which is   
XORed
with the plaintext to give the ciphertext. The algorithm used to generate   
the
stream is as follows:

     x and y are initialized to 0.

     To generate each byte:

1    x = (x + 1) % 256;
2    y = (state[x] + y) % 256;
3    swap_byte(&state[x], &state[y]);   
                                      

4    xorIndex = (state[x] + state[y]) % 256;   
                                 

5    GeneratedByte = state[xorIndex];

One way to exploit our analysis of the state table is to find   
circumstances
under which one or more generated bytes are strongly correlated with a   
small
subset of the key bytes.

Consider what happens when generating the first byte if state[1] == 1.

1    x = (0 + 1) % 256;                  /* x == 1 */
2    y = (state[1] + 0) % 256;           /* y == 1 */
3    swap_byte(&state[1], &state[1]);    /* no effect */
4    xorIndex = (state[1] + state[1]);   /* xorIndex = 2 */
5    GeneratedByte = state[2]

And we know that state[2] is has a high probability of being

     S[2] = K[0] + K[1] + K[2] + 2 (2+1) / 2

Similarly,

     S[1] = K[0] + K[1] + 1 (1+1) / 2

So to make it probable that S[1] == 1, we have:

     K[0] + K[1] == 0 (mod 256)

In which case the most likely value for S[2] is:

     S[2] = K[2] + 3

This allows us to identify a class of weak keys:

C.   Given an RC4 key K[0]..K[N] with K[0] + K[1] == 0 (mod 256), there   
is a
     significant probability that the first byte generated by RC4 will be   

     K[2] + 3 (mod 256).

Note that there are two special cases, caused by "unexpected" swapping   
during
key generation. When K[0]==1, the "expected" output byte is k[2] + 2, and   
when
k[0]==2, the expected value is k[2] + 1.

There are a number of similar classes of "weak keys" which only affect a   
few
keys out of every 65536. However the particular symmetry in this class   
means
that it affects one key in 256, making it the most interesting instance.

Once again I took the easy way out and used simulation to determine the
approximate probability that result C holds for any given key.   
Probabilities
ranged between 12% and 16% depending on the values of K[0] and K[1], with   
a
mean of about 13.8%. All these figures are significantly greater than the   

0.39% which would be expected from an uncorrelated generator. The key   
length
used was again 80 bits. This works the other way around as well: given   
the
first byte B[0] generated by a weak key, the probability that   
K[2]==B[0]-3
(mod 256) is 13.8%.

       

4.  EXPLOITING WEAK KEYS IN RC4

Having found a class of weak keys, we need a practical way to attack RC4   
based
cryptosystems using them. The most obvious way would be to search   
potential
weak keys first during an exhaustive attack. However since only one in   
every
256 keys is weak, the effective reduction in search space is not   
particularly
significant.

The usefulness of weak keys does increase if the opponent is satisfied   
with
recovering only a percentage of the keys subjected to analysis. Given a   
known
generator output which includes the first generated byte, one could   
assume
that the key was weak and search only the weak keys which would generate   
the
known initial byte. Since 1 in 256 keys is weak, and there is a 13.8%   
chance
that the assumed value of K[2] will be correct, there is only a 0.054%   
chance
of finding the key this way. However, you have reduced the search space   
by 16
bits due to the assumed relationship between K[0] and K[1] and the   
assumed
value of K[2], so the work factor per key recovered is reduced by a   
factor of
35, which is equivalent reducing the effective key length by 5.1 bits.

However in particular circumstances, the known relationships between weak   
keys
may provide a much more significant reduction in workload. The remainder   
of
this section describes an attack which, although requiring very specific
conditions, illustrates the potential threat.

As a stream cipher, a particular RC4 key can only be used once. When   
multiple
communications sessions are required, some mechanism must be provided for   

generating a new session key each time. Let us suppose that an   
implementation
chose the simple method of incrementing the previous session key to get   
the
new session key, and that the session key was treated as a "little   
endian"
(least significant byte first) integer for this purpose.

We now have the interesting situation that the session keys will "cycle
through" weak keys in a pattern which repeats every 2^16 keys:

00 00 00 ...    Weak
(510 non-weak keys)
FF 01 00 ...    Weak
(254 non-weak keys)
FE 02 00 ...    Weak
(254 non-weak keys)
FD 03 00 ...    Weak
...
01 FF 00 ...    Weak
(254 non-weak keys)
00 00 01 ...    Weak
(510 non-weak keys)
FF 01 01 ...    Weak

(Least significant byte on the left)

Now while an isolated weak key cannot be identified simply from a known
generator output, this cycle of weak keys at known intervals can be   
identified
using statistical techniques since each of the weak keys has a higher   
than
expected probability of generating the _same_ initial byte. This means   
that an
opponent who knew the initial generated bytes of about 2^16 session keys   
could
identify the weak keys, and would also be able to locate the 510-key gap
between successive cycles of weak keys (although not precisely). Since   
the
510-key gap occurs immediately following a key which begins with 00 00,   
the
opponent not only knows that the keys are weak, but also knows the first   
two
bytes of each key. The third byte of each key can be guessed from the   
first
output byte generated by the key, with a 13.8% chance of a correct guess.   

Assuming that the "510-key gap" is narrowed down to 1 of 8 weak keys, the   

attacker can search a key space which is 24 bits less than the size of   
the
session keys, with a 13.8%/8 chance of success, effectively reducing the   
key
space by approximately 18 bits.

Although this particular attack depends on a very specific set of
circumstances, it is likely that other RC4 based cryptosystems in which   
there
are linear relationships between successive session keys could be   
vulnerable
to similar attacks.


5.  RECOMMENDATIONS

The attacks described in this algorithm result from inadequate "mixing"   
of key
bytes during the generation of the RC4 state table. The following   
measures
could be taken to strengthen cryptosystems based on the RC4 algorithm:

(a) After initializing the algorithm, generate and discard a number of   
bytes.
    Since the algorithm used to generate bytes also introduces additional   

    non-linear dependencies into the state table, this would make   
analysis
    more difficult.

(b) In systems which require multiple session keys, ensure that session   
keys
    are not linearly related to each other.

(c) Avoid using the weak keys described.


6.  CONCLUSION

This preliminary analysis of RC4 shows that the algorithm is vulnerable   
to
analytic attacks based on statistical analysis of its state table. It is
likely that a more detailed analysis of the algorithm will reveal more
effective ways to exploit the weaknesses described.



Andrew Roos <andrewr@vironix.co.za>

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