Cryptanalysis of Multiswap

Nikita Borisov, Monica Chew, Rob Johnson, and David Wagner
UC Berkeley

An anonymous security researcher working under the pseudonym "Beale Screamer" reverse engineered the Microsoft Digital Rights Management subsystem and, by October 20th, the results were available on cryptome.org.  As part of the reverse engineering effort Screamer found an unpublished block cipher, which he dubbed MultiSwap, being used as part of DRM.  Screamer did not need to break the MultiSwap cipher to break DRM, but we thought it would be a fun excercise, and summarize the results of our investigation below.  The attacks described here show weaknesses in the MultiSwap encryption scheme, and could potentially contribute to an attack on DRM.  However, the attack on DRM described by Beale Screamer would be much more practical, so we feel that these weaknesses in MultiSwap do not pose a significant threat to DRM at this time.

We present these results to further the science of computer security, not to promote rampant copying of copyrighted music.

The cipher

The Multswap algorithm takes a 64-bit block consisting of two 32-bit numbers x0 and x1 and encrypts them using the subkeys k0,...,k11 as diagramed below.

Multiswap diagram
Figure 1: The MultiSwap algorithm


All values are 32-bit, and * and + are normal multiplication and addition mod 2^32.  The Swap Symbol symbols represent swapping the two 16-bit halves of a 32-bit value.  When used in a chained mode of operation, s0 and s1 are the inputs from the previous block.  Otherwise they are 0.  Note that the diagram is split into two stages, and the output from the first half, s0' and s1', is fed into the second half.  So if s0=s1=0, then the input is x0 and x1, and the output is c0 and c1.  Observe also that k0,...,k4,k6,...,k10 must all be odd if the cipher is to be invertible.  Thus the cipher has a 2*32 + 10*31 = 374-bit key.

Recovering k5 and k11

Consider the algorithm operating on input (0,x1).  Since s0=s1=0, it's not hard to see that s0'=s1'=k5.  After the second half, regardless of the input x1, the output will satisfy c1=c0+k5.  Thus one can derive k5=c1-c0 with one chosen-plaintext message of the form (0,x1).

Given k5, k11 can be recovered with one additional message.  Observe that with input (0,-k5), it will still be the case that s0'=s1'=k5.  In the second half, though, adding x1=-k5 to s0' will yield another 0, which will propogate through the multiplications and swaps as before.  Thus the output will be c0=k11 and c1=k5+k11.

So k5 and k11 can be exposed with a 2-message adaptive chosen-plaintext attack.

Recovering the rest of the key

We first present a chosen-plaintext attack, and then describe how to convert this to a known-plaintext attack.  We will present the attack assuming that s0=s1=0, but it also works when they are non-zero but known: just use x0=-s0 instead of x0=0.

The chosen-plaintext attack makes use of the keys k5 and k11 recovered above.  With these keys, one can control the input to the second half of the encryption.  For example, to force the value multiplied by k6 to be w, simply query the encryption oracle with plaintext (0,w-k5).  As for the output, since k11 is known, given c0 one can partially decrypt to find the intermediate value computed immediately after the multiplication by k10.  This reduces the problem to the following system for which the input, w, can be controlled and the output, v, can be observed.  The goal is to recover k6,...,k10.

Second Round
Figure 2: The second round

For the rest of this section, "input" and "output" will refer to the input and output of the above cipher fragment, not the cipher as a whole.  As mentioned above, this is fine since one can perfectly control and observer these inputs and outputs.  

The attack uses differential cryptanalysis.  The differential is not an additive or xor-differential, it is a multiplicative differential.  Suppose the above fragment is given inputs w and 2w.  Then clearly this differential is preserved across multiplication by k6.  But when  is it preserved by swapping the two halves of k6*w and k6*2w?  In other words, for arbitrary z, when is
2(swap z)=swap (2z)
It's not hard to see that, for 32-bit numbers this occurs precisely when bits 15 and 31 of z are 0. (Note: the LSB of z is bit 0).  This occurs with probability 1/4.  So the probability that a pair of inputs w and 2w produce outputs y and 2y is the probability that the differential is preserved at every swapping step.  Since this happens with probabilitity 1/4 at each swap, we expect this to occur with probability (1/4)^4=1/256.

So suppose the input pair (w,2w) produces output pair (v,2v).  We call the pair (w,2w) a right pair.  Then with high probability bits 15 and 31 of k6*w are 0.  This is a two-bit condition on k6*w that one can use to filter the set of potential values of k6; 1/4 of all k6 values will pass this test.  One can repeat this test for 16 right input pairs (w1,2*w1)...(w16,2*w16) chosen uniformly at random, and the probability of a given k6 value surviving all 16 tests is roughly (1/4)^16 = 2^-32, so we expect about one value of k6 to survive.

We now show that, having determined k6, an attacker can determine k7 using very few additional queries.  Note that if (w,2w) is a right pair, then swap (k6*w) and swap (k6(2w)) form a right pair for determining k7.  Thus the right pairs used to determine k6 can be used to determine k7, too.  In the rare case that the right pairs from k6 do not completely determine k7, one may need to make a few more queries.  But since  k6 is known, an attacker can control the input to the cipher fragment starting with multiplication by k7.  This yields a differential with probability 1/64.  So we may very pessimistically estimate that k7 can be determined with an additional 1024 queries.

Once k7 has been determined, the right pairs can be applied to determing k8, and so on.  Continuing in this way, we see that k6,...,k10 can all be determined with high probability using fewer than 8192 chosen-plaintexts.  An attacker can then apply the same trick to k0,...,k4.  Thus the whole cipher can be broken with about 2^14 chosen-plaintexts.  This is surprisingly small considering the large key size.

We should now mention that the work factor of breaking the cipher is quite low, as well.  Suppose an attacker has right pairs (w1,2*w1),...,(w16,2*w16) which determine k6.  By definition of being right, bits 15 and 31 of k6*wi are 0 for all i.  These constraints can be translated into nonlinear equations on the bits of k6.  Unfortunately, the degree of the equations is as large as 31, so solving them directly is impossible. One could iterate over all possible values of k6, throwing out the ones that don't satisfy the equations, but this will require testing 2^32 keys.  Observe that bit 15 of k6*wi is independent of bits 16,...,31 of k6, though.  Thus an attacker can try all possible values for the low 16 bits of k6, checking whether they satisfy this equation.  After discovering the lower 16 bits, he can then do the same thing for the upper 16 bits.  Since we have to test each half of a key against each right pair, the total number of tests performed is 2*2^16*2^4=2^21.  Repeating for k7,...,k10, and then again for k0,...,k4 yields that the whole cipher can be broken with10*2^19~=2^25 tests.  But how expensive is each test?  Testing the lower or upper 16 bits of a key against a wi involves multiplying by wi, masking bit 15 (or 31), and testing for 0.  This is about 1/8th as expensive as the MultiSwap encryption, which requires 10 multiplies, 10 swaps, and 6 adds.  So the work factor is about (2^25)/8=2^22 encryptions.

Converting to known-plaintext attack

Recall there are two stages to the attack: recover k5 and k11, and recover the rest of the key.  The attack on k5 and k11 can be converted to a known-plaintext attack as follows.  Referring to Figures 1 and 2, observe that w=c1-c0+x1.  With probability 2^-32, this value is 0, and that situation can be detected.  When this happens, c0=k11.  So a set of 2^32 known-plaintexts should suffice to recover k11.  Similarly, s0'=c1-c0-s1.  Out of a set of 2^32 known-plaintexts, on average one plaintext should satisfy x0+s0=0, in which case s0'=k5.  Since these two events are roughly independent, an attacker should be able to recover k5 and k11 with 2^32 known-plaintexts.

One can also convert the second stage of the attack to use known-plaintexts.  We first have to see that the inputs and outputs of the two halves of the cipher can be isolated.  So suppose an attacker knows k5 and k11.  First observe that c1 = c0 + s0'.  So the input to Figure 2 can be computed as w=c1 - c0 + x1.  Since he knows k11, an attacker can also obviously compute the output, v,  of the fragment in Figure 2.  For the first half of the cipher, the input is x0 (or x0 + s0, which is known, if used in a chaining mode).  The output (immediately after multiplication by k4) is swap (c0-c1-k5).

All that's left is to figure out the number of messages one needs to capture before expecting to have 8192 pairs (w,2w) for the second round and 8192 pairs (w,2w) for the first round.  With 2^22.5 known-plaintexts, we get 2^44 pairs, and the probability that any one of these pairs is of the form (w,2w) is 1/2^31.  Hence we expect to have 2^13 such pairs.  Experiments confirm this estimate.  Thus with 2^22.5 known plaintexts, we expect that the 2^22.5 inputs to the second round will contain about 2^13 pairs, enough to recover k6,...,k10.  But these same messages yield 2^22.5 inputs to the first round, which should also contain 2^13 pairs.  Since these events are independent, one should be able to break the system with 2^22.5 known plaintexts.  Detecting the pairs in a set of known plaintexts is easy if the pairs are stored in a hash-table, so the work factor is just 2^25, as above.

A better known-plaintext attack

The known-plaintext attack described above requires 2^32 texts, which seems like a waste since those texts are only required to recover k5 and k11.  By performing both halves of the attack simultaneously, one can get by with just 2^22.5 known-plaintexts.

Recall that, even without knowledge of k5 and k11, we can derive the input to the cipher fragment in Figure 2 via w=c1-c0+x1.  If we extend our differential through the additional swap immediately preceding the addition of k11, we get a differential (w,2w) -> (v,2v) -> (v+k11,2v+k11) with probability 1/2^10.  Given such a right pair with outputs c0 and c0', we can compute k11=c0'-2*c0.

So collect 2^22.5 known-plaintexts, and collect from them 2^13 pairs whose input to Figure 2 is (w,2w).  Each such pair suggest a candidate for k11=c0'-2*c0.  The right value of k11 will be suggested once for each right pair, or 2^13/2^10=8 times.  Wrong pairs will suggest a random value for k11, and so no other value for k11 should be suggested more than once or twice.

With k11, one can use the previously described attack to recover k6,...,k10.  This attack can then be repeated for k5, and then for k0,...,k4.  The total work factor is about the same as for the previous attacks.  The storage is also quite small, since we don't have two keep a counter for every possible value of k11, only the ones suggested by a pair.  Since we use only about 2^13 pairs, the storage requirement is about 2^16 bytes.

Conclusion

We have seen that MultiSwap can be broken with a 2^14 chosen-plaintext attack or a 2^22.5 known-plaintext attack, requiring 2^25 work.  We believe this shows that MultiSwap is not safe for any use.