# Cryptanalysis of Carl Ellison's des|tran|des|tran|des

Carl Ellison proposes a large-block variant of triple-DES based on an unkeyed reversible mixing function, tran, which takes an 8192 byte (65536-bit) input block, seeds a PRNG with a histogram of the bytes in the block, and shuffles the bytes based on the output of the PRNG using a series of swaps. Because the histogram is unchanged by the shuffle, the state of the PRNG can be reconstructed and the shuffle reversed.

This is then used to construct a form of inner chaining for large-block triple-DES: three ECB-mode DES encryptions of the 65536-bit block with three different keys are interleaved with two unkeyed TRAN stages. This gives a 168-bit keyspace; we refer to Layers 1 - 5, where odd numbered layers are keyed DES transformations, and even numbered are unkeyed tran mixing layers. David Wagner proposed an attack on this scheme, but it turns out that this attacks a variant on the scheme based on a bug in an implementation of tran.

However, an attack requiring around 2^71 trial DES encryptions is possible. Consider a plaintext chosen such that the input to each of the DES blocks in the Layer 1 is the same. Since all the outputs are the same, at most eight distinct bytes are available for Layer 2 tran to shuffle, and so there are only 8^8 possible inputs to each DES block in Layer 3. Since there are 2^10 actual inputs, the probability that two will collide is roughly 2^-5; this will result in two identical 64-bit blocks in the output of Layer 3, an event that would otherwise occur with probability around 2^-45.

The attack proceeds as follows. First, generate 128 blocks of plaintext-ciphertext pairs matching the special property above. With probability about 0.91, at least two of these blocks will have collisions in the output of Layer 3, though of course we don't yet know which two. We find out as follows: take the first of the blocks and make a guess at the key for Layer 5; decrypt Layer 5 of the ciphertext and then the unkeyed Layer 4, and look for coincidences in the DES blocks of the resulting block. A key which suggests such a collision we call a "collision candidate".

Most collision candidates (all but one in 2^16) will be false positives. We can test whether a candidate is genuine by using it to decrypt all the other ciphertexts; if it is the genuine key then it's likely to suggest a collision in another block as well. Most likely, after exhaustive search of the keyspace for the Layer 5 key, the first block will turn out not to contain a collision that suggests the genuine key; we discard it and move onto the next one.

Expressed in a C-like pseudocode:

```    int candidate(key, i) {
return contains_coincidence(untran(decrypt(key, ctext[i])));
}
for (i = 0; i < 127; i++) {
for (key = 0; key < 2^56; key++) {
if (candidate(key, i)) {
/* This block is hit on average 2^11 times per i
* Try some other blocks to check for false positives */
for (j = i + 1; j < 128; j++) {
if (candidate(key, j)) {
/* Found it */
return key;
}
}
/* Just another false positive, try again */
}
}
}

```

On average we will discard around 2^5 pairs in this way before finding one with a collision that suggests the genuine key to find one containing a collision; each test involves 2^56 * 2^10 trial decryptions, so the overall cost of finding the Layer 3 key is around 2^71 trials. Discarding false positives takes 2^16 * 128 * 2^10 = 2^33 extra work.

This cost dwarfs the cost of finding the other two keys once this is done. We exhaustively search for the Layer 1 key by looking for a key which generates the Layer 3 input collisions corresponding to our output collision; these trials are much cheaper since every plaintext input block is the same, so the expected cost is 2^55 trials. Then a further 2^55 trials will find the Layer 3 key by normal means.

This certainly lends weight to the general conclusion that inner chaining usually leads to a weaker cipher, but this conclusion isn't as useful as it could be in a world that needs but doesn't have widely trusted large block ciphers. The significance I read from this attack is that large block ciphers tend to be vulnerable to plaintexts with special structures, and so any secure large block cipher will almost certainly have good whitening of both plaintext and ciphertext despite the expense, which is something the Mercy paper anticipated (see Section 5.7: Whitening of the HTML version).

Thanks to both Carl Ellison and David Wagner for discussion and clarification of this attack.