Meet-in-the-middle attack

The Meet-in-the-middle attack is a cryptographic attack which, like the birthday attack, makes use of a space-time tradeoff. While the birthday attack attempts to find two values in the domain of a function that map to the same value in its range, the meet-in-the-middle attack attempts to find a value in each of the ranges and domains of the composition of two functions such that the forward mapping of one through the first function is the same as the inverse image of the other through the second function—quite literally meeting in the middle of the composed function.
It was first developed as an attack on an attempted expansion of a block cipher by Diffie and Hellman in 1977. When trying to improve the security of a block cipher, one might get the idea to simply use two independent keys to encrypt the data twice and think that this would square the security of the double-encryption scheme. Certainly, an exhaustive search of all possible combination of keys would take formula_1 attempts if each key is n bits long, compared to the formula_2 attempts required for a single key. However, Diffie and Hellman, discovered a time-memory tradeoff that could break the scheme in only double the time to break the single-encryption scheme. The attack works by encrypting from one end and decrypting from the other end, thus meeting in the middle.
Assume the attacker knows a set of plaintext and ciphertext: "P" and "C". That is,
where E is the encryption function (cipher), and "K"1 and "K"2 are the two keys.
The attacker can then compute "EK"("P") for all possible keys "K" and store the results in memory. Afterwards he can decrypt the ciphertext by computing "DK"("C") for each "K". Any matches between these two resulting sets are likely to reveal the correct keys. (To speed up the comparison, the "EK"("P") set is stored in an in-memory lookup table, then each "DK"("C") can be matched against the values in the lookup table to find the candidate keys.)
Once the matches are discovered, they can be verified with a second test-set of plaintext and ciphertext. If the keysize is "n", this attack uses only formula_4 encryptions (and formula_5 space) in contrast to the naive attack, which needs formula_1 encryptions (but only formula_7 space).