Meet-in-the-middle attack

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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 range and domain 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. Naively, one might think that this would double the security of the double-encryption scheme. Certainly, an exhaustive search of all possible combination of keys would take <math>2^{2n}</math> attempts if each key is n bits long, compared to the <math>2^n</math> attempts required for a single key.

Diffie and Hellman, however, devised a time-memory tradeoff that could break the scheme in only double the time to break the single-encryption schemeTemplate:Ref. 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,

<math>

 C=E_{K_2}(E_{K_1}(P))
 </math>,

where K₁ and K₂ are the two keys.

The attacker can then compute E_K(P) for all possible keys K and store the results in memory. Afterwards he can compute D_K(C) for each K and compare with the table in memory. If he gets a match it is likely that he has discovered the two keys and he can verify it with a second set of plaintext and ciphertext. If the keysize is n, this attack uses only <math>2^{n+1}</math> encryptions in contrast to the naive attack, which needs <math>2^{2n}</math> encryptions.

See also

• Triple DES
• Birthday attack
• Man in the middle attack

References