Security amplification by composition: The case of doubly-iterated,   ideal ciphers
  We investigate, in the Shannon model, the security of constructions corresponding to double and (two-key) triple DES. That is, we consider F_{k1}(F_{k2}(.)) and F_{k1}(F_{k2}^{-1}(F_{k1}(.))) with the component functions being ideal ciphers. This models the resistance of these constructions to ``generic'' attacks like meet in the middle attacks. We obtain the first proof that composition actually increases the security of these constructions in some meaningful sense. We compute a bound on the probability of breaking the double cipher as a function of the number of computations of the base cipher made, and the number of examples of the composed cipher seen, and show that the success probability is the square of that for a single key cipher. The same bound holds for the two-key triple cipher. The first bound is tight and shows that meet in the middle is the best possible generic attack against the double cipher. 
William Aiello, Mihir Bellare, Giovanni Di Crescenzo and Ramarathnam   Venkatesan