While generally impractical due to the storage requirement, the attack of Fact 7.39 isreferred to as a certificational attack on two-key triple encryption, demonstrating it to beweaker than triple encryption. This motivates consideration of triple-encryption with threeindependent keys, although a penalty is a third key to manage.Fact 7.40, stated specifically for DES (n = 64, k = 56), indicates that for the priceof additional computation, the memory requirement in Fact 7.39 may be reduced and thechosen-plaintext condition relaxed to known-plaintext. The attack, however, appears impracticaleven with extreme parallelization; for example, for lg t = 40, the number of operationsis still 280
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