DES Has No Per Round Linear Factors

Interest in the cryptanalysis of the National Bureau of Standards' Data Encryption Standard (DES) has been strong since its announcement. Here we describe an attack on a class of ciphers like DES based on linear factors.If DES had any non trivial factors, these factors would provide an easier attack than one based on complete enumeration. Basically, a factor of order n reduces the cost of a solution from 256 to 2n+256-n At worst (n-1 or 55), this reduces the cost of a Diffie-Hellman search machine from 20 million dollars to 10 million dollars: a 10 million dollar savings. At best (n-28), even without iteration, the method could reduce the cost from 256 to 228+ 228: a computation well within the reach of a personal computer.Alas, DES has no such linear factors.