This work is motivated by the observation that in DES-like ciphexs it is possible to choose the round functions in such a way that every non-trivial one-round characteristic has small probability. This gives rise to the following definition. A mapping is called differentially uniform if for every non-mro input difference and any output difference the number of possible inputs has a uniform upper bound. The examples of differentially uniform mappings provided in this paper have also other desirable cryptographic properties: large distance from ailbe functions, high nonlinear order and efficient computability.
The results of this paper give the theoretical fundaments on which Matsui's linear cryptanalysis of the DES is based. As a result we obtain precise information on the assumptions expficitely or impficitely stated in [2] and show that the success of Algorithm 2 is underestimated in [2]. We also derive a formula for the strength of Algorithm 2 for DES-fike ciphers and see what is its dependence on the plaintext distribution. Finally, it is shown how to achieve proven resistance against linear cryptanalysis.
Linear Cryptanalysis of a DES-like CipherWe consider a DES-like iterated cipher consisting of r rounds of iterationat the rounds i = 1,2,...,r-1, and
eL = XL(r) -]-f(E(XR(r)) + Kr) cR = xR(r)
Here we have denoted by Ki the round key used at the ith round and by X(i) ----(XL(i), XR(i)) the input to the ith round with its left and right halves. Hence X(1) = P = (PL, PR) is the plaintext and C = (CL, CR) is the ciphertext.In [2] M. Matsui introduces the linear cryptanalysis method to recover with high probability certain key bits using sufficient large number of known plaintextciphertext pairs. The main part of this attack is a procedure called Algorithm 2 which can be used to recover 12 bits of a DES-key. Let us give a short description of this procedure. First the round function is analyzed to find linear approximations of the function f of the form
b(i). f(Z) = ci " Z
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