A new criterion for nonlinearity of block ciphers

For years, the cryptographic community has searched for good nonlinear functions. Bent functions, almost perfect nonlinear functions, and similar constructions have been suggested as a good base for cryptographic applications due to their highly nonlinear nature. In the first part of this paper we study these functions as block ciphers, and present several distinguishers between almost perfect nonlinear permutations and random permutations. The data complexity of the best distinguisher is O(2^n/3) and its time complexity is O(2^2n/3) for an n-bit block size, independent of the key size. In the second part of the paper we suggest a criterion to measure the effective linearity of a given block cipher. We devise a distinguisher for general block ciphers based on their effective linearity. Finally, we show that for several constructions, our distinguishing attack is better than previously known techniques.