This paper investigates some security properties of basic substitution-permutation encryption networks (SPNs) by studying the nonlinearity distribution and the XOR table distribution. Based on the idea that mixing small weak transformations results in a large strong cipher, we provide some evidence which shows that a basic SPN converges to a randomly generated s-box with the same dimensions as the SPN after enough number of rounds. We also present a new differential-like attack which is easy to implement and outperforms the classical differential cryptanalysis on the basic SPN structure. In particular, it is shown that 64-bit SPNs with 8 × 8 s-boxes are resistant to our attack after 12 rounds. All of above effort may be regarded as the first step towards provable security for SPN cryptosystems.