Linear and differential cryptanalysis of small-sized random (n, m)-S-boxes

S-boxes are used in cryptography in order to provide non-linearity in the design of cryptographic primitives such as block ciphers and hash functions. Some cryptographic primitives use bijective S-boxes as in the Advanced Encryption Standard (AES), and others use surjective S-boxes as in the Data Encryption Standard (DES). That is, S-boxes can have inputs and outputs of the same length as in the (8,8)-S-box of AES, or alternatively the input length can be larger than the output as in the (6, 4)-S-boxes of DES. In this paper, we perform a statistical study of linear and differential properties of randomly generated (n, m)-S-boxes, where m ≤ n. We show that certain S-boxes with well-behaved linear and differential properties can be feasibly obtained via random search. We show further that certain types of S-boxes with specific desirable linear and differential properties are improbable.