A generalized methodology for designing non-linear elements in symmetric cryptographic primitives

Symmetric ciphers are widely used for a variety of security purposes. These cryptographic primitives are ubiquitous. The primary source of non-linearity in any symmetric cryptographic primitive is the substitution box, or s-box. While there have been many proposals for varying ways to construct s-box's, this paper proposes a generalized methodology that can be used with any symmetric cipher, either in the creation of new s-boxes, during cipher design, the evaluation of the efficacy of existing s-boxes, or in the modification of s-boxes for existing ciphers. The current paper describes a generalized methodology for designing and testing s-boxes for symmetric ciphers. This methodology includes the application of three well-established tests that should be utilized for designing or testing every s-box. The study also indicates that at least some mathematical methods for designing s-boxes, while secure, are no more secure than non-mathematical methods, but are more computationally intensive.