Perfect nonlinear S-boxes on the real-line

The objective of this contribution is to introduce an analogue to the classical secret-key block ciphers, such as DES, IDEA or AES, in the nondenumerable setting, namely where cleartexts, plaintexts and keys are real numbers. The nonlinear part of traditional secret-key block ciphers, the S-boxes, is designed to provide confusion i.e. to resist to several kind of cryptanalysis such as algebraic, differential or linear attacks. By analogy we construct S-boxes in the uncountable setting which provide the best resistance to a classical or modified version of the differential attack. Since our S-boxes are real-valued functions defined on the real-line, we also need to prevent possible new attacks based on real analysis (such as continuity and derivability), which are ignored since impossible in the finite case: we must hide the topological structure. So we introduce a new kind of Discontinuous-boxes for this purpose.