Probabilistic Analysis Methods of S-Boxes and Their Applications

The relations between cryptographic properties of S-boxand its component functions are researched in this paper. First, theconcepts of independence and probability product function ofS-box are introduced, the inversion formula and its fast transformationof output probability between an S-box and the linear combination of itscomponent functions are presented, which are very useful tools inanalyzing cryptographic properties of S-boxes. Then, the necessary andsufficient conditions of S-boxes with balancedness or correlationimmunity are investigated. Finally, the necessary and sufficientconditions of independence of S-boxes are given and proved.