On the complexity of specifying a symmetric group of permutations of degree 2$^{n}$ in a threshold basis on a promising element base

The appeal to the threshold method of setting substitutions reflects the current trends towards increasing the speed of information processing and transmission connected with the possibility of implementing threshold functions directly in the signal carrier medium, primarily in optics or on other carriers related to the field of nanotechnology. In addition, the actively developing direction of building neurocomputers also requires the development of information protection systems using the basic operations of neurocomputers-threshold elements. The aim of the study was to find a way to construct a symmetric group of substitutions of degree 2$^{n}$ in the threshold basis. For this purpose, a method for implementing transpositions is proposed, with the help of which any transposition can be constructed, which allows us to say that it is possible to implement the entire symmetric group of substitutions of degree 2$^{n}$. From a computational point of view, the provisions of the article are of exceptional interest due to the simplicity of the algorithm for implementing substitutions.