An analysis of bent function properties using the transeunt triangle and the SRC-6 reconfigurable computer

Abstract

Linear attacks against cryptosystems can be defeated when combiner functions are composed of highly nonlinear Boolean functions. The highest nonlinearity Boolean functions, or bent functions, are not common- especially when they have many variables-bent functions are difficult to find. Understanding what properties are common to bent functions will help ease the search for them. Using the SRC-6 reconfigurable computer, functions can be generated or tested at a rate much higher than a PC. This thesis uses the SRC-6 to characterize data for functions with 4, 5 and 6 variables. The data compiled showed trends based on the order, homogeneity, balance, and symmetry of Boolean functions. The transeunt triangle is used to convert a Boolean function into Algebraic Normal Form, so that the properties are easily determined. The first known proof that the transeunt triangle correctly converts between the two Boolean functions' representations is included. The SRC-6, while capable of pipelining code so that it runs up to six thousand times faster than a PC, is limited by the speed of the FPGA, 100 MHz. Functions with up to six variables were tested. Predictions on this data, as well as ways to improve the computing capability of the SRC-6, are included.