Research of boomerang uniformity of quadratic permutations

The boomerang attack, proposed in 1999, is a variation of the difference attack. Its advantage is that even in the presence of low differential uniformity the cipher can still be vulnerable to boomerang attack. This paper is devoted to such a parameter of a vector Boolean function as boomerang uniformity, which characterizes the function's resistance to the boomerang attack. Quadratic permutations are considered and the dependence of the boomerang characteristic on the differential characteristic for this class has been studied. The main result is an expression connecting the boomerang uniformity of a function with the values of its DDT table and obtained using the matrix approach for working with quadratic functions, as well as the known properties of differential and boomerang characteristics In addition, for the boomerang characteristic, some constructions of quadratic substitutions in a small number of variables have been studied and other properties have been established.