Permutation Polynomials With Low Boomerang Uniformity

In block ciphers,Substitution-box occupies a very important position.With the endless variety of cryptanalytic attacks,the security of Substitution-box faces great challenges.In order to measure its resistance to attacks,many security indicators（such as nonlinearity,differential uniformity,etc.）are defined.Boomerang attack is a stronger attack found on the basis of differential attack.In recent years,with the proposal of its security indicator,boomerang uniformity,many scholars have devoted themselves to finding permutation polynomials with low boomerang uniformity.Most of these studies start from the existing4-differential uniformity permutation polynomials and the 4-differential uniformity permutation polynomials derived from the inverse function,and then search for permutation polynomials with low boomerang uniformity.At present,the research on the boomerang uniformity is facing the difficulty of tedious solution and the scarcity of the permutation polynomials with low boomerang uniformity.This thesis conducts further research on the boomerang uniformity from these two aspects,and mainly obtained the following results:1.This thesis simplified the formula proposed by Kangquan Li for the boomerang uniformity,and then obtained a new formula.The new formula embodies the relationship between differential uniformity and boomerang uniformity,and through this formula,the relationship between the boomerang connectivity table and the difference distribution table of the commonly used 2 and 4 boomerang uniformity permutation polynomials is further analyzed.In order to verify the simplification of the new formula for solving the boomerang uniformity,this thesis used Kangquan Li’s formula and the new formula to solve the boomerang uniformity of the same class of permutation polynomials.Classify the situation,and more directly locate the point where the uniformity of the boomerang may be the largest for further analysis.During this period,this thesis also proved that（x^d+x ³+x）(x+x^2k)^2n-1+x ³+xis a class of permutation polynomials with the boomerang uniformity not greater than 10.2.This thesis analyzed the boomerang uniformity of a class of permutation polynomials （?）mwith a differential uniformity of 4 modified from the inverse function by applying Kangquan Li’s formula.The effect of the value of γ on the boomerang uniformity is discussed during the analysis process.According to the results of the discussion,the boomerang uniformity of the permutation polynomials is reduced by 8by appropriately constraining the value of γ.