Constructions Of Two Classes Of Cryptographic Functions With Low Differential Uniformity

In the design of block ciphers,S-boxes are the most important computing elements,which are the heart of block ciphers.In order to resist all kinds of common attacks(such as differential attacks,linear attacks and etc.),the designed S-boxes should have good cryp-tographic properties,which are of great importance.When designing block ciphers,if an S-box with both low differential uniformity and high nonlinearity is used,the designed block cipher can resist not only linear attack but also differential attack.In this thesis,several cryp-tographic functions with low differential uniformity and high nonlinearity are studied.The following results are obtained:1)A method for constructing functions with low differential uniformity and high non-linearity is given.Two new classes of cryptographic functions with low differential unifor-mity are obtained by changing the values of the monomial APN functions(i.e.APN power function)-the Niho function x22k+2k-1,X22k+1+23k+2-1 and the inverse function x2n-2 on the subfield of the finite field with even characteristics.The nonlinearity of the constructed functions is proved by calculating the Walsh spectra value of the constructed functions.Fi-nally,the algebraic degree of the constructed functions is calculated,and from the point of view of the algebraic degree,the extended affine inequivalence(EA inequivalence)between the constructed functions and the existing functions is discussed.2)The above method is extended so that it can be used for the binomial cryptograph-ic functions with low differential uniformity.And a construction method of permutation polynomials with low differential uniformity and high nonlinearity is given.A new class of permutation polynomial with low differential uniformity is obtained by changing the val-ues of the binomial function x2k+1+x2k+1+1 on the subfield of the finite field with even characteristics.The nonlinearity and the algebraic degree of the constructed function are calculated.And from the point of view of the algebraic degree,the extended affine inequiv-alence(EA inequivalence)between the constructed functions and the existing functions is discussed.