Construction And Analysis Of K-Rotation Symmetric Boolean Functions With Good Cryptographic Properties

Cryptographic functions consist of Boolean functions and vectorial Boolean functions.They can be used as important components for block ciphers,sequence ciphers and Hash functions.As the extension of rotation symmetric Boolean functions(RSBF),the k-rotation symmetric Boolean functions(k-RSBF)have all the excellent cryptographic properties of the rotation symmetric Boolean functions.This thesis mainly studies the constructions of k-rotation symmetric bent functions,k-rotation symmetric semi-bent functions and vectorial resilient k-rotation symmetric Boolean functions.The following results have been obtained:(1)In this thesis,several bent and semi-bent functions are presented.The first method is to construct a class of 2m variable k-rotation symmetric bent functions and two kinds of 2m variable 2k-rotation symmetric bent functions by modifying the truth table of Rothaus's bent function or its affine equivalent function,where k|m.By comparing this method with some existing methods of constructing rotation symmetric bent functions,it's proved that the latter are the special cases of the former.The algebraic normal forms of the functions generated by these constructions are given,and three examples are obtained using the above construction method.The second method is to construct two kinds of 2m variable k-rotation symmetric bent and semi-bent functions by using the relevant conclusions of the M-M functions.(2)This thesis presents the orbital distributions of vectorial rotation symmetric and krotation symmetric Boolean functions.It also gives a method to compute the number of orbits of the same size in two functions and extends the conclusion that only some special orbital counts are given on the RSBFs and k-RSBFs.It is proved that there exists balanced(n,m)k-RSBF for suitable k,where n=pr,n=2pr,n=2r,and some instances are constructed.The existence of resilient(n,m)k-RSBF is also proved,and some specific functions are constructed,including 1-resilient(2r,m)2-RSBF,where 2?m?2r-r,r?3 and 1-resilient(pr,m)p-RSBF,where 2?m?p-1,r?2,and p is an odd prime.This confirms that the(n,m)k-RSBFs have properties that the(n,m)-RSBFs do not have.Finally two examples of the 1-resilient(n,m)k-RSBFs are obtained using the two methods.