Toward a general correlation theorem

In 2001, Nyberg proved three important correlation theorems and applied them to several cryptanalytic contexts. We continue the work of Nyberg in a more theoretical direction. We consider a general functional form and obtain its Walsh transform. Two of Nyberg's correlation theorems are seen to be special cases of our general functional form. S-box lookup, addition modulo 2/sup 2k/, and X-OR are three frequently occurring operations in the design of symmetric ciphers. We consider two methods of combining these operations and in each apply our main result to obtain the Walsh transform.