Symmetric Boolean Function Algebra Immunodeficiency

To resist algebraic attacks, Boolean functions should have high algebraic immunity. Symmetric Boolean functions are an important class of Boolean functions. And each symmetric function can be written into a linear combination of homogeneous symmetric functions. In chapter 1, we introduce the basic concepts about Boolean functions, and get some easy generizations of some properties. In chapter 2, from the definition of Boolean function and the properties of zero factors, we get some results on the algebraic immunity of symmetric Boolean functions, some examples and corollaries are given. Under affine transformations, a symmetric Boolean function need not map into another, in chapter 3, we study the affine transformations under which symmetric Boolean functions preserve the symmetric character, give two theorems, and construct some examples. On the study of the algebraic immunity of Boolean functions, we have great interests on the problem to get maximum algebraic immunity, in chapter 4, on the odd and even conditions, we construct a series of Boolean functions having maximum algebraic immunity.