Symmetric Boolean functions and their extension to finite fields
CRYPTOGRAPHY is the study of secret writing. A CIPHER is a way of hiding ordinary text, called PLAINTEXT, by transforming it into CIPHERTEXT. BLOCK and STREAM ciphers are widely used now. In order to encrypt the plaintext, a "random" sequence is needed. An important method to produce such sequences is the use of Boolean functions. Also, Boolean functions are widely used in various cryptosystems. Hence, there is a need for Boolean functions with "good" properties to defend against various attacks. There are many measures to judge how good a Boolean function is. This research area is very active since thousands of papers have been published. In chapter 1, we describe a method to find k -th order symmetric SAC functions ( SSAC(k) ). In this chapter, we determined all the SSAC(k) n -variable functions for n ≤ 30, k = 1, 2, ..., n - 2. In chapter 2, we prove that there are exactly 4 n -variable symmetric PC(k) functions for k = 2, 3, ..., 2[ n /2]. In chapter 3, we extend the concept SAC to finite fields GF(p) . A necessary and sufficient condition is given by using spectral analysis. Also, based on an interesting permutation polynomial theorem, we prove various facts about ( n - 1)-th order SAC functions on GF(p) . We also construct many such functions. In chapter 4, it is shown that nonlinear symmetric functions over finite fields GF(p) have no linear structures other than equal component vectors. In chapter 5, when n is not divided by p , we give a lower bound of the number of n -variable balanced symmetric polynomials over finite fields GF(p) . The existence of nonlinear balanced symmetric polynomials is an immediate corollary of this bound. For the convenience of reading, each chapter is written independently from the others. Actually, each chapter is based on a joint paper with T. W. Cusick.