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.