Chapter 3 Primes, primitive roots and sequences

Abstract

Publisher SummaryThis chapter discusses primes, primitive roots, and sequences. The study presented in the chapter discusses searching for those pairs (N, GF(q)) such that every sequence of period N over GF(q) has both large linear and sphere complexity when the Hamming weight of one period of the sequence is neither too large nor too small. Such pairs (N, GF(q)) are called good partner pairs, because they work in harmony. This chapter discusses about cyclotomic polynomials. Cyclotomic polynomials have close relations with coding theory. The linear complexity and period of sequences as well as their stability are also closely related to cyclotomic polynomials. This chapter discusses only the linear and sphere complexity aspect of sequences. Other cryptography-related topics of number theory discussed in this chapter are Euler's function, Carmichael function, primitive roots, least primitive roots, common primitive roots, Artin's conjectures, Fermat's last theorem, order, Wieferich and non-Wieferich primes, Stern primes, Sophie Germain primes, o-primes, e-primes, Tchebychef primes, and primes of other forms as well as the Chinese Remainder Theorem.