Generalized Cyclotomic Numbers And Cyclotomic Sequences

Autocorrelation values of cyclotomic sequences are very important cryptography indexs.Cyclotomic sequences with a few correlation values have wide applications in communicationsystems and cryptography. Cyclotomic numbers are important Mathematical tools to calculateproperties of cyclotomic sequences.Firstly, we assume thatp₁,p₂,…,p_t are distinct primes and gcd（p_i－1,p_j－1）＝2ifi≠j. We generalize the Whiteman’s generalized cyclotomic numbers from2primes to t primes.Also we will give the recurrence formulas of Whiteman’s generalized cyclotomic numbers withrespect to p₁p₂…p_t. In especial, we obtain the Whiteman’s generalized cyclotomic numbers withrespect top1p2p3.Secondly, we construct a nearly equidistance cyclotomic sequence of order six over GF（3）, anduse cyclotomic number to calculate its autocorrelation values. Moreover, we find conditions of psuch that its autocorrelation values have three-level.Thirdly, let p, qbe distinct primes with gcd（p－1,q－1）＝2and p＋2≡q≡3（mod4）. Inthis paper, we calculate the autocorrelation values of a balanced and generalized cyclotomic binarysequence with respect to p and q.