The 4-Adic Complexity of A Class of Quaternary Cyclotomic Sequences with Period 2p

In cryptography, we hope a sequence over ${\mathbb{Z}}_m$ with period $N$ having larger $m$-adic complexity. Compared with the binary case, the computation of 4-adic complexity of knowing quaternary sequences has not been well developed. In this paper, we determine the 4-adic complexity of the quaternary cyclotomic sequences with period 2$p$ defined in [6]. The main method we utilized is a quadratic Gauss sum $G_p$ valued in ${\mathbb{Z}}_{4^N-1}$ which can be seen as a version of classical quadratic Gauss sum. Our results show that the 4-adic complexity of this class of quaternary cyclotomic sequences reaches the maximum if $5\nmid p-2$ and close to the maximum otherwise.