A lower bound on the 2-adic complexity of the modified Jacobi sequence

Abstract

Let p, q be distinct primes satisfying gcd(p −  1, q −  1) = d and let D_i, i =  0, 1, · · · ,d −  1, be Whiteman’s generalized cyclotomic classes with $\mathbb {Z}_{pq}^{\ast }=\cup _{i = 0}^{d-1}D_{i}$. In this paper, we give the values of Gauss periods based on the generalized cyclotomic sets $D_{0}^{\ast }=\cup _{i = 0}^{\frac {d}{2}-1}D_{2i}$ and $D_{1}^{\ast }=\cup _{i = 0}^{\frac {d}{2}-1}D_{2i + 1}$. As an application, we determine a lower bound on the 2-adic complexity of the modified Jacobi sequence. Our result shows that the 2-adic complexity of the modified Jacobi sequence is at least pq − p − q − 1 with period N = pq. This indicates that the 2-adic complexity of the modified Jacobi sequence is large enough to resist the attack of the rational approximation algorithm (RAA) for feedback with carry shift registers (FCSRs).