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

Let $p,q$ be distinct primes satisfying $\mathrm{g}\mathrm{c}\mathrm{d}(p-1,q-1)=d$ and let $D_i$, $i=0,1,\cdots ,d-1$, be Whiteman’s generalized cyclotomic classes with $Z_{pq}^{\ast }={\cup }_{i=0}^{d-1}{D}_i$. In this paper, we give the values of Gauss periods based on the generalized cyclotomic sets $D_0^{\ast }={\sum }_{i=0}^{\frac{d}{2}-1}{D}_{2i}$ and $D_1^{\ast }={\sum }_{i=0}^{\frac{d}{2}-1}{D}_{2i+1}$. As an application, we determine a lower bound on the 2-adic complexity of modified Jacobi sequence. Our result shows that the 2-adic complexity of modified Jacobi sequence is at least $pq-p-q-1$ with period $N=pq$. This indicates that the 2-adic complexity of modified Jacobi sequence is large enough to resist the attack of the rational approximation algorithm (RAA) for feedback with carry shift registers (FCSRs).