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

Abstract: Let $p,q$ be distinct primes satisfying $\mathrm{gcd}(p-1,q-1)=d$ and let $D_i$, $i=0,1,\cdots,d-1$, be Whiteman's generalized cyclotomic classes with $Z_{pq}<sup>{\ast}=\cup_{i=0}<sup>{d-1}D_i$. In this paper, we give the values of Gauss periods based on the generalized cyclotomic sets $$D_0<sup>{\ast}=\sum_{i=0}<sup>{\frac{d}{2}-1}D_{2i}$$ and $$D_1<sup>{\ast}=\sum_{i=0}<sup>{\frac{d}{2}-1}D_{2i+1}$$. 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).