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

TL;DR

This paper establishes a lower bound on the 2-adic complexity of modified Jacobi sequences using Gauss periods and cyclotomic classes, demonstrating their robustness against certain cryptanalytic attacks.

Contribution

It provides the first explicit lower bound on the 2-adic complexity of modified Jacobi sequences based on generalized cyclotomic sets.

Findings

2-adic complexity is at least pq - p - q - 1

The result shows high resistance to RAA attacks

Gauss periods are explicitly calculated for specific cyclotomic sets

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 $Z_{pq}^{\ast}=\cup_{i=0}^{d-1}D_i$. 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).