On the 4-adic complexity of the two-prime quaternary generator

TL;DR

This paper investigates the 4-adic complexity of two-prime quaternary sequences, providing formulas for its possible values and demonstrating its robustness against rational approximation attacks.

Contribution

It extends the analysis of 2-adic complexity to 4-adic complexity for two-prime generators in quaternary sequences, offering new formulas and security insights.

Findings

4-adic complexity exceeds pq - log_4(pq^2) - 1 when p<q

The 4-adic complexity can attain high values close to the period

Sequences are resistant to rational approximation attacks

Abstract

R. Hofer and A. Winterhof proved that the 2-adic complexity of the two-prime (binary) generator of period $pq$ with two odd primes $p\neq q$ is close to its period and it can attain the maximum in many cases. When the two-prime generator is applied to producing quaternary sequences, we need to determine the 4-adic complexity. We present the formulae of possible values of the 4-adic complexity, which is larger than $pq-\log_4(pq^2)-1$ if $p<q$. So it is good enough to resist the attack of the rational approximation algorithm.