# Algebraic dependence in generating functions and expansion complexity

## TL;DR

This paper advances the understanding of expansion complexity in cryptographic sequences by analyzing its algebraic properties, providing bounds for random sequences, and examining sequences from differential equations, including the inversive generator.

## Contribution

It introduces algebraic methods to analyze expansion complexity, establishes bounds for random sequences, and explores sequences generated by differential equations.

## Key findings

- Expansion complexity relates to Gröbner bases of polynomial ideals.
- Bounds on expansion complexity for random sequences are established.
- Sequences from differential equations, like the inversive generator, are analyzed for expansion complexity.

## Abstract

In 2012, Diem introduced a new figure of merit for cryptographic sequences called expansion complexity. Recently, a series of paper has been published for analysis of expansion complexity and for testing sequences in terms of this new measure of randomness. In this paper, we continue this analysis. First we study the expansion complexity in terms of the Gr\"obner basis of the underlying polynomial ideal. Next, we prove bounds on the expansion complexity for random sequences. Finally, we study the expansion complexity of sequences defined by differential equations, including the inversive generator.