Expansion complexity and linear complexity of sequences over finite fields

TL;DR

This paper compares the traditional linear complexity with the newer expansion complexity for sequences over finite fields, showing that expansion complexity can be a stronger cryptographic unpredictability measure especially for shorter or nonperiodic sequences.

Contribution

It establishes the relationship between linear and expansion complexity, demonstrating the latter's strength in certain sequence analyses and providing probabilistic insights for random sequences.

Findings

For periodic sequences, both complexities are comparable.

Expansion complexity offers a stronger test for nonperiodic or short sequences.

Probabilistic behavior of expansion complexity in random sequences is characterized.

Abstract

The linear complexity is a measure for the unpredictability of a sequence over a finite field and thus for its suitability in cryptography. In 2012, Diem introduced a new figure of merit for cryptographic sequences called expansion complexity. We study the relationship between linear complexity and expansion complexity. In particular, we show that for purely periodic sequences both figures of merit provide essentially the same quality test for a sufficiently long part of the sequence. However, if we study shorter parts of the period or nonperiodic sequences, then we can show, roughly speaking, that the expansion complexity provides a stronger test. We demonstrate this by analyzing a sequence of binomial coefficients modulo $p$. Finally, we establish a probabilistic result on the behavior of the expansion complexity of random sequences over a finite field.