The Eigenvalue Complexity of Sequences in the Real Domain

Lingfeng Liu,Renzhi Li,Hanping Hu,Hongyue Xiang

doi:10.3390/e21121194

Lingfeng Liu, Renzhi Li + Show 2 more

Open Access

https://doi.org/10.3390/e21121194

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Abstract

The eigenvalue is one of the important cryptographic complexity measures for sequences. However, the eigenvalue can only evaluate sequences with finite symbols—it is not applicable for real number sequences. Recently, chaos-based cryptography has received widespread attention for its perfect dynamical characteristics. However, dynamical complexity does not completely equate to cryptographic complexity. The security of the chaos-based cryptographic algorithm is not fully guaranteed unless it can be proven or measured by cryptographic standards. Therefore, in this paper, we extended the eigenvalue complexity measure from the finite field to the real number field to make it applicable for the complexity measurement of real number sequences. The probability distribution, expectation, and variance of the eigenvalue of real number sequences are discussed both theoretically and experimentally. With the extension of eigenvalue, we can evaluate the cryptographic complexity of real number sequences, which have a great advantage for cryptographic usage, especially for chaos-based cryptography.

Highlights

Sequence complexity can be regarded as a series of measures that depicts the different characteristics of sequences
The eigenvalue was provided from a similar aspect as well, while the eigenvalue profile more closely reflected the rate of vocabulary growth than the LZ complexity
The relationship between LZ complexity and nonlinear complexity was studied in [8], which shows that these two complexity measures are converse in a sense

Summary

Introduction

Sequence complexity can be regarded as a series of measures that depicts the different characteristics of sequences. For the shortest Linear Feedback Shift Register (LFSR), it is referred to as the linear complexity These two measures have been studied for many decades [1,2,3,4,5,6]. The relationship between LZ complexity and nonlinear complexity was studied in [8], which shows that these two complexity measures are converse in a sense For all these complexities, there exists a premise, which is that the measured sequences should be on the finite field. We should extend the cryptographic complexity from the finite field to the real number field. We mainly focused on the eigenvalue complexity, extending this measure from the finite field to the real number field to evaluate the cryptographic properties of real number sequences.

Eigenvalue for Binary Sequences

Eigenvalue of Sequences in the Real Domain

Eigenvalue of Uniformly Distributed Random Sequence

Eigenvalue of Logistic

Measure the Complexity of Chaotic Sequences

Conclusions