Chaos theory is a branch of mathematics that deals with studying non-linear dynamical systems that exhibit sensitivity to initial conditions. Chaotic conditions exist in nature. A decent and typical example of such is the weather.

Chaos theory has had many applications in meteorology, anthropology, economics, and computer science, among others, since its inception. Chaotic systems can be characterized by sensitivity to initial conditions, periodic orbit density, and topological mixing. Its attributes, such as determinism and sensitivity, make chaotic systems attractive for cryptography.

### Table of contents

- Prerequisite
- Characteristics of chaotic systems
- Chaos cryptographic primitives
- Practical problems of Chaos-Based Cryptography
- Possible solutions
- Conclusion
- Further reading

### Prerequisite

To follow this article, the reader should have a basic understanding of cryptography concepts.

### Characteristics of chaotic systems

The following are characteristics of chaotic systems.

#### 1. Sensitivity to initial conditions

This means that a slight change to initial conditions creates an unexpected behavior. The most common analogy used to express this idea is the butterfly effect, an essential property in creating chaotic cryptographic algorithms and hash functions comparable to the diffusion property.

#### 2. Periodic orbit density

Periodic orbit density means that any chaotic point in the trajectory can approach other points due to the strange attractor phenomenon. This characteristic helps create affine transformations that can be pretty useful in cryptography.

#### 3. Topological mixing

Topological mixing means that the chaotic trajectory generated from a random region of the phase space can cover the rest of the phase space as the trajectory evolves, which is an attractive feature in chaos since it is analogous to the **uniform distribution property** in cryptography.

#### 4. Deterministic

Deterministic can be considered an extension of the sensitivity to initial conditions characteristics. Provided the initial conditions remain unchanged, the chaotic path followed will always be the same. This property is comparable to the deterministic pseudo-randomness property of cryptographic schemes.

### Chaos cryptographic primitives

Academics and researchers over the past decades have designed many chaos-based cryptographic schemes. The following sub-section will be a brief overview of some of their works.

#### 1. Hash function

A *hash function* is a function that takes an input of any size and produces a fixed-length output. One of the chaos-based hash algorithms developed is the Chaos Hash Algorithm (CHA-1). CHA-1 is a cryptographic hash algorithm developed by two researchers which use the logistic map as its basis for pseudo-randomness. Its comparison to SHA-1 shows a better security factor but lower performance.

#### 2. Encryption algorithm

Encryption aims to serve data confidentiality, whereas chaos is particularly used in image encryption. One such instance is the image encryption and compression algorithm development, which relies on the wavelet transform and 1-D discrete chaotic maps. Its design uses a 97-bit key combination, increasing its security. Statistical and correlation tests reveal that its large keyspace and chaotic nature make it infeasible for brute force attacks.

#### 3. Digital signature algorithm

Researchers have proposed a digital signature algorithm that relies on the quadratic residue problem and Chebyshev chaotic map. This signature scheme is a one-to-one interaction between the signer and the verifier. This scheme is more efficient because it does not require modular exponentiation or scalar multiplication of conventional digital signature schemes.

### Practical problems of chaos-based cryptography

Many primitives use finite precision floating-point representation. There will be no issues if the same operating system, hardware platform, and mathematical representation are used in verification. It becomes difficult to verify the algorithm if there is a mismatch in these factors, making it difficult to reproduce the results.

Efficiency problems may arise due to complex design algorithms meaning that most chaos cryptographic primitives are slower than conventional cryptosystems. For example, CHA-1 had a better security factor than SHA-1 but was slower. The use of weak keys is another problem. Special care needs to be taken to avoid secret vital values that will lead to non-chaotic behavior. In a cryptosystem, this would be a huge flaw.

The lack of proper security tests via cryptanalysis is a significant problem. Most of the chaos cryptographic primitives rely on statistical tests for security justifications. Statistical tests are essential. However, they should not be the only basis for security tests. Other cryptanalytic tests should be done. These tests might be difficult to perform due to the over-complexity of the chaos cryptographic primitives.

### Possible solutions

Using secure but straightforward design principles in chaos-based algorithms is a possible solution. The design of convoluted and complex algorithms is a double-edged sword. It may make analysis by adversaries difficult. It also makes it difficult for researchers to verify its security claims. This solution may bring confidence in the practical application of chaos cryptographic primitives.

The performance of chaos cryptographic systems should be measured against conventional cryptosystems. Comparisons should be based on metrics such as execution time and security factors.

Mathematical representations used by chaos cryptographic primitives should be rigorously defined, which helps to guarantee reproducibility. Floating-point representations suffer from dynamic degradation. Addressing this issue would be a significant step in producing verifiable, consistent algorithms.

### Conclusion

Chaos theory possesses characteristics that suit cryptographic usage. There have been great strides in harnessing these characteristics. There is still a long way to go before realizing a practical chaos-based cryptosystem. Chaos theory has the potential to be useful for cryptography in the future. For now, however, it stays confined to the academic domain.

Happy learning!

### Further reading

- Implementations and practical problems of chaos-based cryptography revisited
- New hash function based on chaos theory (CHA-1)
- A New Chaos-Based Image-Encryption and Compression Algorithm
- A new digital signature scheme based on chaotic maps and quadratic residue problems

Peer Review Contributions by: Briana Nzivu