# A cubic map chaos criterion theorem with applications in generalized synchronization based pseudorandom number generator and image encryption

## Abstract

This paper sets up a chaos criterion theorem on a kind of cubic polynomial discrete maps. Using this theorem, Zhou-Song's chaos criterion theorem on quadratic polynomial discrete maps and generalized synchronization (GS) theorem construct an eight-dimensional chaotic GS system. Numerical simulations have been carried out to verify the effectiveness of theoretical results. The chaotic GS system is used to design a chaos-based pseudorandom number generator (CPRNG). Using FIPS 140-2 test suit/Generalized FIPS 140-2, test suit tests the randomness of two 1000 key streams consisting of 20 000 bits generated by the CPRNG, respectively. The results show that there are 99.9%/98.5% key streams to have passed the FIPS 140-2 test suit/Generalized FIPS 140-2 test. Numerical simulations show that the different keystreams have an average 50.001% same codes. The key space of the CPRNG is larger than 2{sup 1345}. As an application of the CPRNG, this study gives an image encryption example. Experimental results show that the linear coefficients between the plaintext and the ciphertext and the decrypted ciphertexts via the 100 key streams with perturbed keys are less than 0.00428. The result suggests that the decrypted texts via the keystreams generated via perturbed keys of the CPRNG are almost completely independent onmore »

- Authors:

- Schools of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083 (China)

- Publication Date:

- OSTI Identifier:
- 22402555

- Resource Type:
- Journal Article

- Journal Name:
- Chaos (Woodbury, N. Y.)

- Additional Journal Information:
- Journal Volume: 25; Journal Issue: 5; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1054-1500

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; CHAOS THEORY; COMPUTER CODES; COMPUTERIZED SIMULATION; CRYPTOGRAPHY; IMAGES; MAPS; POLYNOMIALS; RANDOMNESS; STREAMS; SYNCHRONIZATION

### Citation Formats

```
Yang, Xiuping, E-mail: yangxiuping-1990@163.com, Min, Lequan, E-mail: minlequan@sina.com, and Wang, Xue, E-mail: wangxue-20130818@163.com.
```*A cubic map chaos criterion theorem with applications in generalized synchronization based pseudorandom number generator and image encryption*. United States: N. p., 2015.
Web. doi:10.1063/1.4917380.

```
Yang, Xiuping, E-mail: yangxiuping-1990@163.com, Min, Lequan, E-mail: minlequan@sina.com, & Wang, Xue, E-mail: wangxue-20130818@163.com.
```*A cubic map chaos criterion theorem with applications in generalized synchronization based pseudorandom number generator and image encryption*. United States. doi:10.1063/1.4917380.

```
Yang, Xiuping, E-mail: yangxiuping-1990@163.com, Min, Lequan, E-mail: minlequan@sina.com, and Wang, Xue, E-mail: wangxue-20130818@163.com. Fri .
"A cubic map chaos criterion theorem with applications in generalized synchronization based pseudorandom number generator and image encryption". United States. doi:10.1063/1.4917380.
```

```
@article{osti_22402555,
```

title = {A cubic map chaos criterion theorem with applications in generalized synchronization based pseudorandom number generator and image encryption},

author = {Yang, Xiuping, E-mail: yangxiuping-1990@163.com and Min, Lequan, E-mail: minlequan@sina.com and Wang, Xue, E-mail: wangxue-20130818@163.com},

abstractNote = {This paper sets up a chaos criterion theorem on a kind of cubic polynomial discrete maps. Using this theorem, Zhou-Song's chaos criterion theorem on quadratic polynomial discrete maps and generalized synchronization (GS) theorem construct an eight-dimensional chaotic GS system. Numerical simulations have been carried out to verify the effectiveness of theoretical results. The chaotic GS system is used to design a chaos-based pseudorandom number generator (CPRNG). Using FIPS 140-2 test suit/Generalized FIPS 140-2, test suit tests the randomness of two 1000 key streams consisting of 20 000 bits generated by the CPRNG, respectively. The results show that there are 99.9%/98.5% key streams to have passed the FIPS 140-2 test suit/Generalized FIPS 140-2 test. Numerical simulations show that the different keystreams have an average 50.001% same codes. The key space of the CPRNG is larger than 2{sup 1345}. As an application of the CPRNG, this study gives an image encryption example. Experimental results show that the linear coefficients between the plaintext and the ciphertext and the decrypted ciphertexts via the 100 key streams with perturbed keys are less than 0.00428. The result suggests that the decrypted texts via the keystreams generated via perturbed keys of the CPRNG are almost completely independent on the original image text, and brute attacks are needed to break the cryptographic system.},

doi = {10.1063/1.4917380},

journal = {Chaos (Woodbury, N. Y.)},

issn = {1054-1500},

number = 5,

volume = 25,

place = {United States},

year = {2015},

month = {5}

}