# Using the Logistic Coupled Map for Public Key Cryptography under a Distributed Dynamics Encryption Scheme

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{1}(x) means to use the logistic map of Equation (1) with μ

_{1}and f

_{2}(x) with μ

_{2}. Multiple chaotic attractors are observed in this system improving the unimodal shortcoming of the simple logistic map [8]. Figure 1 and Figure 2 show examples of chaotic attractors for the non-symmetric logistic coupled map (NLCM). The NLCM has a well documented chaotic range for 3.63 ≤ μ ≤ 4 and 0 ≤ α ≤ 1 [11].

## 2. Encryption

_{T}+ D

_{R}into two parts with D

_{T}transmitter variables t(n) = [t

_{1}(n); …; t

_{DT}(n)], and D

_{R}receiver variables r(n) = [r

_{1}(n) ; …; r

_{DR}(n)]. The receiver receives the scalar signal s

_{t}(n) from the transmitter, and the transmitter receives the scalar signal s

_{r}(n) from the receiver:

## 3. Decryption

## 4. Cryptanalysis

_{T}is the range of the data transmitted and L

_{q}is the quantization in the signal. Equation (11) is still useful in our case because it has been derived in general for any CML. For our implementation this number is around 4 × 10

^{7}which shows why Figure 10 can resolve the decision curve and how the security of this implementation is of the same level as the original DDE proposal.

## 5. Communication Scenario

_{1}, μ

_{2}, and α. Additionally, Bob has his own private key: the parameter A, which is selected in the interval accepted by Alice.

- (a)
- Obtain Alice’s authentic public key (Equation (8) and μ
_{1}). - (b)
- Represent the message as binary code.
- (c)
- Obtain Alice’s first 50 X data.
- (d)
- Compute 50 Y data for the first bit, using Equation (8), μ
_{1}, X data, and parameter A. - (e)
- Send Y data to Alice.
- (f)
- Repeat for the next bits with a new 50 X data from Alice.

- (a)
- Pair the first X data sent to Bob with the Y data received from Bob.
- (b)
- Take just the last 12 pairs.
- (c)
- For each pair calculate the Euclidian distance to each point of the long (X,Y) list of the simulation and preserve the minimum value of the distance.
- (d)
- If the average of the 12 minimum pairs is greater than the tolerance, it is a 1-bit, otherwise it is a 0-bit.
- (e)
- Repeat for the next 50 Y data from Bob.

_{1}, μ

_{2}, and α. Recall that they must lead to chaotic conditions (she could verify this with help of the Lyapunov exponent of NLCM from [11]). Let us say she uses μ

_{1}= 3.1, μ

_{2}= 2.9, and α = 0.3314, the same attractor from Figure 1. She will announce publicly Equation (8) with μ

_{1}and α from the previous selection and keep secret Equation (9) and μ

_{2}. Bob will take this public information to transmit the message. Alice makes the simulation offline using Equations (8) and (9), and it will produce a long list of (X,Y) points. Equation (9), with its parameters, is the private key; it does not travel by channel. To start the communication from the long list that Alice has, she sends 50 X data to Bob. He will take his message and convert it to binary, he could use ASCII, so W will be 01010111, in eight bits. He takes the first bit, 0, and using Equation (8) recalculates a new pair Y for the fifty received from Alice and send back to her, in this Equation (8) it is the parameter A which Bob actually chooses, it is better if it is random, also to compute Equation he needs a Y

_{0}starting value which he selects also randomly. Eve the evil genius, who is listening in the channel from the data sent by Bob, cannot reconstruct the first bit from the 50 numbers thanks to the private keys of Bob and the fact that Equation (8) is in a chaotic state. Eve will need to wait until having enough data to use the attack described in the previous section. Alice receives the 50 Y numbers and using the last 12, pairs them with the last 12 of X that she sent and calculates the distance to every point in the long list from the simulation that she has and takes the minimum distances for the 12 pairs. If the average of the 12 minimum distances are lower than the tolerance (the minimum value that A can be) is a 0-bit, and if it is greater than the tolerance, it is a 1-bit. In this case, Alice will see a 0-bit. Now, the process is repeated for the next bits. Alice does not need to send adjacent X data to Bob for the transmission of the message.

## 6. Computer Experiment

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Katz, J.; Menezes, A.J.; Van Oorschot, P.C.; Vanstone, S.A. Handbook of Applied Cryptography; CRC Press: Boca Raton, FL, USA, 1996; ISBN 9780849385230. [Google Scholar]
- Wang, X.; Gong, X.; Zhan, M.; Lai, C.H. Public-key encryption based on generalized synchronization of coupled map lattices. Chaos
**2005**, 15, 023109. [Google Scholar] [CrossRef] [PubMed] - Kocarev, L.; Makraduli, J.; Amato, P. Public-key encryption based on Chebyshev polynomials. Circ. Syst. Signal Process.
**2005**, 24, 497–517. [Google Scholar] [CrossRef] - Zhen, P.; Zhao, G.; Min, L.; Li, X. A survey of chaos-based cryptography. In Proceedings of the 2014 Ninth International Conference on P2P, Parallel, Grid, Cloud and Internet Computing, Guangdong, China, 8–10 November 2014; pp. 237–244. [Google Scholar]
- Devaney, R.L. An Introduction to Chaotic Dynamical Systems; CRC Press: Boca Raton, FL, USA, 1996; ISBN 9780201130461. [Google Scholar]
- Arroyo, D.; Amigo-Garcia, J.M.; Li, S.; Alvarez, G. On the Inadequacy of Unimodal Maps for Cryptographic Applications; RECSI: Tarragona, Spain, 2010; ISBN 9788469333044. [Google Scholar]
- Lawnik, M. Generalized logistic map and its application in chaos based cryptography. J. Phys. Conf. Ser.
**2017**, 936, 012017. [Google Scholar] [CrossRef][Green Version] - Kaneko, K. Theory and Applications of Coupled Map Lattices; Wiley: New York, NY, USA, 1993; Volume 159, ISBN 978-0471937418. [Google Scholar]
- Lloyd, A.L. The coupled logistic map: A simple model for the effects of spatial heterogeneity on population dynamics. J. Theor. Biol.
**1995**, 173, 217–230. [Google Scholar] [CrossRef] - Schult, R.L.; Creamer, D.B.; Henyey, F.S.; Wright, J.A. Symmetric and nonsymmetric coupled logistic maps. Phys. Rev. A
**1987**, 35, 3115–3118. [Google Scholar] [CrossRef] - Zhang, Y.Q.; Wang, X.Y. Spatiotemporal chaos in Arnold coupled logistic map lattice. Nonlinear Anal. Model. Control
**2013**, 18, 526–541. [Google Scholar] - Tenny, R.; Tsimring, L.S.; Larson, L.; Abarbanel, H.D. Using distributed nonlinear dynamics for public key encryption. Phys. Rev. Lett.
**2003**, 90, 047903. [Google Scholar] [CrossRef] [PubMed] - Xiao, D.; Liao, X.; Deng, S. A novel key agreement protocol based on chaotic maps. Inf. Sci.
**2007**, 177, 1136–1142. [Google Scholar] [CrossRef] - Alvarez, G.; Li, S. Some Basic Cryptographic Requirements for Chaos-Based Cryptosystems. Int. J. Bifurc. Chaos
**2006**, 16, 2129–2151. [Google Scholar] [CrossRef] - Tenny, R.; Tsimring, L.; Abarbanel, H.; Larson, L. Security of chaos-based communication and encryption. In Digital Communications Using Chaos and Nonlinear Dynamics; Larson, L., Tsimring, L., Liu, J.-M., Eds.; Institute for Nonlinear Science, Springer: New York, NY, USA, 2006; pp. 191–229. ISBN 978-0387297873. [Google Scholar]
- Elhadj, Z.; Sprott, J.C. Chaotifying 2-D piecewise-linear maps via a piecewise-linear controller function. Nonlinear Oscill.
**2011**, 13, 352–360. [Google Scholar] [CrossRef][Green Version] - Li, S.; Mou, X.; Cai, Y.; Ji, Z.; Zhang, J. On the security of a chaotic encryption scheme: Problems with computerized chaos in finite computing precision. Comput. Phys. Commun.
**2003**, 153, 52–58. [Google Scholar] [CrossRef]

**Figure 1.**Chaotic attractor for the non-symmetric logistic coupled map μ = 3.1, μ = 2.9 and α = 0.3314.

**Figure 3.**A message encrypted over the fully-known dynamic of Figure 1.

**Figure 4.**A longer message over the full dynamic of Figure 2.

**Figure 5.**An example of the data, which can be captured from the communication channel by an unauthorized receiver.

**Figure 6.**An example of the data, which can be captured from the communication channel when the attractor is not chaotic.

**Figure 7.**The decision surface (solid line) obtained from the attack when the attractor is not chaotic.

**Figure 8.**The decision surface (solid line) obtained from the attack when the attractor of Figure 1 has a low number of observations.

**Figure 9.**The decision surface (solid line) obtained from the attack for when the attractor of Figure 1 has a medium number of observations.

**Figure 10.**The decision surface (solid line) obtained from the attack when the attractor is the same as from Figure 1 and enough observations are known.

No. Bits | Encryption Time (s) | Decryption Time (s) |
---|---|---|

8 | 0.045 | 141.1081 |

16 | 0.0997 | 293.1638 |

32 | 0.2017 | 600.4974 |

64 | 0.3199 | 1318.4119 |

128 | 0.5973 | 2567.0675 |

256 | 1.7431 | 5297.5988 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Solís-Sánchez, H.; Barrantes, E.G. Using the Logistic Coupled Map for Public Key Cryptography under a Distributed Dynamics Encryption Scheme. *Information* **2018**, *9*, 160.
https://doi.org/10.3390/info9070160

**AMA Style**

Solís-Sánchez H, Barrantes EG. Using the Logistic Coupled Map for Public Key Cryptography under a Distributed Dynamics Encryption Scheme. *Information*. 2018; 9(7):160.
https://doi.org/10.3390/info9070160

**Chicago/Turabian Style**

Solís-Sánchez, Hugo, and E. Gabriela Barrantes. 2018. "Using the Logistic Coupled Map for Public Key Cryptography under a Distributed Dynamics Encryption Scheme" *Information* 9, no. 7: 160.
https://doi.org/10.3390/info9070160