Cryptography is a science to maintain the security of the message by changing data or information into a different form, so the message cannot be recognized. Today, many algorithms have been proposed for image encryption, but the chaotic encryption methods have a good combination of speed and high security. In recent years, the chaos based cryptographic algorithms have suggested some new and efficient ways to develop secure image encryption techniques. The chaos-based encryption schemes are composed of two steps: chaotic confusion and pixel diffusion. In the chaotic confusion stage, a combination of the chaotic maps is used to realize the confusion of all pixels. In this paper, we first give a brief introduction into chaotic image encryption and then we investigate some important properties and behaviour of the logistic map. The logistic map, aperiodic trajectory, or random-like fluctuation, could not be obtained with some choice of initial condition. Therefore, a noisy logistic map with an additive system noise is introduced. The proposed scheme is based on the extended map of the Clifford strange attractor, where each dimension has a specific role in the encryption process. Two dimensions are used for pixel permutation and the third dimension is used for pixel diffusion. In order to optimize the Clifford encryption system we increase the space key by using the noisy logistic map and a novel encryption scheme based on the Clifford attractor and the noisy logistic map for secure transfer images is proposed. This algorithm consists of two parts: the noisy logistic map shuffle of the pixel position and the pixel value. We use times for shuffling the pixel position and value then we generate the new pixel position and value by the Clifford system. To illustrate the efficiency of the proposed scheme, various types of security analysis are tested. It can be concluded that the proposed image encryption system is a suitable choice for practical applications.

Abstract
1. Introduction
2. Noisy logistic map
3. Methodology
4. Simulation results and statistical analysis
    4.1. Key space analysis
    4.2. Encryption and histogram test
    4.3. Chi-square test
    4.4. Analysis for correlation of adjacent pixels
    4.5. Analysis for correlation of plain image and cipherimage
    4.7. Sensitivity analysis
5. Conclusions
References

• Mohadeseh KANAFCHIAN(Dep. of Mathematics, University of Guilan)
• Behrouz FATHI-VAJARGAH(Dep. of Statistics, University of Guilan) Corresponding Author