Chaotic Tent Map Research Articles

Tent Chaotic Map and Population Classification Evolution Strategy-Based Dragonfly Algorithm for Global Optimization

Dragonfly algorithm (DA) is a recently proposed optimization algorithm based on swarm intelligence, which has been successfully applied in function optimization, feature selection, parameter adjustment, etc. However, it fails to take individual optimal position into consideration but only relies on population optimal position and 5 behaviours to update individual position, leading to low accuracy, slow convergence, and local optima. To overcome these drawbacks, Tent Chaotic Map and Population Classification Evolution Strategy-Based Dragonfly Algorithm (TPDA) is proposed. Tent chaotic map is used to initialize the population, making individuals distributed more uniformly in search space to improve population diversity and search efficiency. Population is classified according to individual fitness value, and different position update methods are adopted for different types of individuals to guide the search process and improve the ability of TPDA to jump out of local optima, thus realizing a balance between exploration and exploitation. The efficiency of TPDA has been validated by tests on 18 basic unconstrained benchmark functions. A comparative performance analysis between TPDA, Particle Swarm Optimization (PSO), DA, and Adaptive Learning Factor and Differential Evolution-Based Dragonfly Algorithm (ADDA) has been carried out. Experimental and statistical results demonstrate that TPDA gives significantly better performances compared with PSO, DA, and ADDA on the average and standard deviation in all 18 functions. The global optimization capability of TPDA on high-dimensional functions and the comparison of the time complexity of TPDA and other swarm intelligence algorithms is also verified in the paper. The results indicate that TPDA is able to perform better on optimizing functions without consuming more computational time.

Task Offloading Strategy of Vehicular Networks Based on Improved Bald Eagle Search Optimization Algorithm

To reduce computing delay and energy consumption in the Vehicular networks, the total cost of task offloading, namely delay and energy consumption, is studied. A task offloading model combining local vehicle computing, MEC (Mobile Edge Computing) server computing, and cloud computing is proposed. The model not only considers the priority relationship of tasks, but also considers the delay and energy consumption of the system. A computational offloading decision method IBES based on an improved bald eagle search optimization algorithm is designed, which introduces Tent chaotic mapping, Levy Flight mechanism and Adaptive weights into the bald eagle search optimization algorithm to increase initial population diversity, enhance local search and global convergence. The simulation results show that the total cost of IBES is 33.07% and 22.73% lower than that of PSO and BES, respectively.

Equalization Optimizer-Based LSTM Application in Reservoir Identification.

In recent years, the use of long short-term memory (LSTM) has made significant contributions to various fields and the use of intelligent optimization algorithms combined with LSTM is also one of the best ways to improve model shortcomings and increase classification accuracy. Reservoir identification is a key and difficult point in the process of logging, so using LSTM to identify the reservoir is very important. To improve the logging reservoir identification accuracy of LSTM, an improved equalization optimizer algorithm (TAFEO) is proposed in this paper to optimize the number of neurons and various parameters of LSTM. The TAFEO algorithm mainly employs tent chaotic mapping to enhance the population diversity of the algorithm, convergence factor is introduced to better balance the local and global search, and then, a premature disturbance strategy is employed to overcome the shortcomings of local minima. The optimization performance of the TAFEO algorithm is tested with 16 benchmark test functions and Wilcoxon rank-sum test for optimization results. The improved algorithm is superior to many intelligent optimization algorithms in accuracy and convergence speed and has good robustness. The receiver operating characteristic (ROC) curve is used to evaluate the performance of the optimized LSTM model. Through the simulation and comparison of UCI datasets, the results show that the performance of the LSTM model based on TAFEO has been significantly improved, and the maximum area under the ROC curve value can get 99.43%. In practical logging applications, LSTM based on an equalization optimizer is effective in well-logging reservoir identification, the highest recognition accuracy can get 95.01%, and the accuracy of reservoir identification is better than other existing identification methods.

STRUCTUREAL DEASIGN OF SECURE E-COMMERCE WEBSITES EMPLOYING MULTI-AGENT SYSTEM


 
 
 
 The vast growth in electronic trading and monetary transactions results to increase the importance of E-commerce applications rapidly. In order to accomplish the fully functioning, reliable, scalable, secure, efficient and user-friendly E-commerce applications, sufficient system analysis and design processes are important. Shopping over the internet requires offering secure payment method.
 Providing security for each individual online consumer at the same time (particularly for large websites) is a time-consuming effort that can result in a variety of issues, including response delays, customer orders being lost, and system freeze or crash, all of which impair system performance.
 This work seeks to present a new multi-agent system prototype structure that solves the challenge of security while avoiding issues that might degrade system performance. By creating integrated system taking into account the three parties the e-commerce, the e-bank and the customer, the main concentration is to create an archeticture that helps in decreasing the website traffic and supportng the security with effective performance. To simulate the real systems a recommendation system is created based on the customer behaviors to increase system performance in helping the customers to find their favorite products and solve problems like cold start, sparsity of user‐item matrix, scalability, and changes in user interest.
 This is done by creating a commercial environment (e-commerce and e-bank websites) and a software agent that is installed on the client's device to handle the purchase and ciphering procedures without the need for the consumer to intervene. The security process involve preprocessing phase followed by a modified lightweight AES employing chaotic tent map to generate encryption key.
 
 
 

Detecting Starch-Head and Mildewed Fruit in Dried Hami Jujubes Using Visible/Near-Infrared Spectroscopy Combined with MRSA-SVM and Oversampling.

Dried Hami jujube has great commercial and nutritional value. Starch-head and mildewed fruit are defective jujubes that pose a threat to consumer health. A novel method for detecting starch-head and mildewed fruit in dried Hami jujubes with visible/near-infrared spectroscopy was proposed. For this, the diffuse reflectance spectra in the range of 400–1100 nm of dried Hami jujubes were obtained. Borderline synthetic minority oversampling technology (BL-SMOTE) was applied to solve the problem of imbalanced sample distribution, and its effectiveness was demonstrated compared to other methods. Then, the feature variables selected by competitive adaptive reweighted sampling (CARS) were used as the input to establish the support vector machine (SVM) classification model. The parameters of SVM were optimized by the modified reptile search algorithm (MRSA). In MRSA, Tent chaotic mapping and the Gaussian random walk strategy were used to improve the optimization ability of the original reptile search algorithm (RSA). The final results showed that the MRSA-SVM method combined with BL-SMOTE had the best classification performance, and the detection accuracy reached 97.22%. In addition, the recall, precision, F1 and kappa coefficient outperform other models. Furthermore, this study provided a valuable reference for the detection of defective fruit in other fruits.

Parameter Identification of Lithium Battery Model Based on Chaotic Quantum Sparrow Search Algorithm

An accurate battery model is of great importance for battery state estimation. This study considers the parameter identification of a fractional-order model (FOM) of the battery, which can more realistically describe the reaction process of the cell and provide more precise predictions. Firstly, an improved sparrow search algorithm combined with the Tent chaotic mapping, quantum behavior strategy and Gaussian variation is proposed to regulate the early population quality, enhance its global search ability and avoid trapping into local optima. The effectiveness and superiority are verified by comparing the proposed chaotic quantum sparrow search algorithm (CQSSA) with the particle swarm optimization (PSO), genetic algorithm (GA), grey wolf optimization algorithm (GWO), Dingo optimization algorithm (DOA) and sparrow search algorithm (SSA) on benchmark functions. Secondly, the parameters of the FOM battery model are identified using six algorithms under the hybrid pulse power characterization (HPPC) test. Compared with SSA, CQSSA has 4.3%, 5.9% and 11.5% improvement in mean absolute error (MAE), root mean square error (RMSE) and maximum absolute error (MaAE), respectively. Furthermore, these parameters are used in the pulsed discharge test (PULSE) and urban dynamometer driving schedule (UDDS) test to verify the adaptability of the proposed algorithm. Simulation results show that the model parameters identified by the CQSSA algorithm perform well in terms of the MAE, RMSE and MaAE of the terminal voltages under all three different tests, demonstrating the high accuracy and good adaptability of the proposed algorithm.

Simulation and Prediction of Injection Pressure in CO2 Geological Sequestration Based on Improved LSO Algorithm and BP Neural Networks

In order to avoid the occurrence of caprock integrity damage and gas escape due to injection pressure overrun in CO2 sequestration, an optimized back propagation (BP) neural network model based on Monte Carlo simulation (MCS) and an improved lion swarm optimization (ILSO) algorithm is proposed for the maximum sustainable injection pressure prediction. Firstly, a hydromechanical model is constructed to simulate the damage changes of the reservoir caprock during injection by ABAQUS. Secondly, in view of the uncertainties of formation parameters that could lead to deviations between the model calculation results and actual geological condition, the MCS method is used, and then, the probability distribution interval of the maximum injection pressure with high probability of caprock failure under different formation parameters is obtained by MATLAB. This effectively reduces the uncertainty and improves the calculation accuracy. Finally, based on the numerical simulation results, the maximum injection pressure prediction model is constructed. Aiming at the problems of the sensitivity of the BP neural network to initial weights and its poor convergence, tent chaotic mapping and difference mechanism are introduced to improve the LSO algorithm. Following this, the neural network is optimized by ILSO algorithm whose superiority is verified through 8 benchmark functions, and the maximum injection pressure is effectively predicted according to porosity, permeability, and other parameters. Experimental results show that, compared to the other three optimization methods, the ILSO_BP model has a faster convergence speed, higher prediction accuracy, and stability, which can provide a powerful guide for the safe injection of CO2 and efficient sequestration management.

Membrane Fouling Prediction Based on Tent-SSA-BP.

In view of the difficulty in obtaining the membrane bioreactor (MBR) membrane flux in real time, considering the disadvantage of the back propagation (BP) network in predicting MBR membrane flux, such as the local minimum value and poor generalization ability of the model, this article introduces tent chaotic mapping in the standard sparrow search algorithm (SSA), which improves the uniformity of population distribution and the searching ability of the algorithm (used to optimize the key parameters of the BP network). The tent sparrow search algorithm back propagation network (Tent-SSA-BP) membrane fouling prediction model is established to achieve accurate prediction of membrane flux; compared to the BP, genetic algorithm back propagation network (GA-BP), particle swarm optimization back propagation network (PSO-BP), sparrow search algorithm extreme learning machine(SSA-ELM), sparrow search algorithm back propagation network (SSA-BP), and Tent particle swarm optimization back propagation network (Tent–PSO-BP) models, it has unique advantages. Compared with the BP model before improvement, the improved soft sensing model reduces MAPE by 96.76%, RMSE by 99.78% and MAE by 95.61%. The prediction accuracy of the algorithm proposed in this article reaches 97.4%, which is much higher than the 48.52% of BP. It is also higher than other prediction models, and the prediction accuracy has been greatly improved, which has some engineering reference value.

Encryption method of annual bad debt accumulation information and reimbursement based on ant colony algorithm

In the enterprise financial information management, the information security of reimbursement and annual bad debt accumulation cannot be ignored. In order to solve the problem that the plaintext exceeds the byte limit in information encryption and affects the encryption calculation time, an annual bad debt accumulation information and reimbursement encryption method is proposed based on ant colony algorithm. We first define all the variables including encryption method, organizations’ information and reimbursement encryption, Ant colony algorithm and then we utilized elliptic curve digital signature to process the data and to prevent the annual bad debt accumulation information data from being tampered with or replaced in the transmission process. Furthermore, one dimensional Logistic chaotic map and Tent chaotic map with good randomness are selected to generate chaotic sequences as keys, and they are grouped based on ant colony algorithm. Each group is XOR with the previous group of ciphertext, which shortens the plaintext bytes. The design of the encryption method for the accumulated information of annual bad debt accrual is completed through the operation and the second round of encryption. The plaintext data files with file sizes of 10MB, 50MB, 100MB and 200MB are selected for serial and parallel encryption calculation. It shows that the average speedup ratio of this method is 3.90, which is 0.44 and 0.46 higher than the encryption methods based on genetic algorithm and particle swarm optimization algorithm, so it has better encryption performance.

High-Performance Computing Analysis and Location Selection of Logistics Distribution Center Space Based on Whale Optimization Algorithm.

As a meta-heuristic algorithm based on swarm intelligence, the WOA algorithm has few control parameters and searches for the optimal solution by encircling the prey, searching for the prey, and attacking the bubble net. During the whole process, only two internal parameters A and C are utilized for the control of the exploration and development process. BWOA is simple to implement. In the process of algorithm execution, the initial population, global exploration, and local development stages have shortcomings. Therefore, it is necessary to optimize the WOA algorithm. Based on WOA, this study conducts a high-performance computing analysis and location selection of logistics distribution center space. It is concluded that: (1) by using the combination of direct logistics distribution and hierarchical logistics distribution, the WOA algorithm optimizes the cross selection strategy, the population fitness S-LO is improved, the quality of LA is guaranteed, and the chaotic S-LO mapping eliminates inferior individuals in the population. Direct distribution is carried out for bulky goods and important distribution customers, and hierarchical logistics distribution is used for customers in intensive logistics distribution destinations. (2) WOA uses the second reverse learning, chaotic mapping, and logistic chaotic mapping to improve the location update mode. The direct distribution method is mostly used for the logistics business with short journeys, fixed distribution points, and more goods delivered at one time, and logistics enterprises do not need to store and distribute goods. The uniform ergodicity of the Tent chaotic map and logistic chaotic map is improved. Ka adaptive inertia weights are a good complement to optimize the limitations of the Ao whale algorithm. (3) The inertia weight of the levy flight behavior can play a powerful role in balancing the global exploration ability and optimization performance of the intelligent algorithm. The long-term short-distance search of HED and the long-distance jump of KVAR are combined. Variant individuals undergo vector synthesis. It reduces the construction and operation costs of logistics sites and is suitable for logistics distribution under specific conditions.

An Effective Color Image Encryption Based on Henon Map, Tent Chaotic Map, and Orthogonal Matrices.

In the last decade, the communication of images through the internet has increased. Due to the growing demands for data transfer through images, protection of data and safe communication is very important. For this purpose, many encryption techniques have been designed and developed. New and secured encryption schemes based on chaos theory have introduced methods for secure as well as fast communication. A modified image encryption process is proposed in this work with chaotic maps and orthogonal matrix in Hill cipher. Image encryption involves three phases. In the first phase, a chaotic Henon map is used for permuting the digital image. In the second phase, a Hill cipher is used whose encryption key is generated by an orthogonal matrix which further is produced from the equation of the plane. In the third phase, a sequence is generated by a chaotic tent map which is later XORed. Chaotic maps play an important role in the encryption process. To deal with the issues of fast and highly secured image processing, the prominent properties of non-periodical movement and non-convergence of chaotic theory play an important role. The proposed scheme is resistant to different attacks on the cipher image. Different tests have been applied to evaluate the proposed technique. The results of the tests such as key space analysis, key sensitivity analysis, and information entropy, histogram correlation of the adjacent pixels, number of pixel change rate (NPCR), peak signal to noise ratio (PSNR), and unified average changing intensity (UCAI) showed that our proposed scheme is an efficient encryption technique. The proposed approach is also compared with some state-of-the-art image encryption techniques. In the view of statistical analysis, we claim that our proposed encryption algorithm is secured.

Node Localization Algorithm Based on Modified Archimedes Optimization Algorithm in Wireless Sensor Networks

Node localization information plays an important role in wireless sensor networks (WSNs). To solve the problem of low localization accuracy of distance vector hop (DV-Hop) localization algorithm in wireless sensor networks, an improved localization algorithm called MAOADV-Hop based on the modified Archimedes optimization algorithm (MAOA) and DV-Hop is proposed, which can achieve the balance between the localization speed and the localization precision. Firstly, tent chaotic mapping and particle swarm optimization (PSO) algorithm are introduced into Archimedes optimization algorithm to improve the initial population diversity and change the update rules of density and volume, which improve the global convergence ability and convergence speed of the algorithm. Secondly, the MAOA is used to replace the least square part of the DV-Hop localization algorithm to improve the localization accuracy of the algorithm. Finally, MAOADV-Hop is verified through four different network environments and compared with DE_DV-Hop, BOA_DV-Hop, and DV-Hop. The simulation results show that the localization speed of the proposed approach is faster than that of DE_DV-Hop and BOA_DV-Hop, and the localization error is less than that of DV-Hop, DE_DV-Hop, and BOA_DV-Hop.

IHSSAO: An Improved Hybrid Salp Swarm Algorithm and Aquila Optimizer for UAV Path Planning in Complex Terrain

In this paper, we propose a modified hybrid Salp Swarm Algorithm (SSA) and Aquila Optimizer (AO) named IHSSAO for UAV path planning in complex terrain. The primary logic of the proposed IHSSAO is to enhance the performance of AO by introducing the leader mechanism of SSA, tent chaotic map, and pinhole imaging opposition-based learning strategy. Firstly, the tent chaotic map is utilized to substitute the randomly generated initial population in the original algorithm to increase the diversity of the initial individuals. Secondly, we integrate the leader mechanism of SSA into the position update formulation of the basic AO, which enables the search individuals to fully utilize the optimal solution information and enhances the global search capability of AO. Thirdly, we introduce the pinhole imaging opposition-based learning in the proposed IHSSAO to enhance the capability to escape from the local optimization. To verify the effectiveness of the proposed IHSSAO algorithm, we tested it against SSA, AO, and five other advanced meta-heuristic algorithms on 23 classical benchmark functions and 17 IEEE CEC2017 test functions. The experimental results indicate that the proposed IHSSAO is superior to the other seven algorithms in most cases. Eventually, we applied the IHSSAO, SSA, and AO to solve the UAV path planning problem. The experimental results verify that the IHSSAO is superior to the basic SSA and AO for solving the UAV path planning problem in complex terrain.

A Novel Chimp Optimization Algorithm with Refraction Learning and Its Engineering Applications

The Chimp Optimization Algorithm (ChOA) is a heuristic algorithm proposed in recent years. It models the cooperative hunting behaviour of chimpanzee populations in nature and can be used to solve numerical as well as practical engineering optimization problems. ChOA has the problems of slow convergence speed and easily falling into local optimum. In order to solve these problems, this paper proposes a novel chimp optimization algorithm with refraction learning (RL-ChOA). In RL-ChOA, the Tent chaotic map is used to initialize the population, which improves the population’s diversity and accelerates the algorithm’s convergence speed. Further, a refraction learning strategy based on the physical principle of light refraction is introduced in ChOA, which is essentially an Opposition-Based Learning, helping the population to jump out of the local optimum. Using 23 widely used benchmark test functions and two engineering design optimization problems proved that RL-ChOA has good optimization performance, fast convergence speed, and satisfactory engineering application optimization performance.

Diagnosis of unbalanced rolling bearing fault sample based on adaptive sparse contrative Auto-encoder and IGWO-USELM

At present, numerous deep learning techniques are being used in fault diagnosis with good effects. However, obtaining a sample of a fault may be difficult in certain circumstances. Moreover, the distribution of the sample is highly unbalanced. Typically, when the sample size is unbalanced, deep learning methods exhibit the phenomenon of overfitting, which results in a lack of generalizability and precision. In this paper, we proposed an adaptive sparse contractive auto-encoder (ASCAE) and an improved gray wolf optimization unsupervised extreme learning machine (IGWO-USELM) model for diagnosing unbalanced rolling bearing faults. First, CAE was improved by employing sparse graph embedding. Additionally, homotopy regularization was utilized to optimize sparse coefficients. Consequently, the effect of fault feature extraction was improved. The IGWO model was improved using Tent chaotic mapping to compensate for the fault sample’s sparse information. Subsequently, we used the IGWO to dynamically optimize USELM's performance. Accordingly, the IGWO-USELM model was developed. The ASCAE was used to extract multi-layer features from the vibration signal. Additionally, the extracted features were fed into the IGWO-USELM fault diagnosis model. Lastly, we performed experimental analysis on the fault dataset from Case Western Reserve University and the two real datasets. The results demonstrate that our method is capable of extracting time–frequency features of high faults. The proposed method is the most accurate when the fault samples are unbalanced. It has demonstrated excellent performance, particularly on the real-world fault dataset. When noise is present, the proposed method exhibits excellent noise immunity and a short run time. A further advantage is that it can be used to perform real-time bearing fault diagnosis.

Variable frequency wind speed trend prediction system based on combined neural network and improved multi-objective optimization algorithm

Trend prediction data with low measurement frequency has always been needed in wind power station, but the traditional multi-step prediction method has caused error accumulation and led to poor prediction accuracy, in order to solve this problem, a new wind speed trend prediction system is proposed which includes data preprocessing (Fuzzy Information Granulation), combined neural network prediction and an improved multi-objective manta rays foraging optimization based on Tent chaotic map and T-distribution perturbation operator (IMOMRFO). The algorithm not only has a good ability to escape from the local optimal solution, but also proves theoretically that the Pareto optimal solution is obtained. Through the simulation of four groups of experiments, it is obvious that the stability, generalization and accuracy of the model are satisfactory. It is confirmed that the model greatly improves the accuracy of trend prediction and makes a certain contribution to solve the problem of wind speed prediction, through the test of the ability of point prediction and interval prediction of the model.

Modified Harris Hawks Optimization Algorithm with Exploration Factor and Random Walk Strategy.

One of the most popular population-based metaheuristic algorithms is Harris hawks optimization (HHO), which imitates the hunting mechanisms of Harris hawks in nature. Although HHO can obtain optimal solutions for specific problems, it stagnates in local optima solutions. In this paper, an improved Harris hawks optimization named ERHHO is proposed for solving global optimization problems. Firstly, we introduce tent chaotic map in the initialization stage to improve the diversity of the initialization population. Secondly, an exploration factor is proposed to optimize parameters for improving the ability of exploration. Finally, a random walk strategy is proposed to enhance the exploitation capability of HHO further and help search agent jump out the local optimal. Results from systematic experiments conducted on 23 benchmark functions and the CEC2017 test functions demonstrated that the proposed method can provide a more reliable solution than other well-known algorithms.

Research on Coverage Optimization in a WSN Based on an Improved COOT Bird Algorithm.

To address the problems of uneven distribution and low coverage of wireless sensor network (WSN) nodes in random deployment, a node coverage optimization strategy with an improved COOT bird algorithm (COOTCLCO) is proposed. Firstly, the chaotic tent map is used to initialize the population, increase the diversity of the population, and lay the foundation for the global search for the optimal solutions. Secondly, the Lévy flight strategy is used to perturb the individual positions to improve the search range of the population. Thirdly, Cauchy mutation and an opposition-based learning strategy are fused to perturb the optimal solutions to generate new solutions and enhance the ability of the algorithm to jump out of the local optimum. Finally, the COOTCLCO algorithm is applied to WSN coverage optimization problems. Simulation results show that COOTCLCO has a faster convergence speed and better search accuracy than several other typical algorithms on 23 benchmark test functions; meanwhile, the coverage rate of the COOTCLCO algorithm is increased by 9.654%, 13.888%, 6.188%, 5.39%, 1.31%, and 2.012% compared to particle swarm optimization (PSO), butterfly optimization algorithm (BOA), seagull optimization algorithm (SOA), whale optimization algorithm (WOA), Harris hawks optimization (HHO), and bald eagle search (BES), respectively. This means that in terms of coverage optimization effect, COOTCLCO can obtain a higher coverage rate compared to these algorithms. The experimental results demonstrate that COOTCLCO can effectively improve the coverage rate of sensor nodes and improve the distribution of nodes in WSN coverage optimization problems.

A Color Image Scrambling Method Based on Zigzag Transform and Cross-channel Permutation

It is a challenging task to maintain the privacy and security of color images that are shared over public networks. To address this challenge, image encryption methods can be used to protect the confidentiality of the shared image. As a part of spatial domain-based image encryption, the scrambling process aims to reduce the correlation between adjacent pixels. Furthermore, the correlation between different channels of an RGB color image can be decreased by applying a cross-channel permutation. To achieve both decorrelations, this paper presents a new image scrambling method that is based on zigzag transform and cross-channel permutation. Firstly, a color image’s red, green, and blue channels are obtained, and the zigzag transform is applied to each channel, respectively. The transformed channels are concatenated for further shuffling. A chaotic tent map is employed to create the necessary parameters to perform cross-channel permutation. The cross-channel permutation ensures swapping pixels between different channels. Performance measures of the proposed method show that the correlation coefficients of an RGB image can be significantly reduced. The information entropy of each channel is also greatly increased with this method. The average execution time is found to be less than 0.08 s for a 256 x 256 color image, indicating that the proposed method can be used in image encryption applications.

Image encryption using modified perfect shuffle-based bit level permutation and learning with errors based diffusion for IoT devices

Image encryption is important for IoT devices which handle military, medical and other confidential images. In this paper, stream cipher based image diffusion is performed using tent chaotic map and block based encryption is done by taking four bytes every time to multiply the matrix formed using the secret key and error matrix. Pixel permutation is performed in two stages: block level pixel permutation and whole image permutation to resist the statistical attacks. Bit level permutation is performed with piecewise linear chaotic map using modified perfect shuffle technique to overcome differential attacks. Various statistical and differential analyses are performed on the encrypted image and the results obtained using these analyses convey its resistance to statistical and differential attacks.