In this paper, we present a parallel encryption algorithm based on the Tinkerbell and Ikeda chaotic functions in order to meet the multi-core processor. The parallel algorithm is constructed with a data parallel approach. The experimental outputs show that the parallel encryption proposes more efficient performance against the serial execution of the scheme.
The enchanting knowledge in data processing and communication
interchange have created a high demand for the secure of multimedia communication
over the Internet. A major challenge is to protect confidentiality for the informa
tion in the networks. Modern ciphers like RC4 [
Parallel computations are used to make encryption quicker and less expensive. Parallel encryption algorithm for multi-core processor based on logistic map are presented in [9, 10, 11, 12, 13]. In this study, parallel calculations with multicore CPU is used to gain both time execution and security level of the designed encryption scheme. Inspired from [14], the main contributions of our study can be summarized as follows: • we present novel pseudo-random byte generation scheme based on Tinkerbell and Ikeda chaotic functions, which has good statistical characteristics; • we design novel parallel encryption algorithm with high level of security. 2. New pseudo-random byte generator based on Tinkerbell and
The Tinkerbell map [15] is a two-dimensional discrete-time dynamical system given by:
yn+1 = 2xnyn – cxn + dyn, where a = 0.9, b = 0.6013, c = 2.0 and d = 0.50.
xn+1 = x2n – y2n + axn + byn zn+1 = 1 + u(cos tn + wnsin tn) wn+1 = u(znsin tn + wncos tn)
The two-dimensional Ikeda chaotic function is given by [16, 17]: where u is a parameter u = 0.7941 and 6 tn = 0.4 – –1–+––z–2n–+–w––2n– (1) (2) (3) (4) (5)
The proposed generator consists of the next steps: 1. The initial values x0, y0, z0, and w0 from Tinkerbell and Ikeda chaotic maps are determined. 2. The two chaotic maps are iterated for J and K times, respectively. 3. The iteration continues, and as a result, four real fractions are generated and postprocessed as positive bytes. 4. Perform XOR operation between xi, yi, zi, and wi to get a single output byte. 5. Return to Step 2 until the byte output is reached.
In case of serial encryption, the initial key is a set of all initial values of the proposed pseudo-random generator. There are four 64 bits floating-point values by 51 significant bits each [18]. The initial seed of the proposed byte generation algorithm is 204 bits. This is suficiently high against exhaustive search [19].
In order to estimate pseudo-randomness of the output bytes, we used the
statistical programs NIST [
The NIST application consists 15 statistical tests: frequency, block frequency, cumulative sums forward and reverse, runs, longest run of ones, rank, spectral, non-overlapping templates, overlapping templates, universal, approximate entropy, serial first and second, linear complexity, random excursion, and random excursion variant. For the NIST tests, we generated 1000 diferent byte streams of length 125,000 bytes each. The outputs from the tests are given in Table 1.
The minimum pass rate for each statistical test with the exception of the random excursion (variant) test is approximately = 980 for a sample size = 1000 byte streams. The minimum pass rate for the random excursion (variant) test is approximately = 613 for a sample size = 627 byte streams. The designed pseudorandom byte generation algorithm passed successfully all the NIST statistical tests.
The PractRand application [
P-value R= +0.0 R= -4.4 R= -1.4 R= -2.8 R= +2.1 R= -3.3 R= -3.3 R= -1.8 R= +0.8 R= +1.4 R= -0.4 R= -2.0 R= +0.0 R= +1.9 R= +0.9 P-value R= +0.0 R= +0.0 R= +0.4 R= +0.7 R= -2.6 R= -1.4 R= -3.0 R= -1.2 R= -1.8 R= +0.0 R= -1.5 R= -0.0 R= +0.0 R= -5.2
Pass rate “pass” p = 0.970 p = 0.715 p = 0.873 p = 0.190 p = 0.923 p = 0.925 p = 0.770 p = 0.335 p = 0.252 p = 0.499 p = 0.933
“pass” p = 0.171 p = 0.270 Pass rate “pass” p = 0.474 p = 0.401 p = 0.363 p = 0.868 p = 0.696 p = 0.925 p = 0.664 p = 0.906
“pass” p = 0.940 p = 0.493
“pass” p =1-9.6e-3
The ENT software [
Result 7.999998 bits per byte OC would reduce the size of this 125000000-byte file by 0 % For 125000000 samples is 262.30, and randomly would exceed this value 36.33 % of the time 127.5048 (127.5 = random) 3.3.141320114 (error 0.01 %) -0.0000091 (totally uncorrelated = 0.0)
The new pseudo-random byte generator based on Tinkerbell and Ikeda chaotic functions is used in the construction of the parallel encryption algorithm. The proposed parallel encryption algorithm consists of the next steps: 1. The input data file is divided into p fragments of the available cores (threads). 2. The initial values of p pseudo-random generators (from Section 2.2) based on Tinkerbell and Ikeda chaotic functions, 1 ≤ j ≤ p, x0j, y0j, z0j, and w0j are determined. 3. The p diferent chaotic pseudo-random generators are iterated each one on a thread, and as a result p diferent pseudo-random bytes are obtained simultaneously. 4. Perform XOR operation between the p pseudo-random bytes and p bytes from each part of the fragmented input data file. 5. Return to Step 3 until the input byte parts are encrypted.
6. The encrypted parts from all threads are merged in output file.
In case of parallel encryption, the initial key space is a set of p diferent ini tial keys of p byte generation algorithms, Section 2.2. Consequently, the overall key space of the parallel scheme is p × 204 bits. For example, when 8 threads are used, the initial seed will be equal to 1632 bits.
For the performance evaluation in the parallel case, tests with diferent thread
numbers are performed. The laptop CPU in the experiment is Intel(R) Core(TM)
i7-8550U CPU@1.80 GHz 2.00 GHz, 4 cores, 8 threads, hyper-threading
activated; RAM: 8 GB; cache: 8 MB. The speed-up S(p) on p threads of the presented
parallel algorithm is given by the equation [
T(1) E = ––––– (6)
T(p)´
Table 5 presents the encryption results and speed-up as the number of threads increase. The encryption is conducted with p initial keys, as p separate parts are encrypted with a diferent key. Apparently, that with 8 cores the proposed algo rithm achieves its maximum speed-up. The experiments were performed with 9.92 MB data file.
The results of the security and execution outputs are summarized in Table 6. Using the given experimental findings, we can complete that the novel parallel scheme, based on the Tinkerbell and Ikeda maps, has acceptable statistical properties and provide suficient data security. One such use of our novel algorithm in the image and audio encryption techniques. More blocks of the encryption image can be processed at the same time. Another possible application is the real-time selective video encryption. More sensitive layers can be encrypted in parallel with our proposed scheme.
We have proposed a novel pseudo-random byte generation scheme based on the Tinkerbell and Ikeda functions. The parallel algorithm is created with a data parallel method. The experimental outputs show that the parallel encryption presents more eficient outputs against the serial execution of the algorithm.
This work is partially supported by the Scientific research fund of University of Shumen, Bulgaria, under the grant No. RD-08-110/22.02.2022.
scheme for sensor data using a novel logarithmic chaotic map, Journal of Information Security and Applications 47 (2019) 320–328. doi:10.1016/j. jisa.2019.05.017. [8] P. R. Krishna, C. Surya Teja, R. D. S., T. V., A chaos based image encryption using tinkerbell map functions, in: 2018 Second International Conference on Electronics, Communication and Aerospace Technology (ICECA), 2018, pp. 578–582. doi:10.1109/ICECA.2018.8474891. [9] J. Liu, D. Song, Y. Xu, A parallel encryption algorithm for dual-core processor based on chaotic map, in: S. S. Mahmoud, Z. Zeng, Y. Li (Eds.), Fourth International Conference on Machine Vision (ICMV 2011): Computer Vision and Image Analysis; Pattern Recognition and Basic Technologies, volume 8350, International Society for Optics and Photonics, SPIE, 2012, pp. 73 – 79. doi:10.1117/12.920226. [10] W. Wang, X. Wang, D. Song, A parallel chaotic cryptosystem for dual-core processor, in: The 2nd International Conference on Information Science and Engineering, 2010, pp.920–923. doi:10.1109/ICISE.2010.5689747. [11] J. Liu, H. Zhang, D. Song, G. Sun, W. Bi, M. K. Buza, A parallel encryption algorithm of the logistic map for multicore with openmp, in: Ifost, volume 2, 2013, pp. 47–50. doi:10.1109/IFOST.2013.6616857. [12] D. Burak, Parallelization of the block encryption algorithm based on logistic map, Przegląd Elektrotechniczny 88 (2012) 198–200. [13] M. J. Rostami, A. Shahba, S. Saryazdi, H. Nezamabadi-pour, A novel parallel image encryption with chaotic windows based on logistic map, Computers & Electrical Engineering 62 (2017) 384–400. doi:10.1016/j.compeleceng.2017.04.004. [14] Ü. Çavuşoğlu, S. Kaçar, A novel parallel image encryption algorithm based on chaos, Cluster Computing 22 (2019) 1211–1223. doi:10.1007/s10586018-02895-w. [15] K. T. Alligood, T. D. Sauer, J. A. Yorke, D. Chillingworth, Chaos: an introduction to dynamical systems, Springer, New York, 1996. [16] K. Ikeda, Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system, Optics Communications 30 (1979) 257–261. doi:10.1016/0030-4018(79)90090-7. [17] K. Ikeda, H. Daido, O. Akimoto, Optical turbulence: Chaotic behavior of transmitted light from a ring cavity, Physical Review Letters 45 (1980) 709–712. doi:10.1103/PhysRevLett.45.709. [18] F.-P. W. Group, IEEE standard for floating-point arithmetic, IEEE Std 754-2019 (Revision of IEEE 754-2008) (2019) 1–84. doi:10.1109/ IEEESTD.2019.8766229. [19] G. Alvarez, S. Li, Some basic cryptographic requirements for chaos-based cryptosystems, International Journal of Bifurcation and Chaos 16 (2006)