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A highly nonlinear S-box based on a fractional linear transformation

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ABSTRACT

We study the structure of an S-box based on a fractional linear transformation applied on the Galois field \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$GF(2^{8})$$\end{document}GF(28). The algorithm followed is very simple and yields an S-box with a very high ability to create confusion in the data. The cryptographic strength of the new S-box is critically analyzed by studying the properties of S-box such as nonlinearity, strict avalanche, bit independence, linear approximation probability and differential approximation probability. We also apply majority logic criterion to determine the effectiveness of our proposed S-box in image encryption applications.

No MeSH data available.


Encryption with the proposed S-box in noisy environments. a Encryption of Fig. 8a. b Encryption of Fig. 8b. c Encryption of Fig. 8c
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Fig9: Encryption with the proposed S-box in noisy environments. a Encryption of Fig. 8a. b Encryption of Fig. 8b. c Encryption of Fig. 8c

Mentions: We may further test the performance of the proposed method in noisy environments. For this purpose, we consider as a bounded rectangular grid. Let and be the true and noisy images, respectively, such that\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} v(i)=u(i)+n(i),\quad i=(i_1,i_2)\in \Omega , \end{aligned}$$\end{document}v(i)=u(i)+n(i),i=(i1,i2)∈Ω,where u(i) and are the intensities of gray level and n(i) is an independent and identically different Gaussian random noise with zero mean and variance at pixel . The continuous image is interpreted as the Shannon interpolation of the discrete grid of samples v(i) over . The goal here is to test the performance of method on noisy imageV in order to analyse the behaviour of proposed method in comparison with its test on the true image U. For this purpose three different noise levels with , 50 and 75 are considered in Fig. 8 to test the significant application of the proposed algorithm. It can be observed that in case of noisy environment slight variations occur in visual quality and quantitative results as shown in Fig. 9 and Table 6. One can see that the entropy level of noise corrupted pixels is decreasing with increase in the level of Gaussian random noise. It shows most of the pixels are adopting similar grey levels in random data instead of particular arrangement of pixels in the original image. The contrast level also decreases with increasing noise level. Similarly changes in other parameters can be observed. The comparative analysis performed by applying AES S-box at the same noise levels is also shown in Table 7 and Fig. 10. One can observe that, with the increase in noise, the visual and numerical results obtained by the newly designed S-box are better or at least pretty similar to the recent state-of-the-art AES S-box (Daemen and Rijmen 2002).


A highly nonlinear S-box based on a fractional linear transformation
Encryption with the proposed S-box in noisy environments. a Encryption of Fig. 8a. b Encryption of Fig. 8b. c Encryption of Fig. 8c
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Related In: Results  -  Collection

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getmorefigures.php?uid=PMC5037109&req=5

Fig9: Encryption with the proposed S-box in noisy environments. a Encryption of Fig. 8a. b Encryption of Fig. 8b. c Encryption of Fig. 8c
Mentions: We may further test the performance of the proposed method in noisy environments. For this purpose, we consider as a bounded rectangular grid. Let and be the true and noisy images, respectively, such that\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} v(i)=u(i)+n(i),\quad i=(i_1,i_2)\in \Omega , \end{aligned}$$\end{document}v(i)=u(i)+n(i),i=(i1,i2)∈Ω,where u(i) and are the intensities of gray level and n(i) is an independent and identically different Gaussian random noise with zero mean and variance at pixel . The continuous image is interpreted as the Shannon interpolation of the discrete grid of samples v(i) over . The goal here is to test the performance of method on noisy imageV in order to analyse the behaviour of proposed method in comparison with its test on the true image U. For this purpose three different noise levels with , 50 and 75 are considered in Fig. 8 to test the significant application of the proposed algorithm. It can be observed that in case of noisy environment slight variations occur in visual quality and quantitative results as shown in Fig. 9 and Table 6. One can see that the entropy level of noise corrupted pixels is decreasing with increase in the level of Gaussian random noise. It shows most of the pixels are adopting similar grey levels in random data instead of particular arrangement of pixels in the original image. The contrast level also decreases with increasing noise level. Similarly changes in other parameters can be observed. The comparative analysis performed by applying AES S-box at the same noise levels is also shown in Table 7 and Fig. 10. One can observe that, with the increase in noise, the visual and numerical results obtained by the newly designed S-box are better or at least pretty similar to the recent state-of-the-art AES S-box (Daemen and Rijmen 2002).

View Article: PubMed Central - PubMed

ABSTRACT

We study the structure of an S-box based on a fractional linear transformation applied on the Galois field \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$GF(2^{8})$$\end{document}GF(28). The algorithm followed is very simple and yields an S-box with a very high ability to create confusion in the data. The cryptographic strength of the new S-box is critically analyzed by studying the properties of S-box such as nonlinearity, strict avalanche, bit independence, linear approximation probability and differential approximation probability. We also apply majority logic criterion to determine the effectiveness of our proposed S-box in image encryption applications.

No MeSH data available.