A Novel Construction of Perfect Strict Avalanche Criterion S-box using Simple Irreducible Polynomials

Alamsyah Alamsyah


An irreducible polynomial is one of the main components in building an S-box with an algebraic technique approach. The selection of the precise irreducible polynomial will determine the quality of the S-box produced. One method for determining good S-box quality is strict avalanche criterion will be perfect if it has a value of 0.5. Unfortunately, in previous studies, the strict avalanche criterion value of the S-box produced still did not reach perfect value. In this paper, we will discuss S-box construction using selected irreducible polynomials. This selection is based on the number of elements of the least amount of irreducible polynomials that make it easier to construct S-box construction. There are 17 irreducible polynomials that meet these criteria. The strict avalanche criterion test results show that the irreducible polynomial p17(x) =x8 + x7 + x6 + x + 1 is the best with a perfect SAC value of 0.5. One indicator that a robust S-box is an ideal strict avalanche criterion value of 0.5


Strict Avalanche Criterion; S-box; Irreducible Polynomial; Algebraic Technique

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Wu, C. K., & Feng, D. (2016). Boolean functions and their applications in cryptography. Springer Berlin Heidelberg.

Hussain, I., & Shah, T. (2013). Literature survey on nonlinear components and chaotic nonlinear components of block ciphers. Nonlinear Dynamics, 74(4), 869-904.

Daemen, J., & Rijmen, V. (2002). The design of Rijndael (Vol. 2). New York: Springer-verlag.

C. Paar and J. Pelzl. (2010). Understanding Cryptography, 1st ed., vol. 1. Springer-Verlag Berlin Heidelberg.

Alamsyah, Bejo, A., & Adji, T. B. (2017, August). AES S-box construction using different irreducible polynomial and constant 8-bit vector. In 2017 IEEE Conference on Dependable and Secure Computing (pp. 366-369). IEEE.

Williams, H., Webster, A., & Tavares, S. (1986). On the design of s-boxes. In Advances in Cryptology—CRYPTO’85 Proceedings (Vol. 218, pp. 523-534).

Girija, R., & Singh, H. (2018). Enhancing security of double random phase encoding based on random S-Box. 3D Research, 9(2), 15.

Farwa, S., Muhammad, N., Shah, T., & Ahmad, S. (2017). A novel image encryption based on algebraic S-box and Arnold transform. 3D Research, 8(3), 26.

Çavuşoğlu, Ü., Kaçar, S., Pehlivan, I., & Zengin, A. (2017). Secure image encryption algorithm design using a novel chaos based S-Box. Chaos, Solitons & Fractals, 95, 92-101.

Hussain, I., Anees, A., AlKhaldi, A. H., Algarni, A., & Aslam, M. (2018). Construction of chaotic quantum magnets and matrix Lorenz systems S-boxes and their applications. Chinese Journal of Physics, 56(4), 1609-1621.

Belazi, A., Khan, M., El-Latif, A. A. A., & Belghith, S. (2017). Efficient cryptosystem approaches: S-boxes and permutation–substitution-based encryption. Nonlinear Dynamics, 87(1), 337-361.

Özkaynak, F., Çelik, V., & Özer, A. B. (2017). A new S-box construction method based on the fractional-order chaotic Chen system. Signal, Image and Video Processing, 11(4), 659-664.

Özkaynak, F., & Yavuz, S. (2013). Designing chaotic S-boxes based on time-delay chaotic system. Nonlinear Dynamics, 74(3), 551-557.

Liu, G., Yang, W., Liu, W., & Dai, Y. (2015). Designing S-boxes based on 3-D four-wing autonomous chaotic system. Nonlinear Dynamics, 82(4), 1867-1877.

Khan, M., Shah, T., Mahmood, H., Gondal, M. A., & Hussain, I. (2012). A novel technique for the construction of strong S-boxes based on chaotic Lorenz systems. Nonlinear Dynamics, 70(3), 2303-2311.

Khan, M., & Shah, T. (2015). An efficient construction of substitution box with the fractional chaotic system. Signal, Image, and Video Processing, 9(6), 1335-1338.

Lambić, D. (2017). A novel method of S-box design based on a discrete chaotic map. Nonlinear Dynamics, 87(4), 2407-2413.

Khan, M., & Azam, N. A. (2015). S-boxes based on affine mapping and orbit of power function. 3D Research, 6(2), 12.

Çavuşoğlu, Ü., Zengin, A., Pehlivan, I., & Kaçar, S. (2017). A novel approach for strong S-Box generation algorithm design based on chaotic scaled Zhongtang system. Nonlinear Dynamics, 87(2), 1081-1094.

Ullah, A., Jamal, S. S., & Shah, T. (2017). A novel construction of substitution box using a combination of chaotic maps with improved chaotic range. Nonlinear Dynamics, 88(4), 2757-2769.

Isa, H., Jamil, N., & Z’aba, M. R. (2016). Construction of cryptographically strong S-Boxes inspired by bee waggle dance. New generation computing, 34(3), 221-238.

Ul Islam, F., & Liu, G. (2017). Designing S-box based on 4D-4wing hyperchaotic system. 3D Research, 8(1), 9.

Hussain, I., Gondal, M. A., & Hussain, A. (2015). Construction of Substitution Box Based on Piecewise Linear Chaotic Map and S 8 Group. 3D Research, 6(1), 3.

Stallings, W. (2014). Cryptography and network security: principles and practice, international edition: principles and practice. Pearson Higher Ed.

Gangadari, B. R., & Ahamed, S. R. (2015, August). Analysis and algebraic construction of S-Box for AES algorithm using irreducible polynomials. In 2015 Eighth International Conference on Contemporary Computing (IC3) (pp. 526-530). IEEE.

Wang, D., & Sun, S. L. (2008, December). Replacement and Structure of S-boxes in Rijndael. In 2008 International Conference on Computer Science and Software Engineering (Vol. 3, pp. 782-784). IEEE.

Alamsyah, Bejo, A., & Adji, T. B. (2018). The replacement of irreducible polynomial and affine mapping for the construction of a strong S-box. Nonlinear Dynamics, 93(4), 2105-2118.

Alamsyah, Bejo, A., & Adji, T. B. (2018, October). S-box Construction of Highly Strict Avalanche Criterion Using Algebraic Technique. In 2018 Third International Conference on Informatics and Computing (ICIC) (pp. 1-4). IEEE.

Alamsyah, Bejo, A., & Adji, T. B. (2019, October). Enhancement strict avalanche criterion value in robust S-boxes construction using selected irreducible polynomial and affine matrixes. In Journal of Physics: Conference Series (Vol. 1321, No. 3, p. 032020). IOP Publishing.

DOI: https://doi.org/10.15294/sji.v7i1.24006


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