A cryptography using lifting scheme integer wavelet transform over min-max-plus algebra

Yunus, Mahmud ; Firmansyah, Mohamad Ilham Dwi ; Subiono

Kybernetika, Tome 60 (2024) no. 5, pp. 576-602

Abstract (VO)

We propose a cryptographic algorithm utilizing integer wavelet transform via a lifting scheme. In this research, we construct some predict and update operators within the lifting scheme of wavelet transforms employing operations in min-max-plus algebra, termed as lifting scheme integer wavelet transform over min-max-plus algebra (MMPLS-IWavelet). The analysis and synthesis process on MMPLS-IWavelet is implemented for both encryption and decryption processes. The encryption key comprises a sequence of positive integers, where the first element specifies MMPLS-IWavelet type and subsequent elements indicate the levels of each executed transformation. The decryption key involves three components: the original encryption key, a binary encoding of the analyzed signal, and a sequence of non-negative integer representing the length of coefficient signals from the approximation and detail signals. We present a rigorous analysis confirming the correctness of the proposed cryptographic scheme, and evaluate its performance based on various metrics such as correlation value between plaintext and ciphertext, encryption quality, computation time, key sensitivity, entropy analysis, and key space analysis. We also analyze the computational costs of the encryption and decryption processes. The experimental results demonstrate that the proposed algorithms empirically yield satisfactory performance, exhibiting a near zero correlation between plaintext and ciphertext for most of test data, high encryption quality (over 80 percent), substantial key sensitivity, the large key space, and greater randomness in ciphertext compare to plaintext. The algorithm is efficient in terms of computational time and has linear complexity with respect to the number of input characters. The vast key space makes it highly impractical for brute-force approaches to find the decryption key directly.

Détail

DOI : 10.14736/kyb-2024-5-0576

Classification : 15A80, 42C40, 94A60
Keywords: cryptography; lifting scheme; min-max-plus algebra; wavelet

Bibliographie

[1] Arul, J., Venkatesulu, M.: Encryption quality and performance analysis of GKSBC algorithm. J. Inform. Engrg. Appl. 2(10) (2012).

[2] Cahyono, J., Subiono, Adzkiya., D., Davvas, B.: A cryptographic algorithm using wavelet transforms over max-plus algebra. J. King Saud University-Computer Inform. Sci. 34 (2020), 2, 627-635. | DOI

[3] Durcheva, M.: Some applications of idempotent semirings in public key cryptography. ACM Commun. Comput. Algebra 49 (2015), 1, 9. | DOI

[4] Fahim, K., Yunus, M.: Max-plus algebra-based wavelet transforms and their applications in compressed image. Int. J. Tomography Simul. 30 (2017), 1, 118-126.

[5] Fujinoki, K., Ashizawa, K.: Directional Lifting Wavelet Transform for Image Edge Analysis. Signal Process. J. Pre-proof (2023), 118-126. | DOI

[6] Gafsi, M., Abbassi, N., Amdouni, Rim, A., Hajjaji, M. A., Mohamed, A., Mtibaa, A.: Hardware implementation of the Haar 2D discrete wavelet transform with an application to image watermarking. In: 2022 5th International Conference on Advanced Systems and Emergent Technologies (IC_ASET), IEEE 2022, pp. 324-329. | DOI

[7] Gon, A., Mukherjee, A.: FPGA-Based Low-Cost Architecture for R-Peak Detection and Heart-Rate Calculation Using Lifting-Based Discrete Wavelet Transform. Circuits Systems Signal Process. 42 (2022), 1, 580-600. | DOI

[8] Goswami, D., Rahman, N., Biswas, J., Koul, A., Tamang, L. R., Bhattacharjee, A. K.: A discrete wavelet transform based cryptographic algorithm. Int. J. Computer Sci. Network Security 11 (2011), 4, 178-182.

[9] Grigoriev, D., Shpilrain, V.: Tropical cryptography. Commun. Algebra 42 (2014), 6, 2624-2632. | DOI | MR

[10] Heijmans, H. J., Goutsias, J.: Nonlinear multiresolution signal decomposition schemes. II. Morphological wavelets. IEEE Trans. Image Process. 9 (2000), 11, 1897-1913. | DOI | MR

[11] Hellman, M.: New directions in cryptography. IEEE Trans. Inform. Theory 22 (1976), 6, 644-654. | DOI | MR

[12] Kistosil, F., Adzkiya, D., Subiono: Generalized public transportation scheduling using max-plus algebra. Kybernetika 54 (2018), 2, 243-267. | DOI | MR

[13] Klees, R., Haagmans, R.: Wavelets in the Geosciences. Springer, Berlin 1996. | DOI

[14] Mallat, S.: A Wavelet Tour of Signal Processing. Elsevier, New York 2009. | DOI | MR

[15] Merdekawati, D. A., Subiono: Closed Shop Scheduling Optimisation using Max-Plus Automata. J.f Physics: Conference Series 1341 (2019), 4, 042015. | DOI

[16] Naseer, R., Nasim, M., Sohaib, M., Younis, J., Mehmood, A., Alam, M., Massoud, Y.: VLSI architecture design and implementation of 5/3 and 9/7 lifting Discrete Wavelet Transform. Integration 87 (2022), 253-259. | DOI

[17] Nobuhara, H., Trieu, D. B. K., Maruyama, T., Bede, B.: Max-plus algebra-based wavelet transforms and their FPGA implementation for image coding. Inform. Sci. 180 (2010), 12, 3232-3247. | DOI | MR

[18] Parthasarathy, M. B., Srinivasan, B.: Increased security in image cryptography using wavelet transforms. Indian Jo.Sci. Technol. 269 (2014), 21-34. | DOI

[19] Rivest, R. L., Shamir, A., Adleman, L.: A method for obtaining digital signatures and public key cryptosystems. Commun. ACM 21 (1978), 2, 120-126. | DOI | MR

[20] Salehi, S. A., Dhruba, D. D.: Efficient hardware implementation of discrete wavelet transform based on stochastic computing. In: IEEE Computer Society Annual Symposium on VLSI (ISVLSI) 2020, pp. 422-427. | DOI

[21] Silverman, J. H., Pipher, J., Hoffstein, J.: An Introduction to Mathematical Cryptography. Kybernetika, Springer, New York 2008. | DOI | MR

[22] Sweldens, W.: The lifting scheme: A construction of second generation wavelets. SIAM J. Math. Anal. 29 (1998), 2, 511-546. | DOI | MR

[23] Sweldens, W.: ZAMM-Zeitschrift fur Angewandte Mathematik und Mechanik. SIAM J. Math. Anal. 76 (1996), 2, 41-44.

[24] Sweldens, W.: Lifting scheme: a new philosophy in biorthogonal wavelet constructions. Wavelet Appl. Signal Image Process. III 2569 (1995), 68-79. | DOI

[25] Swenson, C.: Modern Cryptanalysis: Techniques for Advanced Code Breaking. John Wiley and Sons, Indianapolis 2008.

[26] Szczesna, A., Switonski, A., Slupik, J., Zghidi, J. H., Josinski, H., Wojciechowski, K.: Quaternion lifting scheme applied to the classification of motion data. Inform. Sci. 575 (2021), 732-746. | DOI | MR

[27] Tao, Y., Wang, C.: Global optimization for max-plus linear systems and applications in distributed systems. Automatica 119 (2020), 109104. | DOI | MR

[28] Tedmori, S., Al-Najdawi, N.: Image cryptographic algorithm based on the Haar wavelet transform. Inform. Sci. 269 (2014), 21-34. | DOI | MR

[29] Ukasha, A.: Double compression efficiency for image data hiding using integer wavelet transform. In: International Conference on Engineering and MIS (ICEMIS), 2022, pp. 1-7. | DOI

[30] Walpole, R.: Introduction to Statistics. New York 1974.

[31] Zarkar, S., Vaidya, S., Bharambe, A., Tadvi, A., Chavan, T.: Secure server verification by using encryption algorithm and visual cryptography. Int. J. Sci. Res. (IJSR) (2014), 310-313.

[32] Zhang, H., Tao, Y., Yuegang., Zhang, Z.: Strong solvability of interval max-plus systems and applications to optimal control. Systems Control Lett. 96 (2016), 88-94. | DOI | MR