Generalizing some properties of differential uniformity and nonlinearity of permutations on finite commutative rings
*Sonu PalCorresponding authorsonupal923@gmail.comDepartment of MathematicsUniversity of DelhiNew Delhi, Delhi, 110007, India0009-0008-9199-9337View full profile → , Anupama Panigrahiapanigrahi@maths.du.ac.inDepartment of MathematicsUniversity of DelhiNew Delhi, Delhi, 110007, India0000-0003-4637-3702View full profile → , Saibal. K. Palskptech@yahoo.comDefence Research and Development OrganizationMetcalfe House, New Delhi, Delhi, 110054, India0000-0003-3297-1605View full profile →
* Corresponding author · click or hover a name for details
- Received:
- 01 Oct 2025
- Published Online:
- 05 Oct 2026
- Article type:
- Research Article
- Language:
- EN
- Article no.:
- JDMSC-2670
- Pages:
- 1–12
Abstract
This study examines two essential characteristics of permutations within algebraic frameworks. The initial emphasis is on the full differential uniformity of permutations within a finite commutative ring possessing unity. We generalize the permutation results examined by Gupta, Mishra, and Gaur [10] in 2021 about Zn to encompass broader finite commutative rings. Expanding upon the research conducted by Canteaut, Duval, and Leurent, [3] in 2015 about differential uniformity bounds for functions derived from Feistel and Misty architectures, we introduce novel bounds for functions formulated using Lai- Massey structures.
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References
[1] T. Beth and C. Ding, “On almost perfect nonlinear permutations,” in Advances in Cryptology—EUROCRYPT’93, Lofthus, Norway, pp. 65-76 (May 1993).
[2] A. Bogdanov, L. R. Knudsen, G. Leander, C. Paar, A. Poschmann, M. J. Robshaw, and C. Vikkelsoe, “PRESENT: An ultra-lightweight block cipher,” in Cryptographic Hardware and Embedded Systems-CHES 2007, Vienna, Austria, pp. 450-466 (Sep. 2007).
[3] A. Canteaut, S. Duval, and G. Leurent, “Construction of Lightweight S-boxes Using Feistel and MISTY Structures,” in International Conference on Selected Areas in Cryptography (SAC 2015), Cham, Switzerland: Springer International Publishing, pp. 373-393 (Aug. 2015).
[4] C. Carlet and C. Ding, “Highly nonlinear mappings,” J. Complexity, vol. 20, no. 2-3, pp. 205-244 (2004).
[5] M. Calderini, M. Sala, and I. Villa, “A note on APN permutations in even dimension,” Finite Fields Appl., vol. 46, pp. 1-16 (2017).
[6] Y. Crama and P. L. Hammer, Eds., Boolean Models and Methods in Mathematics, Computer Science, and Engineering, vol. 2. Cambridge, U.K.: Cambridge Univ. Press (2010).
[7] T. W. Cusick and P. Stanica, Cryptographic Boolean Functions and Applications. Amsterdam, The Netherlands: Academic Press (2017).
[8] J. Daemen and V. Rijmen, “AES proposal: Rijndael,” Nat. Inst. Standards Technol., Comput. Security Resource Center (CSRC) (1999).
[9] K. Gowdhaman, T. Gulliver, C. Mohab, and D. Chinnapillai, “Constacyclic codes over the non-chain finite commutative ring Z4 [u,v]/〈u2–u, v2, uv〉,” J. Discrete Math. Sci. Cryptogr., vol. 27, no. 6, pp. 1867-1885 (2024).
[10] P. Gupta, P. R. Mishra, and A. Gaur, “On differential uniformity and nonlinearity of permutations on Zn,” in Proc. 7th Int. Conf. Mathematics and Computing (ICMC 2021), Singapore, pp. 627-636 (2022).
[11] P. Gupta, P. R. Mishra, and A. Gaur, “Bounds on the maximum nonlinearity of permutations on the rings Zp and Z2p,” Applicable Algebra Eng. Commun. Comput., pp. 1-16 (2023).
[12] Y. Kumar, P. R. Mishra, N. R. Pillai, and R. K. Sharma, “Affine equivalence and non-linearity of permutations over Zn,” Applicable Algebra Eng. Commun. Comput., vol. 28, pp. 257-279 (2017).
[13] Y. Kumar, P. R. Mishra, and R. K. Sharma, “Nonlinearity of k-cycle permutations on Zn,” Asian-Eur. J. Math., vol. 11, no. 02, Art. no. 1850020 (2018).
[14] Y. Li and M. Wang, “Constructing S-boxes for lightweight cryptography with Feistel structure,” in Cryptographic Hardware and Embedded Systems-CHES 2014, pp. 127-146 (Sep. 2014).
[15] O. A. Logachev, A. A. Salnikov, and V. V. Yashchenko, Boolean Functions in Coding Theory and Cryptography, vol. 241. Providence, RI, USA: Amer. Math. Soc. (2012).
[16] P. R. Mishra, I. Gupta, and N. R. Pillai, “Non-linearity and affine equivalence of permutations,” Cryptology ePrint Archive (2014). [Online]. Available: https://ia.cr/2014/974
[17] P. R. Mishra, P. Gupta, and A. Gaur, “On full differential uniformity of permutations on the ring of integers modulo n,” Applicable Algebra Eng. Commun. Comput., pp. 1-19 (2021).
[18] P. R. Mishra, Y. Kumar, N. R. Pillai, and R. K. Sharma, “On non-linearity and affine equivalence of permutations over an arbitrary finite commutative ring with unity,” Cryptologia, vol. 42, no. 1, pp. 81-94 (2018).
[19] K. Nyberg, “Differentially uniform mappings for cryptography,” in Advances in Cryptology—EUROCRYPT’93, Lofthus, Norway, pp. 55-64 (May 1993).
[20] G. Paul and S. Maitra, RC4 Stream Cipher and Its Variants. Boca Raton, FL, USA: CRC Press (2011).
[21] B. Selikh, A. Chillali, D. Mihoubi, and N. Ghadbane, “A new public key cryptosystem based on the non-commutative ring R,” J. Discrete Math. Sci. Cryptogr., vol. 27, no. 1, pp. 75-93 (2024).
[22] B. Zoltak, “VMPC one-way function and stream cipher,” in Fast Software Encryption-FSE 2004, Delhi, India, pp. 210-225 (Feb. 2004).
