Abstract
Differential uniformity of permutation polynomials has been studied intensively in recent years due to the differential cryptanalysis of S-boxes. The boomerang attack is a variant of differential cryptanalysis which combines two differentials for the upper part and the lower part of the block cipher. The boomerang uniformity measures the resistance of block ciphers to the boomerang attack. In this paper, by using the resultant elimination method, we study the boomerang uniformity of normalized permutation polynomials of the low degree over finite fields. As a result, we determine the boomerang uniformity of all normalized permutation polynomials of degree up to six over the finite field \({\mathbb {F}}_{q}\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ballantine, C., Roberts, J.: A simple proof of Rolle’s theorem for finite fields. Am. Math. Mon. 109, 72–74 (2002)
Biham, E., Shamir, A.: Differential cryptanalysis of DES-like cryptosystems. J. Cryptol. 4, 3–72 (1991)
Boura, C., Canteaut, A.: On the boomerang uniformity of cryptographic S-boxes. IACR Trans. Symmetr. Cryptol. 3, 290–310 (2018)
Cid, C., Huang, T., Peyrin, T., Sasaki, Y., Song, L.: Boomerang connectivity table: a new cryptanalysis tool. In: Nielsen, J., Rijmen, V. (eds.) Advances in Cryptology-EUROCRYPT 2018. Lecture Notes in Computer Science, vol. 10821, pp. 683–714. Springer, Berlin (2018)
Dickson, L.E.: Criteria for the irreducibility of functions in a finite field. Bull. Am. Math. Soc. 13, 1–8 (1906)
Feng, G., Rao, T.R.N., Berg, G.A., Zhu, J.: Generalized Bézout’s theorem and its applications in coding theory. IEEE Trans. Inf. Theory 43, 1799–1810 (1997)
Leonard, P.A., Williams, K.S.: Quartics over \({\mathbb{F}}_{2^n}\). Proc. Am. Math. Soc. 36, 347–350 (1972)
Li, J., Chandler, D.B., Xiang, Q.: Permutation polynomials of degree \(6\) or \(7\) over finite fields of characteristic \(2\). Finite Fields Appl. 16, 406–419 (2010)
Li, K., Qu, L., Sun, B., Li, C.: New results about the boomerang uniformity of permutation polynomials. IEEE Trans. Inf. Theory 65, 7542–7553 (2019)
Lidl, R., Niederreiter, H.: Finite Fields. Encyclopedia of Mathematics and Its Applications, vol. 20. Cambridge University Press, Cambridge (1997)
Mesnager, S., Tang, C., Xiong, M.: On the boomerang uniformity of quadratic permutations over \(\mathbb{F}_{2^n}\). Cryptography (2019) (eprint:1812.11812)
Panario, D., Santana, D., Wang, Q.: Ambiguity, deficiency and differential spectrum of normalized permutation polynomials over finite fields. Finite Fields Appl. 47, 330–350 (2017)
Perrin, D.: Algebraic Geometry: An Introduction. Springer, Berlin (2008)
Shallue, C.J., Wanless, I.M.: Permutation polynomials and orthomorphism polynomials of degree six. Finite Fields Appl. 20, 84–92 (2013)
Song, X.: The cubic equations over a field of \(p^k(p>3)\) elements. J. Math. Pract. Theory (Chinese) 4, 33–38 (1986)
Wagner, D.: The boomerang attack. In: Knudsen, L. (ed.) Fast Software Encryption. FSE 1999. Lecture Notes in Computer Science, vol. 1636, pp. 156–170. Springer, Berlin (1999)
Wan, Z.: Lectures on Finite Fields and Galois Rings. World Scientific Publishing Company, Singapore (2003)
Acknowledgements
The authors would like to thank the anonymous referees and Professor Zhengbang Zha for their valuable comments and helpful suggestions which improved both the quality and presentation of this paper. This work was supported in part by the China Scholarship Council, the Fundamental Research Funds for the Central Universities and the Innovation Fund of Xidian University, the National Natural Science Foundation of China (Grant 61972303, 61672414, 61602361), the National Cryptography Development Fund (Grant MMJJ20170113) and the NSERC of Canada.
Author information
Authors and Affiliations
Corresponding authors
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Wang, YP., Wang, Q. & Zhang, WG. Boomerang uniformity of normalized permutation polynomials of low degree. AAECC 31, 307–322 (2020). https://doi.org/10.1007/s00200-020-00431-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00200-020-00431-1