Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Designs, Codes and Cryptography
Published in: Designs, Codes and Cryptography 5/2023

06-01-2023

The differential spectrum and boomerang spectrum of a class of locally-APN functions

Authors: Zhao Hu, Nian Li, Linjie Xu, Xiangyong Zeng, Xiaohu Tang

Published in: Designs, Codes and Cryptography | Issue 5/2023

Login to get access

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

In this paper, we study the boomerang spectrum of the power mapping \(F(x)=x^{k(q-1)}\) over \({\mathbb {F}}_{q^2}\), where \(q=p^m\), p is a prime, m is a positive integer and \(\gcd (k,q+1)=1\). We first determine the differential spectrum of F(x) and show that F(x) is locally-APN. This extends a result of (IEEE Trans. Inf. Theory 57(12):8127-8137, 2011) from \((p,k)=(2,1)\) to general (p, k). We then determine the boomerang spectrum of F(x) by making use of its differential spectrum, which shows that the boomerang uniformity of F(x) is 4 if \(p=2\) and m is odd and otherwise it is 2. Our results not only generalize the results in Hasan et al. (Des Codes Cryptogr 89:2627–2636, 2021) and Yan et al. (Adv Math Commun 16(4):1111–1120, 2022) but also extend the example \(x^{45}\) over \({\mathbb F}_{2^8}\) in Hasan et al. (Des Codes Cryptogr 89:2627–2636, 2021) into an infinite class of power mappings with boomerang uniformity 2.
previous article Grassl–Rötteler cyclic and consta-cyclic MDS codes are generalised Reed–Solomon codes
next article An improved method for predicting truncated multiple recursive generators with unknown parameters
Literature
1.
2.
3.
Blondeau C., Canteaut A., Charpin P.: Differential properties of power functions. Int. J. Inf. Coding Theory 1(2), 149–170 (2010).MathSciNetMATH
4.
Blondeau C., Canteaut A., Charpin P.: Differential properties of \({x\mapsto x^{2^{t}-1}}\). IEEE Trans. Inf. Theory 57(12), 8127–8137 (2011).CrossRefMATH
5.
Boura C., Canteaut A.: On the boomerang uniformity of cryptographic S-boxes. IACR Trans. Symmetric Cryptol. 2018(3), 290–310 (2018).CrossRef
6.
7.
Charpin P., Peng J.: Differential uniformity and the associated codes of cryptographic functions. Adv. Math. Commun. 13(4), 579–600 (2019).MathSciNetCrossRefMATH
8.
Choi S.-T., Hong S., No J.-S., Chung H.: Differential spectrum of some power functions in odd prime characteristic. Finite Fields Appl. 21, 11–29 (2013).MathSciNetCrossRefMATH
9.
Cid C., Huang T., Peyrin T., Sasaki Y., Song L.: Boomerang Connectivity Table: a new cryptanalysis tool. In: Nielsen J.B., Rijmen V. (eds.) EUROCRYPT 2018. LNCS, vol. 10821, pp. 683–714. Springer, Cham (2018).
10.
Dobbertin H., Helleseth T., Kumar P.V., Martinsen H.: Ternary \(m\)-sequences with three-valued cross-correlation function: new decimations of Welch and Niho type. IEEE Trans. Inf. Theory 47(4), 1473–1481 (2001).MathSciNetCrossRefMATH
11.
Eddahmani S., Mesnager S.: Explicit values of the DDT, the BCT, the FBCT, and the FBDT of the inverse, the gold, and the Bracken–Leander S-boxes. Cryptogr. Commun. 14, 1301–1344 (2022).MathSciNetCrossRefMATH
12.
13.
Hasan S.U., Pal M., Stănică P.: The binary Gold function and its c-boomerang connectivity table. Cryptogr. Commun. 14, 1257–1280 (2022).MathSciNetCrossRefMATH
14.
Jiang S., Li K., Li Y., Qu L.: Differential and boomerang spectrums of some power permutations. Cryptogr. Commun. 14, 371–393 (2022).MathSciNetCrossRefMATH
15.
Kim K.H., Mesnager S., Choe J.H., Lee D.N., Lee S., Jo M.C.: On permutation quadrinomials with boomerang uniformity \(4\) and the best-known nonlinearity. Des. Codes Cryptogr. 90, 1437–1461 (2022).MathSciNetCrossRefMATH
16.
Lei L., Ren W., Fan C.: The differential spectrum of a class of power functions over finite fields. Adv. Math. Commun. 15(3), 525–537 (2021).MathSciNetCrossRefMATH
17.
Li K., Qu L., Sun B., Li C.: New results about the boomerang uniformity of permutation polynomials. IEEE Trans. Inf. Theory 65(11), 7542–7553 (2019).MathSciNetCrossRefMATH
18.
Li N., Wu Y., Zeng X., Tang X.: On the differential spectrum of a class of power functions over finite fields. Preprint (2020). arXiv:​2012.​04316.
19.
20.
Li K., Li C., Helleseth T., Qu L.: Cryptographically strong permutations from the butterfly structure. Des. Codes Cryptogr. 89, 737–761 (2021).MathSciNetCrossRefMATH
21.
Li N., Hu Z., Xiong M., Zeng X.: A note on “Cryptographically strong permutations from the butterfly structure’’. Des. Codes Cryptogr. 90, 265–276 (2022).MathSciNetCrossRefMATH
22.
Lidl R., Niederreiter H.: Finite Fields, Encyclopedia of Mathematics, vol. 20. Cambridge University Press, Cambridge (1997).
23.
Man Y., Xia Y., Li C., Helleseth T.: On the differential properties of the power mapping \(x^{p^m+2}\). Finite Fields Appl. 84, 102100 (2022).CrossRefMATH
24.
Mesnager S., Tang C., Xiong M.: On the boomerang uniformity of quadratic permutations. Des. Codes Cryptogr. 88(10), 2233–2246 (2020).MathSciNetCrossRefMATH
25.
Mesnager S., Mandal B., Msahli M.: Survey on recent trends towards generalized differential and boomerang uniformities. Cryptogr. Commun. 14, 691–735 (2022).MathSciNetCrossRefMATH
26.
Nyberg K.: Differentially uniform mappings for cryptography. In: Helleseth T. (ed.) EUROCRYPT 1993, LNCS, vol. 765, pp. 134–144. Springer, Berlin (1994).
27.
28.
Tang C., Ding C., Xiong M.: Codes, differentially \(\delta \)-uniform functions, and \(t\)-designs. IEEE Trans. Inf. Theory 66(6), 3691–3703 (2020).MathSciNetCrossRefMATH
29.
Tu Z., Zeng X.: A class of permutation trinomials over finite fields of odd characteristic. Cryptogr. Commun. 11, 563–583 (2019).MathSciNetCrossRefMATH
30.
31.
Wagner D.: The boomerang attack. In: Knudsen L.R. (ed.) FSE 1999. LNCS, vol. 1636, pp. 156–170. Springer, Berlin (1999).
32.
Xia Y., Zhang X., Li C., Helleseth T.: The differential spectrum of a ternary power mapping. Finite Fields Appl. 64, 101660 (2020).MathSciNetCrossRefMATH
33.
Xiong M., Yan H.: A note on the differential spectrum of a differentially \(4\)-uniform power function. Finite Fields Appl. 48, 117–125 (2017).MathSciNetCrossRefMATH
34.
Xiong M., Yan H., Yuan P.: On a conjecture of differentially \(8\)-uniform power functions. Des. Codes Cryptogr. 86(8), 1601–1621 (2018).MathSciNetCrossRefMATH
35.
Yan H., Li C.: Differential spectra of a class of power permutations with characteristic 5. Des. Codes Cryptogr. 89, 1181–1191 (2021).MathSciNetCrossRefMATH
36.
Yan H., Zhou Z., Wen J., Weng J., Helleseth T., Wang Q.: Differential spectrum of Kasami power permutations over odd characteristic finite fields. IEEE Trans. Inf. Theory 65(10), 6819–6826 (2019).MathSciNetCrossRefMATH
37.
Yan H., Li Z., Song Z., Feng R.: Two classes of power mappings with boomerang uniformity 2. Adv. Math. Commun. 16(4), 1111–1120 (2022).MathSciNetCrossRef
38.
Yan H., Xia Y., Li C., Helleseth T., Xiong M., Luo J.: The differential spectrum of the power mapping \(x^{p^n-3}\). IEEE Trans. Inf. Theory 68(8), 5535–5547 (2022).CrossRefMATH
39.
Yan H., Zhang Z., Li Z.: Boomerang spectrum of a class of power functions. In: 10th International Workshop on Signal Design and Its Applications in Communications (IWSDA), pp. 1–4 (2022).
40.
Yan H., Zhang Z., Zhou Z.: A class of power mappings with low boomerang uniformity, accepted by WAIFI (2022).
41.
Zha Z., Hu L.: The boomerang uniformity of power permutations \(x^{2^{k}-1}\) over \({\mathbb{F}}_{2^n}\). In: Ninth International Workshop on Signal Design and Its Applications in Communications (IWSDA), pp. 1–4 (2019).
Metadata
Title
The differential spectrum and boomerang spectrum of a class of locally-APN functions
Authors
Zhao Hu
Nian Li
Linjie Xu
Xiangyong Zeng
Xiaohu Tang
Publication date
06-01-2023
Publisher
Springer US
Published in
Designs, Codes and Cryptography / Issue 5/2023
Print ISSN: 0925-1022
Electronic ISSN: 1573-7586
DOI
https://doi.org/10.1007/s10623-022-01161-w

Other articles of this Issue 5/2023

Designs, Codes and Cryptography 5/2023 Go to the issue

OriginalPaper

Short attribute-based signatures for arbitrary Turing machines from standard assumptions

OriginalPaper

An improved method for predicting truncated multiple recursive generators with unknown parameters

OriginalPaper

Cyclic codes of length 5p with MDS symbol-pair

OriginalPaper

Polar sampler: A novel Bernoulli sampler using polar codes with application to integer Gaussian sampling

OriginalPaper

-Veronese variety and some applications

OriginalPaper

Constructions of multi-permutation codes correcting a single burst of deletions

Premium Partner

    Image Credits