Geodesic
Distance between vectorial Boolean functions and affine analogues (following the Eighth International Olympiad in Cryptography)
Matematičeskie voprosy kriptografii, Tome 15 (2024) no. 1, pp. 127-142 Cet article a éte moissonné depuis la source Math-Net.Ru

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We study the Hamming distance from a vectorial Boolean function to a set of affine mappings (the nonlinearity of a vectorial function). New upper bound on the nonlinearity of vectorial functions and lower bound on the nonlinearity of mappings with a given differential uniformity are obtained, which refine the previously known ones. The dependence of the Hamming distance between a vectorial function and an affine mapping on the Walsh – Hadamard coefficients of nonzero linear combinations of coordinates of the vectorial function is found, which makes it possible to give estimates of nonlinearity in terms of these coefficients. 
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V. G. Ryabov. Distance between vectorial Boolean functions and affine analogues (following the Eighth International Olympiad in Cryptography). Matematičeskie voprosy kriptografii, Tome 15 (2024) no. 1, pp. 127-142. http://geodesic.mathdoc.fr/item/MVK_2024_15_1_a6/

[1] Gorshkov S. P., Dvinyaninov A. V., “Nizhnyaya i verkhnyaya otsenki poryadka affinnosti preobrazovanii prostranstv bulevykh vektorov”, Prikl. diskr. matem., 2(20) (2013), 14–18 | Zbl

[2] Ryabov V. G., “O priblizhenii vektornykh funktsii nad konechnymi polyami i ikh ogranichenii na lineinye mnogoobraziya affinnymi analogami”, Diskretnaya matematika, 34:2 (2022), 83–105 | DOI

[3] Ryabov V. G., “K voprosu o priblizhenii vektornykh funktsii nad konechnymi polyami affinnymi analogami”, Matematicheskie voprosy kriptografii, 13:4 (2022), 125–146 | DOI | MR | Zbl

[4] Ryabov V.G., “O chisle podstanovok vektornogo prostranstva nad konechnym polem, imeyuschikh affinnye priblizheniya zadannoi tochnosti”, Mater. XIV Mezhdunar. sem. “Diskretnaya matematika i ee prilozheniya” imeni akademika O.B. Lupanova, IPM im. M.V. Keldysha RAN, M., 2022, 276–279

[5] Ryabov V. G., “Novye granitsy nelineinosti PN-funktsii i APN-funktsii nad konechnymi polyami”, Diskretnaya matematika, 35:3 (2023), 45–59 | DOI

[6] Solodovnikov V. I., “Bent-funktsii iz konechnoi abelevoi gruppy v konechnuyu abelevu gruppu”, Diskretnaya matematika, 14:1 (2002), 99–113 | DOI | Zbl

[7] Browning K.A., Dillon J.F., McQuistan M.T., Wolfe A.J., “An APN permutation in dimension six”, IX Int. Conf. on finite fields and their appl., Contemp. Math., 518, Amer. Math. Soc., 2010, 33–42 | DOI | MR | Zbl

[8] Carlet C., Boolean functions for cryptography and coding theory, Cambridge Univ. Press, Cambridge, 2021, 574 pp. | MR | Zbl

[9] Carlet C., “Bounds on the nonlinearity of differentially uniform functions by means of their image set size, and on their distance to affine functions”, IEEE Trans. Inf. Theory, 67:12 (2021), 8325–8334 | DOI | MR | Zbl

[10] Carlet C., Ding C., “Nonlinearities of S-boxes”, Finite Fields and Their Appl., 13:1 (2007), 121–135 | DOI | MR | Zbl

[11] Carlet C., Ding C., Yuan J., “Linear codes from perfect nonlinear mappings and their secret sharing schemes”, IEEE Trans. Inf. Theory, 51:6 (2005), 2089–2102 | DOI | MR | Zbl

[12] Chen L., Fu F., “On the nonlinearity of multi-output Boolean functions”, Acta Sci. Natur. Univ. Nankaiensis, 34:4 (2001), 28–33

[13] Gorodilova A.A., Tokareva N.N., Agievich S.V. et al., “An overview of the Eight International Olympiad in Cryptography “Non-Stop University CRYPTO””, Sib. elektr. matem. izv., 19:1 (2022), A.9–A.37 | MR

[14] Liu J., Chen L., “On nonlinearity of the second type of multi-output Boolean functions”, Chinese J. Engin. Math., 31:1 (2014), 9–22 | DOI | MR | Zbl

[15] Liu J., Mesnager S., Chen L., “On the nonlinearity of $S$-boxes and linear codes”, Cryptogr. Communic., 9:1 (2017), 345–361 | DOI | MR | Zbl

[16] Nyberg K., “Perfect nonlinear S-boxes”, EUROCRYPT 1991, Lect. Notes Comput. Sci., 547, 1991, 378–386 | DOI | MR | Zbl

[17] Nyberg K., “On the construction of highly nonlinear permutations”, EUROCRYPT 1992, Lect. Notes Comput. Sci., 658, 1993, 92–98 | DOI | MR | Zbl

[18] Nyberg K., “Differentially uniform mappings for cryptography”, EUROCRYPT 1993, Lect. Notes Comput. Sci., 765, 1994, 54–64 | MR

[19] Ryabov V. G., “Kharakteristiki nelineinosti vektornykh funktsii nad konechnymi polyami”, Matematicheskie voprosy kriptografii, 14:2 (2023), 123–136 | DOI | MR | Zbl

[20] Ryabov V. G., “Nelineinost APN-funktsii: sravnitelnyi analiz i otsenki”, Prikladnaya diskretnaya matematika, 61 (2023), 15–27

[21] Nagy G. P., Thin Sidon sets and the nonlinearity of vectorial Boolean functions, 2022, 12 pp., arXiv: 2212.05887