{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,11]],"date-time":"2026-06-11T05:22:41Z","timestamp":1781155361981,"version":"3.54.1"},"reference-count":113,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2021,12,3]],"date-time":"2021-12-03T00:00:00Z","timestamp":1638489600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2021,12,3]],"date-time":"2021-12-03T00:00:00Z","timestamp":1638489600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Cryptogr. Commun."],"published-print":{"date-parts":[[2022,7]]},"DOI":"10.1007\/s12095-021-00551-6","type":"journal-article","created":{"date-parts":[[2021,12,3]],"date-time":"2021-12-03T09:02:58Z","timestamp":1638522178000},"page":"691-735","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":33,"title":["Survey on recent trends towards generalized differential and boomerang uniformities"],"prefix":"10.1007","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4008-2031","authenticated-orcid":false,"given":"Sihem","family":"Mesnager","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Bimal","family":"Mandal","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mounira","family":"Msahli","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2021,12,3]]},"reference":[{"issue":"1","key":"551_CR1","doi-asserted-by":"publisher","first-page":"51","DOI":"10.1016\/j.ffa.2010.10.002","volume":"17","author":"A Akbary","year":"2011","unstructured":"Akbary, A., Ghioca, D., Wang, Q.: On constructing permutations of finite fields. Finite Fields Applic. 17(1), 51\u201367 (2011)","journal-title":"Finite Fields Applic."},{"key":"551_CR2","doi-asserted-by":"publisher","first-page":"187","DOI":"10.1007\/s10801-019-00899-2","volume":"52","author":"D Bartoli","year":"2020","unstructured":"Bartoli, D., Timpanella, M.: On a generalization of planar functions. Journal of Algebraic Combinatorics 52, 187\u2013213 (2020)","journal-title":"Journal of Algebraic Combinatorics"},{"key":"551_CR3","doi-asserted-by":"crossref","unstructured":"Bartoli, D., Calderini, M.: On construction and (non)existence of $$c$$-(almost) perfect nonlinear functions. arXiv:2008.039536v1 (2020)","DOI":"10.1016\/j.ffa.2021.101835"},{"key":"551_CR4","doi-asserted-by":"crossref","unstructured":"Bartoli, D., Calderini, M., Riera, C., St\u0103nic\u0103, P.: Low c-differential uniformity for functions modified on subfields. In The 6th International Workshop on Boolean Functions and their Applications BFA (2021)","DOI":"10.1007\/s12095-022-00554-x"},{"key":"551_CR5","doi-asserted-by":"publisher","first-page":"313","DOI":"10.1007\/978-3-030-17653-2_11","volume":"11476","author":"A Bar-On","year":"2019","unstructured":"Bar-On, A., Dunkelman, O., Keller, N., Weizman, A.: DLCT: a new tool for differential-linear cryptanalysis. Eurocrypt 2019, LNCS 11476, 313\u2013342 (2019)","journal-title":"Eurocrypt 2019, LNCS"},{"key":"551_CR6","doi-asserted-by":"publisher","first-page":"1133","DOI":"10.1007\/s12095-020-00434-2","volume":"12","author":"C Beierle","year":"2020","unstructured":"Beierle, C., Leander, G.: $$4$$-uniform permutations with null nonlinearity. Cryptogr. Commun. 12, 1133\u20131141 (2020)","journal-title":"Cryptogr. Commun."},{"key":"551_CR7","doi-asserted-by":"publisher","first-page":"553","DOI":"10.1016\/S0019-9958(67)91016-9","volume":"10","author":"ER Berlekamp","year":"1967","unstructured":"Berlekamp, E.R., Rumsey, H., Solomon, G.: On the solutions of algebraic equations over finite fields. Information and Control 10, 553\u2013564 (1967)","journal-title":"Information and Control"},{"issue":"9","key":"551_CR8","doi-asserted-by":"publisher","first-page":"4160","DOI":"10.1109\/TIT.2006.880036","volume":"52","author":"T Berger","year":"2006","unstructured":"Berger, T., Canteaut, A., Charpin, P., Laigle-Chapuy, Y.: On almost perfect nonlinear functions over $$\\mathbb{F}_{2^m}$$. IEEE Transactions on Information Theory 52(9), 4160\u20134170 (2006)","journal-title":"IEEE Transactions on Information Theory"},{"key":"551_CR9","doi-asserted-by":"crossref","unstructured":"Biryukov, A.: The boomerang attack on 5 and 6-round reduced AES. AES4, LNCS 3373, 11\u201315 (2004)","DOI":"10.1007\/11506447_2"},{"key":"551_CR10","doi-asserted-by":"crossref","unstructured":"Biryukov, A., Nikoli\u0107, I., Roy, A.: Boomerang attacks on BLAKE-32. FSE 2011, LNCS 6733, 218\u2013237 (2011)","DOI":"10.1007\/978-3-642-21702-9_13"},{"key":"551_CR11","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1007\/BF00630563","volume":"4","author":"E Biham","year":"1991","unstructured":"Biham, E., Shamir, A.: Differential cryptanalysis of DES-like cryptosystems. Journal of Cryptology 4, 3\u201372 (1991)","journal-title":"Journal of Cryptology"},{"key":"551_CR12","doi-asserted-by":"crossref","unstructured":"Biham, E., Shamir, A.: Differential cryptanalysis of the full 16-round DES. Crypto 1992, LNCS 740, 487\u2013496 (1992)","DOI":"10.1007\/3-540-48071-4_34"},{"key":"551_CR13","doi-asserted-by":"crossref","unstructured":"Biham, E., Dunkelman, O., Keller, N.: Enhancing differential-linear cryptanalysis. Asiacrypt 2002, LNCS 2501, 254\u2013266 (2002)","DOI":"10.1007\/3-540-36178-2_16"},{"key":"551_CR14","doi-asserted-by":"crossref","unstructured":"Bracken, C., Leander, G.: A highly nonlinear differentially $$4$$ uniform power mapping that permutes fields of even degree Finite Fields Appl. 16, 231\u2013242 (2010)","DOI":"10.1016\/j.ffa.2010.03.001"},{"key":"551_CR15","doi-asserted-by":"crossref","unstructured":"Browning, K.A., Dillon, J.F., McQuistan, M.T., Wolfe, A.J.: An APN permutation in dimension six. Finite Fields Appl. Contemp. Math. Amer. Math. Soc. 518, 3\u201342 (2010)","DOI":"10.1090\/conm\/518\/10194"},{"key":"551_CR16","first-page":"17","volume":"2365","author":"N Borisov","year":"2002","unstructured":"Borisov, N., Chew, M., Johnson, R., Wagner, D.: Multiplicative differentials. FSE, LNCS 2365, 17\u201333 (2002)","journal-title":"FSE, LNCS"},{"issue":"1","key":"551_CR17","doi-asserted-by":"publisher","first-page":"331","DOI":"10.46586\/tosc.v2020.i1.331-362","volume":"2020","author":"H Boukerrou","year":"2020","unstructured":"Boukerrou, H., Huynh, P., Lallemand, V., Mandal, B., Minier, M.: On the feistel counterpart of the boomerang connectivity table introduction and analysis of the FBCT. IACR Transactions on Symmetric Cryptology 2020(1), 331\u2013362 (2020)","journal-title":"IACR Transactions on Symmetric Cryptology"},{"issue":"3","key":"551_CR18","doi-asserted-by":"publisher","first-page":"290","DOI":"10.46586\/tosc.v2018.i3.290-310","volume":"2018","author":"C Boura","year":"2018","unstructured":"Boura, C., Canteaut, A.: On the boomerang uniformity of cryptographic S-boxes. IACR Transactions on Symmetric Cryptology 2018(3), 290\u2013310 (2018)","journal-title":"IACR Transactions on Symmetric Cryptology"},{"key":"551_CR19","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-12991-4","volume-title":"Construction and Analysis of Cryptographic Functions","author":"L Budaghyan","year":"2014","unstructured":"Budaghyan, L.: Construction and Analysis of Cryptographic Functions. Springer, Berlin (2014)"},{"issue":"1","key":"551_CR20","doi-asserted-by":"publisher","first-page":"19","DOI":"10.1007\/s10623-020-00803-1","volume":"89","author":"L Budaghyan","year":"2021","unstructured":"Budaghyan, L., Calderini, M., Carlet, C., Coulter, R.S., Villa, I.: Generalized isotopic shift construction for APN functions. Des. Codes Cryptogr. 89(1), 19\u201332 (2021)","journal-title":"Des. Codes Cryptogr."},{"key":"551_CR21","doi-asserted-by":"publisher","first-page":"1161","DOI":"10.1007\/s12095-020-00439-x","volume":"12","author":"M Calderini","year":"2020","unstructured":"Calderini, M., Villa, I.: On the boomerang uniformity of some permutation polynomials. Cryptography and Communications 12, 1161\u20131178 (2020)","journal-title":"Cryptography and Communications"},{"key":"551_CR22","doi-asserted-by":"publisher","first-page":"33","DOI":"10.1007\/s10623-020-00807-x","volume":"89","author":"M Calderini","year":"2021","unstructured":"Calderini, M.: Differentially low uniform permutations from known $$4$$-uniform functions. Des. Codes Cryptogr. 89, 33\u201352 (2021)","journal-title":"Des. Codes Cryptogr."},{"key":"551_CR23","doi-asserted-by":"crossref","unstructured":"Canteaut, A., Naya-Plasencia, M.: Structural weaknesses of permutations with a low differential uniformity and generalized crooked functions. In Finite Fields: Theory and Applications- FQ9 - Contemporary Mathematics, AMS, number 518, pp. 55\u201371 (2010)","DOI":"10.1090\/conm\/518\/10196"},{"issue":"11","key":"551_CR24","doi-asserted-by":"publisher","first-page":"7575","DOI":"10.1109\/TIT.2017.2676807","volume":"63","author":"A Canteaut","year":"2017","unstructured":"Canteaut, A., Duval, S., Perrin, L.: A generalisation of Dillon\u2019s APN permutation with the best known differential and nonlinear properties for all fields of size $$2^{4k+2}$$. IEEE Transactions on Information Theory 63(11), 7575\u20137591 (2017)","journal-title":"IEEE Transactions on Information Theory"},{"key":"551_CR25","doi-asserted-by":"crossref","unstructured":"Carlet, C.: Boolean functions for cryptography and error correcting codes. Chapter of the monography Boolean Models and Methods in Mathematics, Computer Science, and Engineering, Y. Crama and P. Hammer (eds), Cambridge University Press, pp. 257\u2013397 (2010)","DOI":"10.1017\/CBO9780511780448.011"},{"key":"551_CR26","doi-asserted-by":"crossref","unstructured":"Carlet, C.: Vectorial Boolean Functions for Cryptography . Chapter of the monography \u201cBoolean Models and Methods in Mathematics\u201d. Computer Science, and Engineering, Y. Crama and P. Hammer (eds), Cambridge University Press, pp. 398\u2013469 (2010)","DOI":"10.1017\/CBO9780511780448.012"},{"issue":"9","key":"551_CR27","doi-asserted-by":"publisher","first-page":"6443","DOI":"10.1109\/TIT.2017.2761392","volume":"64","author":"C Carlet","year":"2018","unstructured":"Carlet, C.: Characterizations of the differential uniformity of vectorial functions by the walsh transform. IEEE Trans. Inf. Theory 64(9), 6443\u20136453 (2018)","journal-title":"IEEE Trans. Inf. Theory"},{"key":"551_CR28","doi-asserted-by":"crossref","unstructured":"Carlet, C.: On APN exponents, characterizations of differentially uniform functions by the Walsh transform and related cyclic-difference-set-like structures. Designs, Codes and Cryptography (87), 203\u2013224 (2019)","DOI":"10.1007\/s10623-018-0512-3"},{"key":"551_CR29","volume-title":"Boolean Functions for Cryptography and Coding Theory","author":"C Carlet","year":"2021","unstructured":"Carlet, C.: Boolean Functions for Cryptography and Coding Theory. Cambridge University Press, Cambridge (2021)"},{"key":"551_CR30","doi-asserted-by":"publisher","first-page":"125","DOI":"10.1023\/A:1008344232130","volume":"15","author":"C Carlet","year":"1998","unstructured":"Carlet, C., Charpin, P., Zinoviev, V.: Codes, bent functions and permutations suitable for DES-like cryptosysytem. Designs, Codes and Cryptography 15, 125\u2013156 (1998)","journal-title":"Designs, Codes and Cryptography"},{"key":"551_CR31","doi-asserted-by":"publisher","first-page":"203","DOI":"10.1109\/TIFS.2020.3006399","volume":"16","author":"C Carlet","year":"2021","unstructured":"Carlet, C., de Ch\u00e9risey, E., Guilley, S., Kavut, S., Tang, D.: Intrinsic resiliency of s-boxes against side-channel attacks best and worst scenarios. IEEE Transactions on Information Forensics and Security 16, 203\u2013218 (2021)","journal-title":"IEEE Transactions on Information Forensics and Security"},{"issue":"1","key":"551_CR32","doi-asserted-by":"publisher","first-page":"5","DOI":"10.1007\/s10623-015-0145-8","volume":"78","author":"C Carlet","year":"2016","unstructured":"Carlet, C., Mesnager, S.: Four decades of research on bent functions. Journal Designs, Codes and Cryptography 78(1), 5\u201350 (2016)","journal-title":"Journal Designs, Codes and Cryptography"},{"key":"551_CR33","doi-asserted-by":"crossref","unstructured":"Chabaud, F., Vaudenay, S.: Links between differential and linear cryptanalysis. In Adv. in Crypt \u2013 EUROCRYPT\u2019 94, LNCS 950, pp. 356\u2013365 (1995)","DOI":"10.1007\/BFb0053450"},{"key":"551_CR34","doi-asserted-by":"publisher","first-page":"87","DOI":"10.1515\/9783110621730-007","volume-title":"Finite Fields and their Applications","author":"P Charpin","year":"2020","unstructured":"Charpin, P.: Crooked functions. In: James, A. (ed.) Finite Fields and their Applications, pp. 87\u2013102. De Gruyter, Davis, Berlin, Boston (2020)"},{"key":"551_CR35","unstructured":"Charpin, P.: The Crooked property. 2020. hal-03091422 (2020)"},{"issue":"5","key":"551_CR36","doi-asserted-by":"publisher","first-page":"615","DOI":"10.1016\/j.ffa.2009.07.001","volume":"15","author":"P Charpin","year":"2009","unstructured":"Charpin, P., Kyureghyan, G.: When does $$G(x)+\\gamma \\rm Tr(H(x))$$ permute $$\\mathbb{F}_{p^n}$$. Finite Fields and its Applications 15(5), 615\u2013632 (2009)","journal-title":"Finite Fields and its Applications"},{"key":"551_CR37","doi-asserted-by":"crossref","unstructured":"Cusick, T.W., St\u0103nic\u0103, P.: Cryptographic Boolean functions and applications. Elsevier, Academic Press (2009)","DOI":"10.1016\/B978-0-12-374890-4.00009-4"},{"key":"551_CR38","doi-asserted-by":"crossref","unstructured":"Cid, C., Huang, T., Peyrin, T., Sasaki, Y., Song, L.: Boomerang connectivity table: a new cryptanalysis tool. Eurocrypt 2018, LNCS 10821, 683\u2013714 (2018)","DOI":"10.1007\/978-3-319-78375-8_22"},{"key":"551_CR39","doi-asserted-by":"publisher","first-page":"220","DOI":"10.1016\/j.ffa.2010.12.002","volume":"17","author":"RS Coulter","year":"2011","unstructured":"Coulter, R.S., Matthews, R.W.: On the number of distinct values of a class of functions over a finite field. Finite Fields Applic. 17, 220\u2013224 (2011)","journal-title":"Finite Fields Applic."},{"key":"551_CR40","doi-asserted-by":"publisher","first-page":"167","DOI":"10.1023\/A:1008292303803","volume":"10","author":"RS Coulter","year":"1997","unstructured":"Coulter, R.S., Matthews, R.W.: Planar functions and planes of Lenz-Barlotti class II. Designs, Codes and Cryptography 10, 167\u2013184 (1997)","journal-title":"Designs, Codes and Cryptography"},{"key":"551_CR41","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-04722-4","volume-title":"The Design of Rijndael: AES - The Advanced Encryption Standard","author":"J Daemen","year":"2002","unstructured":"Daemen, J., Rijmen, V.: The Design of Rijndael: AES - The Advanced Encryption Standard. Springer, Heidelberg (2002)"},{"key":"551_CR42","doi-asserted-by":"crossref","unstructured":"Daemen, J., Knudsen, L., Rijmen, V.: The block cipher Square. FSE 1997, LNCS 1267, 149\u2013165 (1997)","DOI":"10.1007\/BFb0052343"},{"key":"551_CR43","doi-asserted-by":"publisher","first-page":"239","DOI":"10.1007\/BF01111042","volume":"103","author":"P Dembowski","year":"1968","unstructured":"Dembowski, P., Ostrom, T.G.: Planes of order n with collineation groups of order $$n^2$$. Mathematische Zeitschrift 103, 239\u2013258 (1968)","journal-title":"Mathematische Zeitschrift"},{"key":"551_CR44","unstructured":"Dillon, J.: Elementary Hadamard difference sets. Ph.D. dissertation, Net. Commun. Lab., Univ. Maryland, College Park, MD, USA (1974)"},{"issue":"1\u20133","key":"551_CR45","doi-asserted-by":"publisher","first-page":"95","DOI":"10.1016\/S0012-365X(02)00606-4","volume":"267","author":"H Dobbertin","year":"2003","unstructured":"Dobbertin, H., Mills, D., Mller, E.N., Pott, A., Willems, W.: APN functions in odd characteristic. Discrete Mathematics 267(1\u20133), 95\u2013112 (2003)","journal-title":"Discrete Mathematics"},{"key":"551_CR46","doi-asserted-by":"crossref","unstructured":"Dunkelman, O., Indesteege, S., Keller, N.: A differential-linear attack on $$12$$-Round Serpent. Indocrypt 2008, LNCS 5365, 308\u2013321 (2008)","DOI":"10.1007\/978-3-540-89754-5_24"},{"key":"551_CR47","doi-asserted-by":"publisher","first-page":"1526","DOI":"10.1016\/j.jcta.2005.10.006","volume":"113","author":"C Ding","year":"2006","unstructured":"Ding, C., Yuan, J.: A new family of skew Paley-Hadamard difference sets. Journal of Combinatorial Theory, Series A 113, 1526\u20131535 (2006)","journal-title":"Journal of Combinatorial Theory, Series A"},{"key":"551_CR48","doi-asserted-by":"crossref","unstructured":"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 functions. In The 6th International Workshop on Boolean Functions and their Applications BFA (2021)","DOI":"10.1007\/s12095-022-00581-8"},{"issue":"9","key":"551_CR49","doi-asserted-by":"publisher","first-page":"5781","DOI":"10.1109\/TIT.2020.2971988","volume":"66","author":"P Ellingsen","year":"2020","unstructured":"Ellingsen, P., Felke, P., Riera, C., St\u0103nic\u0103, P., Tkachenko, A.: $$c$$-differentials, multiplicative uniformity, and (almost) perfect $$c$$-nonlinearity. IEEE Transactions on Information Theory 66(9), 5781\u20135789 (2020)","journal-title":"IEEE Transactions on Information Theory"},{"issue":"2","key":"551_CR50","doi-asserted-by":"publisher","first-page":"864","DOI":"10.1109\/TIT.2007.913505","volume":"54","author":"C Godsil","year":"2008","unstructured":"Godsil, C., Roy, A.: Two characterizations of crooked functions. IEEE Transactions on Information Theory 54(2), 864\u2013866 (2008)","journal-title":"IEEE Transactions on Information Theory"},{"issue":"1","key":"551_CR51","doi-asserted-by":"publisher","first-page":"154","DOI":"10.1109\/TIT.1968.1054106","volume":"14","author":"R Gold","year":"1968","unstructured":"Gold, R.: Maximal recursive sequences with 3-valued recursive cross-correlation functions. IEEE Trans. Inform. Theory 14(1), 154\u2013156 (1968)","journal-title":"IEEE Trans. Inform. Theory"},{"key":"551_CR52","doi-asserted-by":"publisher","first-page":"363","DOI":"10.1007\/s002000050073","volume":"8","author":"T Helleseth","year":"1997","unstructured":"Helleseth, T., Sandberg, D.: Some power mappings with low differential uniformity. Applicable, Algebra in Engineering, Communications and computing 8, 363\u2013370 (1997)","journal-title":"Applicable, Algebra in Engineering, Communications and computing"},{"key":"551_CR53","doi-asserted-by":"publisher","first-page":"475","DOI":"10.1109\/18.748997","volume":"45","author":"T Helleseth","year":"1999","unstructured":"Helleseth, T., Rong, C., Sandberg, D.: New families of almost perfect nonlinear power mappings. IEEE Transactions on Information Theory 45, 475\u2013485 (1999)","journal-title":"IEEE Transactions on Information Theory"},{"key":"551_CR54","doi-asserted-by":"crossref","unstructured":"Huang, T., Tjuawinata, I., Wu, H.: Differential-linear cryptanalysis of ICEPOLE. FSE 2015, LNCS 9054, 243\u2013263 (2015)","DOI":"10.1007\/978-3-662-48116-5_12"},{"key":"551_CR55","doi-asserted-by":"publisher","first-page":"221","DOI":"10.1007\/s10623-020-00812-0","volume":"89","author":"SU Hasan","year":"2021","unstructured":"Hasan, S.U., Pal, M., Riera, C., St\u0103nic\u0103, P.: On the $$c$$-differential uniformity of certain maps over finite fields. Designs, Codes and Cryptography 89, 221\u2013239 (2021)","journal-title":"Designs, Codes and Cryptography"},{"key":"551_CR56","unstructured":"Hasan, S.U., Pal, M., St\u0103nic\u0103, P.: The (generalized) boomerang uniformity of some classes of functions over finite fields. In The 6th International Workshop on Boolean Functions and their Applications BFA (2021)"},{"key":"551_CR57","unstructured":"Hasan, S.U., Pal, M., St\u0103nic\u0103, P.: The $$c$$-differential uniformity and boomerang uniformity of two classes of permutation polynomials. IEEE Transactions on Information Theory (to appear)"},{"key":"551_CR58","doi-asserted-by":"publisher","first-page":"101797","DOI":"10.1016\/j.ffa.2020.101797","volume":"70","author":"KH Kim","year":"2021","unstructured":"Kim, K.H., Choe, J., Mesnager, S.: Solving $$X^{q+1}+X+a=0$$ over finite fields. Finite Fields and Their Applications 70, 101797 (2021)","journal-title":"Finite Fields and Their Applications"},{"key":"551_CR59","doi-asserted-by":"publisher","first-page":"101902","DOI":"10.1016\/j.ffa.2021.101902","volume":"76","author":"KH Kim","year":"2021","unstructured":"Kim, K.H., Choe, J.H., Mesnager, S.: Complete solution over $$\\rm GF({p^n})$$ of the equation $$X^{p^k+1}+X+a=0$$. Finite Fields and Their Applications 76, 101902 (2021)","journal-title":"Finite Fields and Their Applications"},{"key":"551_CR60","doi-asserted-by":"publisher","first-page":"101630","DOI":"10.1016\/j.ffa.2019.101630","volume":"63","author":"KH Kim","year":"2020","unstructured":"Kim, K.H., Mesnager, S.: Solving $$x^{2^k+1}+x+a=0$$ in $$\\rm GF({p^n})$$ with $$\\text{ gcd }(n, k)=1$$. Finite Fields and Their Applications 63, 101630 (2020)","journal-title":"Finite Fields and Their Applications"},{"key":"551_CR61","doi-asserted-by":"crossref","unstructured":"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 (2021) Submitted","DOI":"10.1007\/s10623-022-01047-x"},{"key":"551_CR62","doi-asserted-by":"crossref","unstructured":"Kocher, P.C.: Timing attacks on implementations of Diffie-Hellman, RSA, DSS and other systems. In Advances in Cryptology CRYPTO 96, N. Koblitz (ed.), pp. 104\u2013113 (1996)","DOI":"10.1007\/3-540-68697-5_9"},{"key":"551_CR63","doi-asserted-by":"crossref","unstructured":"Kocher, P., Jae, J., Jun, B.: Dierential power analysis. Annual International Cryptology Conference CRYPTO 1999: Advances in Cryptology, CRYPTO\u2019 99, 388\u2013397 (1999)","DOI":"10.1007\/3-540-48405-1_25"},{"key":"551_CR64","first-page":"112","volume":"2365","author":"L Knudsen","year":"2002","unstructured":"Knudsen, L., Wagner, D.: Integral cryptanalysis (extended abstract) FSE 2002. LNCS 2365, 112\u2013127 (2002)","journal-title":"LNCS"},{"key":"551_CR65","doi-asserted-by":"crossref","unstructured":"Knudsen, L.: Truncated and higher order differentials. FSE 1994, LNCS 1008, 196\u2013211 (1995)","DOI":"10.1007\/3-540-60590-8_16"},{"key":"551_CR66","unstructured":"Knudsen, L.: DEAL\u2013a 128-bit block cipher. Technical report no. 151. Department of Informatics, University of Bergen, Norway"},{"key":"551_CR67","doi-asserted-by":"crossref","unstructured":"Knudsen, L.R.: Partial and higher order differentials and applications to the DES. BRICS Report Series, RS\u201395\u20139 (1995)","DOI":"10.7146\/brics.v2i9.19512"},{"key":"551_CR68","doi-asserted-by":"crossref","unstructured":"Langford, S.K., Hellman, M.E.: Differential-linear cryptanalysis. Crypto 1994, LNCS 839, 17\u201325 (1994)","DOI":"10.1007\/3-540-48658-5_3"},{"key":"551_CR69","doi-asserted-by":"publisher","first-page":"227","DOI":"10.1007\/978-1-4615-2694-0_23","volume":"276","author":"X Lai","year":"1994","unstructured":"Lai, X.: Higher order derivatives and differential cryptanalysis. Communications and Cryptography 276, 227\u2013233 (1994)","journal-title":"Communications and Cryptography"},{"key":"551_CR70","doi-asserted-by":"crossref","unstructured":"Leurent, G.: Improved differential-linear cryptanalysis of 7-round Chaskey with partitioning. Eurocrypt 2016, LNCS 9665, 344\u2013371 (2016)","DOI":"10.1007\/978-3-662-49890-3_14"},{"key":"551_CR71","doi-asserted-by":"publisher","first-page":"115","DOI":"10.1090\/conm\/574\/11419","volume":"574","author":"E Leducq","year":"2012","unstructured":"Leducq, E.: New families of APN functions in characteristic 3 or 5. Contemporary Mathematics 574, 115\u2013123 (2012)","journal-title":"Contemporary Mathematics"},{"key":"551_CR72","doi-asserted-by":"publisher","unstructured":"Li, K., Li, C., Helleseth, T., Qu, L.: Cryptographically strong permutations from the butterfly structure. Designs, Codes and Cryptography 89, 737\u2013761 (2021). https:\/\/doi.org\/10.1007\/s10623-020-00837-5, Version posted in Archive in December 2019: arXiv:1912.02640","DOI":"10.1007\/s10623-020-00837-5"},{"key":"551_CR73","doi-asserted-by":"publisher","first-page":"101690","DOI":"10.1016\/j.ffa.2020.101690","volume":"66","author":"K Li","year":"2020","unstructured":"Li, K., Qu, L., Li, C., Chen, H.: On a conjecture about a class of permutation quadrinomials. Finite Fields and Their Applications 66, 101690 (2020)","journal-title":"Finite Fields and Their Applications"},{"issue":"12","key":"551_CR74","doi-asserted-by":"publisher","first-page":"1915","DOI":"10.1093\/comjnl\/bxaa069","volume":"63","author":"H Li","year":"2020","unstructured":"Li, H., Zhou, Y., Ming, J., Yang, G., Jin, C.: The notion of transparency order revisited. The Computer Journal 63(12), 1915\u20131938 (2020)","journal-title":"The Computer Journal"},{"key":"551_CR75","unstructured":"Lidl, R., Mullen, G.L., Turnwald, G.: Dickson Polynomials (Pitman Monographs and Surveys in Pure and Applied Mathematics). Longman Scientific and Technical, Essex, England, 65 (1993)"},{"key":"551_CR76","unstructured":"NIST Lightweight Cryptography https:\/\/csrc.nist.gov\/projects\/lightweight-cryptography"},{"key":"551_CR77","doi-asserted-by":"publisher","first-page":"16","DOI":"10.1016\/j.ffa.2013.02.004","volume":"22","author":"N Li","year":"2013","unstructured":"Li, N., Helleseth, T., Tang, X.: Further results on a class of permutation polynomials over finite fields. Finite Fields Applic. 22, 16\u201323 (2013)","journal-title":"Finite Fields Applic."},{"issue":"11","key":"551_CR78","doi-asserted-by":"publisher","first-page":"7542","DOI":"10.1109\/TIT.2019.2918531","volume":"65","author":"K Li","year":"2019","unstructured":"Li, K., Qu, L., Sun, B., Li, C.: New results about the boomerang uniformity of permutation polynomials. IEEE Transactions on Information Theory 65(11), 7542\u20137553 (2019)","journal-title":"IEEE Transactions on Information Theory"},{"key":"551_CR79","unstructured":"Li, N., Hu, Z., Xiong, M., Zeng, X.: $$4$$-uniform BCT permutations from generalized buttefly structure. ArXiv:2001.006v1,02\/01\/2020"},{"issue":"7","key":"551_CR80","doi-asserted-by":"publisher","first-page":"4845","DOI":"10.1109\/TIT.2021.3062681","volume":"67","author":"N Li","year":"2021","unstructured":"Li, N., Xiong, M., Zeng, X.: On permutation quadrinomials and $$4$$-uniform BCT. IEEE Transactions on Information Theory 67(7), 4845\u20134855 (2021)","journal-title":"IEEE Transactions on Information Theory"},{"key":"551_CR81","doi-asserted-by":"crossref","unstructured":"Mesnager, S.: Bent Functions\u2013Fundamentals and Results. Springer, Switzerland, pp. 1\u2013544 (2016)","DOI":"10.1007\/978-3-319-32595-8_1"},{"key":"551_CR82","unstructured":"Mesnager, S.: Linear codes from functions. In A Concise Encyclopedia of Coding Theory Press\/Taylor and Francis Group (Publisher) London, New York (94 pages in Chapter 20), W. C. Huffman, J-L Kim and P. Sol\u00e9 (eds) (2021)"},{"key":"551_CR83","unstructured":"Mesnager, S.: Private communication related to French ANR BARRACUDA (2021)"},{"issue":"10","key":"551_CR84","doi-asserted-by":"publisher","first-page":"2233","DOI":"10.1007\/s10623-020-00775-2","volume":"88","author":"S Mesnager","year":"2020","unstructured":"Mesnager, S., Tang, C., Xiong, M.: On the boomerang uniformity of quadratic permutations. Designs, Codes and Cryptography 88(10), 2233\u20132246 (2020)","journal-title":"Designs, Codes and Cryptography"},{"issue":"10","key":"551_CR85","doi-asserted-by":"publisher","first-page":"6916","DOI":"10.1109\/TIT.2021.3081348","volume":"67","author":"S Mesnager","year":"2021","unstructured":"Mesnager, S., Riera, C., Stanica, P., Yan, H., Zhou, Z.: Investigations on $$c$$-(almost) perfect nonlinear functions. IEEE Transactions on Information Theory 67(10), 6916\u20136925 (2021)","journal-title":"IEEE Transactions on Information Theory"},{"issue":"12","key":"551_CR86","doi-asserted-by":"publisher","first-page":"7884","DOI":"10.1109\/TIT.2019.2933832","volume":"65","author":"S Mesnager","year":"2019","unstructured":"Mesnager, S., Qu, L.: On two-to-one mappings over finite fields. IEEE Transaction on Information Theory 65(12), 7884\u20137895 (2019)","journal-title":"IEEE Transaction on Information Theory"},{"key":"551_CR87","doi-asserted-by":"crossref","unstructured":"Matsui, M.: Linear cryptanalysis method for DES cipher. Eurocrypt 1993, LNCS, 765, 386\u2013397 (1993)","DOI":"10.1007\/3-540-48285-7_33"},{"key":"551_CR88","doi-asserted-by":"crossref","unstructured":"Nyberg, K.: Differentially uniform mappings for cryptography. Eurocrypt 1993, Workshop on the Theory and Application of Cryptographic Techniques. Springer, Berlin, Heidelberg, 1993, LNCS, vol. 765, pp. 55\u201364 (1994)","DOI":"10.1007\/3-540-48285-7_6"},{"key":"551_CR89","unstructured":"Nyberg, K.: The extended autocorrelation and Boomerang tables and Links between nonlinearity properties of vectorial Boolean functions. Cryptology ePrint Archive 1381 (2019)"},{"key":"551_CR90","first-page":"215","volume":"231","author":"W Nobauer","year":"1968","unstructured":"Nobauer, W.: Uber eine Klasse von Permutations polynomen und die dadurch dargestellten Gruppen. J. Reine Angew. Math. 231, 215\u2013219 (1968)","journal-title":"J. Reine Angew. Math."},{"key":"551_CR91","doi-asserted-by":"crossref","unstructured":"Perrin, L., Udovenko, A., Biryukov, A.: Cryptanalysis of a Theorem: Decomposing the Only Known Solution to the Big APN Problem. CRYPTO\u201916 (2), pp. 93\u2013122 (2016)","DOI":"10.1007\/978-3-662-53008-5_4"},{"issue":"3","key":"551_CR92","doi-asserted-by":"publisher","first-page":"315","DOI":"10.1007\/BF02125898","volume":"9","author":"L R\u00f3nyai","year":"1989","unstructured":"R\u00f3nyai, L., Szonyi, T.: Planar functions over finite fields. Combinatorica 9(3), 315\u2013320 (1989)","journal-title":"Combinatorica"},{"key":"551_CR93","doi-asserted-by":"publisher","first-page":"300","DOI":"10.1016\/0097-3165(76)90024-8","volume":"20","author":"O Rothaus","year":"1976","unstructured":"Rothaus, O.: On \u2018bent\u2019 functions,. J. Combin. Theory Ser. A 20, 300\u2013305 (1976)","journal-title":"J. Combin. Theory Ser. A"},{"key":"551_CR94","unstructured":"Shannon, C.E.: A mathematical theory of cryptography. Bell System Technical Memo MM 45-110-02, (1945)"},{"key":"551_CR95","unstructured":"Shamir, A.: Impossible differential attacks. Crypto 1998 rump session"},{"key":"551_CR96","unstructured":"St\u0103nic\u0103, P.: Investigations on $$c$$-boomerang uniformity and perfect non-linearity. Discrete Applied Mathematics, 2021. To appear. arxiv.org\/abs\/2004.11859 (2020)"},{"key":"551_CR97","doi-asserted-by":"crossref","unstructured":"St\u0103nic\u0103, P.: Low $$c$$-differential and $$c$$-boomerang uniformity of the swapped inverse function. Discrete Mathematics, 2021.To appear. arXiv:2009.09268 (2020)","DOI":"10.1016\/j.disc.2021.112543"},{"key":"551_CR98","doi-asserted-by":"crossref","unstructured":"St\u0103nic\u0103, P.: Low $$c$$-differential uniformity of the Gold function modified on a subfield. Proc. International Conf. on Security and Privacy, Springer (ICSP 2020), LNEE 744, Springer, pp. 131\u2013137 (2021)","DOI":"10.1007\/978-981-33-6781-4_11"},{"key":"551_CR99","doi-asserted-by":"publisher","DOI":"10.1007\/s00200-021-00520-9","volume-title":"Using double Weil sums in finding the $$c$$-Boomerang Connectivity Table for monomial functions on finite fields","author":"P St\u0103nic\u0103","year":"2021","unstructured":"St\u0103nic\u0103, P.: Using double Weil sums in finding the $$c$$-Boomerang Connectivity Table for monomial functions on finite fields. Applicable Algebra in Engineering, Communication and Computing (2021)"},{"key":"551_CR100","unstructured":"St\u0103ni\u0103, P., Riera, C., Tkachenko, A.: Characters, Weil sums and $$c$$-differential uniformity with an application to the perturbed Gold functions. Cryptography and Communications. To appear. arXiv:2009.07779v1 (2020)"},{"key":"551_CR101","doi-asserted-by":"publisher","first-page":"295","DOI":"10.1007\/s12095-020-00466-8","volume":"13","author":"P St\u0103nic\u0103","year":"2021","unstructured":"St\u0103nic\u0103, P., Geary, A.: The $$c$$-differential behavior of the inverse function under the EA-equivalence. Cryptography and Communications 13, 295\u2013306 (2021)","journal-title":"Cryptography and Communications"},{"key":"551_CR102","doi-asserted-by":"publisher","first-page":"118","DOI":"10.46586\/tosc.v2019.i1.118-141","volume":"1","author":"L Song","year":"2019","unstructured":"Song, L., Qin, X., Hu, L.: Boomerang connectivity table revisited application to SKINNY and AES. IACR Transactions on Symmetric Cryptology 1, 118\u2013141 (2019)","journal-title":"IACR Transactions on Symmetric Cryptology"},{"key":"551_CR103","doi-asserted-by":"crossref","unstructured":"Todo, Y.: Structural evaluation by generalized integral property. Eurocrypt 2015, LNCS 9056, 287\u2013314 (2015)","DOI":"10.1007\/978-3-662-46800-5_12"},{"key":"551_CR104","doi-asserted-by":"crossref","unstructured":"Tang, D., Mandal, B., Maitra, S.: Vectorial Boolean functions with very low differential-linear uniformity using Maiorana\u2013McFarland type construction. Indocrypt 2019, LNCS 11898, 341\u2013360 (2019)","DOI":"10.1007\/978-3-030-35423-7_17"},{"key":"551_CR105","doi-asserted-by":"publisher","first-page":"1959","DOI":"10.1007\/s10623-020-00785-0","volume":"88","author":"S Tian","year":"2020","unstructured":"Tian, S., Boura, C., Perrin, L.: Boomerang uniformity of popular S-box constructions. Designs, Codes and Cryptography 88, 1959\u20131989 (2020)","journal-title":"Designs, Codes and Cryptography"},{"key":"551_CR106","first-page":"156","volume":"1636","author":"D Wagner","year":"1999","unstructured":"Wagner, D.: The boomerang attack. FSE, LNCS 1636, 156\u2013170 (1999)","journal-title":"FSE, LNCS"},{"issue":"1","key":"551_CR107","doi-asserted-by":"publisher","first-page":"142","DOI":"10.46586\/tosc.v2019.i1.142-169","volume":"2019","author":"H Wang","year":"2019","unstructured":"Wang, H., Peyrin, T.: Boomerang switch in multiple rounds. IACR Transactions on Symmetric Cryptology 2019(1), 142\u2013169 (2019)","journal-title":"IACR Transactions on Symmetric Cryptology"},{"key":"551_CR108","unstructured":"Wu, Y., Li, N., Zeng, X.: New P$$c$$N and AP$$c$$N functions over finite fields. arXiv:2010.05396v1 (2020)"},{"key":"551_CR109","doi-asserted-by":"publisher","first-page":"2869","DOI":"10.1007\/s10623-018-0480-7","volume":"86","author":"X Xu","year":"2018","unstructured":"Xu, X., Li, C., Zeng, X., Helleseth, T.: Constructions of complete permutation polynomial. Designs, Codes and Cryptography 86, 2869\u20132892 (2018)","journal-title":"Designs, Codes and Cryptography"},{"issue":"8","key":"551_CR110","doi-asserted-by":"publisher","first-page":"1931","DOI":"10.1007\/s11425-010-3149-x","volume":"53","author":"Z Zha","year":"2010","unstructured":"Zha, Z., Wang, X.: Power functions with low uniformity on odd characteristic finite fields. Science China Mathematics 53(8), 1931\u20131940 (2010)","journal-title":"Science China Mathematics"},{"issue":"7","key":"551_CR111","doi-asserted-by":"publisher","first-page":"4826","DOI":"10.1109\/TIT.2011.2145130","volume":"57","author":"Z Zha","year":"2011","unstructured":"Zha, Z., Wang, X.: Almost perfect nonlinear power functions in odd characteristic. IEEE Transactions on Information Theory 57(7), 4826\u20134832 (2011)","journal-title":"IEEE Transactions on Information Theory"},{"key":"551_CR112","unstructured":"Zha, Z., Hu, L.: Some classes of power functions with low $$c$$-differential uniformity over finite fields. arXiv:2008.12183v1 (2020)"},{"key":"551_CR113","doi-asserted-by":"publisher","first-page":"71","DOI":"10.1007\/s10623-013-9890-8","volume":"75","author":"M Zieve","year":"2015","unstructured":"Zieve, M.: Planar functions and perfect nonlinear monomials over finite fields. Designs, Codes and Cryptography 75, 71\u201380 (2015)","journal-title":"Designs, Codes and Cryptography"}],"container-title":["Cryptography and Communications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12095-021-00551-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s12095-021-00551-6\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12095-021-00551-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,6,7]],"date-time":"2022-06-07T05:13:26Z","timestamp":1654578806000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s12095-021-00551-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,12,3]]},"references-count":113,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2022,7]]}},"alternative-id":["551"],"URL":"https:\/\/doi.org\/10.1007\/s12095-021-00551-6","relation":{},"ISSN":["1936-2447","1936-2455"],"issn-type":[{"value":"1936-2447","type":"print"},{"value":"1936-2455","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,12,3]]},"assertion":[{"value":"5 April 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 November 2021","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 December 2021","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}