{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,2]],"date-time":"2025-12-02T18:45:11Z","timestamp":1764701111781,"version":"3.37.3"},"reference-count":42,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2024,2,21]],"date-time":"2024-02-21T00:00:00Z","timestamp":1708473600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,2,21]],"date-time":"2024-02-21T00:00:00Z","timestamp":1708473600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100001870","name":"Fundacja na rzecz Nauki Polskiej","doi-asserted-by":"publisher","award":["MAB\/2018\/5"],"award-info":[{"award-number":["MAB\/2018\/5"]}],"id":[{"id":"10.13039\/501100001870","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Cryptogr. Commun."],"published-print":{"date-parts":[[2024,7]]},"abstract":"Abstract<\/jats:title>In this paper, we address an open problem posed by Bai and Xia in [2]. We study polynomials of the form $$f(x)=x^{4q+1}+\\lambda _1x^{5q}+\\lambda _2x^{q+4}$$<\/jats:tex-math>\n \n f<\/mml:mi>\n \n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n =<\/mml:mo>\n \n x<\/mml:mi>\n \n 4<\/mml:mn>\n q<\/mml:mi>\n +<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n +<\/mml:mo>\n \n \u03bb<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n \n x<\/mml:mi>\n \n 5<\/mml:mn>\n q<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n +<\/mml:mo>\n \n \u03bb<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n \n x<\/mml:mi>\n \n q<\/mml:mi>\n +<\/mml:mo>\n 4<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> over the finite field $${\\mathbb F}_{5^{k}}$$<\/jats:tex-math>\n \n F<\/mml:mi>\n \n 5<\/mml:mn>\n k<\/mml:mi>\n <\/mml:msup>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, which are not quasi-multiplicative equivalent to any of the known permutation polynomials in the literature. We find necessary and sufficient conditions on $$\\lambda _1, \\lambda _2 \\in {\\mathbb F}_{5^{k}}$$<\/jats:tex-math>\n \n \n \u03bb<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \n \u03bb<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n \u2208<\/mml:mo>\n \n F<\/mml:mi>\n \n 5<\/mml:mn>\n k<\/mml:mi>\n <\/mml:msup>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> so that f<\/jats:italic>(x<\/jats:italic>) is a permutation monomial, binomial, or trinomial of $${\\mathbb F}_{5^{2k}}$$<\/jats:tex-math>\n \n F<\/mml:mi>\n \n 5<\/mml:mn>\n \n 2<\/mml:mn>\n k<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>.<\/jats:p>","DOI":"10.1007\/s12095-024-00705-2","type":"journal-article","created":{"date-parts":[[2024,2,21]],"date-time":"2024-02-21T11:02:26Z","timestamp":1708513346000},"page":"825-841","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Complete characterization of a class of permutation trinomials in characteristic five"],"prefix":"10.1007","volume":"16","author":[{"given":"Markus","family":"Grassl","sequence":"first","affiliation":[]},{"given":"Ferruh","family":"\u00d6zbudak","sequence":"additional","affiliation":[]},{"given":"Buket","family":"\u00d6zkaya","sequence":"additional","affiliation":[]},{"given":"Burcu G\u00fclmez","family":"Tem\u00fcr","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,2,21]]},"reference":[{"key":"705_CR1","unstructured":"Akbary, A., Wang, Q., On polynomials of the form $$x^rf(x^{(q-1)\/l})$$, Int. J. Math. Sci., Art. ID 23408 (2007)"},{"key":"705_CR2","doi-asserted-by":"publisher","first-page":"1023","DOI":"10.1007\/s12095-017-0263-4","volume":"10","author":"T Bai","year":"2018","unstructured":"Bai, T., Xia, Y.: A new class of permutation trinomials constructed from Niho exponents. Cryptogr. Commun. 10, 1023\u20131036 (2018)","journal-title":"Cryptogr. Commun."},{"key":"705_CR3","doi-asserted-by":"publisher","first-page":"30","DOI":"10.1016\/j.ffa.2018.03.003","volume":"52","author":"D Bartoli","year":"2018","unstructured":"Bartoli, D.: On a conjecture about a class of permutation trinomials. Finite Fields Appl. 52, 30\u201350 (2018)","journal-title":"Finite Fields Appl."},{"key":"705_CR4","doi-asserted-by":"crossref","unstructured":"Bartoli, D., Hasse-Weil type theorems and relevant classes of polynomial functions, in K. Dabrowski, M. Gadouleau, N. Georgiou, M. Johnson, G. Mertzios, D. Paulusma (Eds.), Surveys in Combinatorics 2021 (London Mathematical Society Lecture Note Series, 43\u2013102), Cambridge University Press (2021)","DOI":"10.1017\/9781009036214.003"},{"key":"705_CR5","doi-asserted-by":"crossref","unstructured":"Bartoli, D., Bonini, M., A short note on polynomials $$f(X)=X+AX^{1+q^2(q-1)\/4}+BX^{1+3q^2(q-1)\/4}\\in F_{q^2}[X]$$, $$q$$ even, J. Alg. Appl. 22, no. 07, 2350144 (2023)","DOI":"10.1142\/S021949882350144X"},{"key":"705_CR6","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.ffa.2018.01.001","volume":"51","author":"D Bartoli","year":"2018","unstructured":"Bartoli, D., Giulietti, M.: Permutation polynomials, fractional polynomials, and algebraic curves. Finite Fields Appl. 51, 1\u201316 (2018)","journal-title":"Finite Fields Appl."},{"key":"705_CR7","doi-asserted-by":"publisher","DOI":"10.1016\/j.ffa.2020.101781","volume":"70","author":"D Bartoli","year":"2021","unstructured":"Bartoli, D., Timpanella, M.: A family of permutation trinomials over $$\\mathbb{F} _{q^2}$$. Finite Fields Appl. 70, 101781 (2021)","journal-title":"Finite Fields Appl."},{"key":"705_CR8","doi-asserted-by":"publisher","first-page":"925","DOI":"10.1007\/s12095-022-00558-7","volume":"14","author":"D Bartoli","year":"2022","unstructured":"Bartoli, D., Timpanella, M.: On a conjecture on APN permutations. Cryptogr. Commun. 14, 925\u2013931 (2022)","journal-title":"Cryptogr. Commun."},{"key":"705_CR9","doi-asserted-by":"crossref","unstructured":"Bosma W., Cannon J., and Playoust C., The Magma algebra system. I. The user language, J. Symbolic Comput. 24, 1179\u20131260 (1997)","DOI":"10.1006\/jsco.1996.0125"},{"key":"705_CR10","doi-asserted-by":"publisher","DOI":"10.1016\/j.ffa.2019.101626","volume":"62","author":"X Cao","year":"2020","unstructured":"Cao, X., Hou, X., Mi, J., Xu, S.: More permutation polynomials with Niho exponents which permute $$\\mathbb{F} _{q^2}$$. Finite Fields Appl. 62, 101626 (2020)","journal-title":"Finite Fields Appl."},{"key":"705_CR11","doi-asserted-by":"publisher","first-page":"605","DOI":"10.1007\/s10623-014-0017-7","volume":"78","author":"F Caullery","year":"2016","unstructured":"Caullery, F., Schmidt, K.-U., Zhou, Y.: Exceptional planar polynomials. Des. Codes Cryptogr. 78, 605\u2013613 (2016)","journal-title":"Des. Codes Cryptogr."},{"key":"705_CR12","doi-asserted-by":"crossref","unstructured":"Cox, D., Little, D., O\u2019Shea, D., Ideals, Varieties, and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra, Undergraduate Texts in Mathematics, Springer, Cham (2015)","DOI":"10.1007\/978-3-319-16721-3"},{"key":"705_CR13","doi-asserted-by":"publisher","first-page":"65","DOI":"10.2307\/1967217","volume":"11","author":"LE Dickson","year":"1896","unstructured":"Dickson, L.E.: The analytic representation of substitutions on a power of a prime number of letters with a discussion of the linear group. Ann. Math. 11, 65\u2013120 (1896)","journal-title":"Ann. Math."},{"issue":"08","key":"705_CR14","doi-asserted-by":"publisher","first-page":"2350163","DOI":"10.1142\/S0219498823501633","volume":"22","author":"R Gupta","year":"2023","unstructured":"Gupta, R., Rai, A.: A note on a class of permutation trinomials. Journal of Algebra and Its Applications 22(08), 2350163 (2023)","journal-title":"Journal of Algebra and Its Applications"},{"key":"705_CR15","doi-asserted-by":"publisher","first-page":"89","DOI":"10.1016\/j.ffa.2016.05.004","volume":"41","author":"R Gupta","year":"2016","unstructured":"Gupta, R., Sharma, R.K.: Some new classes of permutation trinomials over finite fields with even characteristic. Finite Fields Appl. 41, 89\u201396 (2016)","journal-title":"Finite Fields Appl."},{"key":"705_CR16","doi-asserted-by":"publisher","first-page":"65","DOI":"10.1007\/s00200-019-00394-y","volume":"31","author":"R Gupta","year":"2020","unstructured":"Gupta, R., Sharma, R.K.: Determination of a type of permutation binomials and trinomials. Appl. Algebra Engrg. Comm. Comput. 31, 65\u201386 (2020)","journal-title":"Appl. Algebra Engrg. Comm. Comput."},{"key":"705_CR17","unstructured":"Hermite, Ch., Sur les fonctions de sept lettres, C.R. Acad. Sci. Paris 57, 750\u2013757 (1863)"},{"key":"705_CR18","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10711-013-9926-2","volume":"173","author":"F Hernando","year":"2014","unstructured":"Hernando, F., McGuire, G., Monserrat, F.: On the classification of exceptional planar functions over $$\\mathbb{F} _p$$. Geom. Dedicata 173, 1\u201335 (2014)","journal-title":"Geom. Dedicata"},{"key":"705_CR19","doi-asserted-by":"publisher","first-page":"82","DOI":"10.1016\/j.ffa.2014.10.001","volume":"32","author":"X Hou","year":"2015","unstructured":"Hou, X.: Permutation polynomials over finite fields-a survey of recent advances. Finite Fields Appl. 32, 82\u2013119 (2015)","journal-title":"Finite Fields Appl."},{"key":"705_CR20","doi-asserted-by":"publisher","first-page":"16","DOI":"10.1016\/j.ffa.2015.03.002","volume":"35","author":"X Hou","year":"2015","unstructured":"Hou, X.: Determination of a type of permutation trinomials over finite fields. Finite Fields Appl. 35, 16\u201335 (2015)","journal-title":"Finite Fields Appl."},{"key":"705_CR21","doi-asserted-by":"crossref","unstructured":"Hou, X., A survey of permutation binomials and trinomials over finite fields. (English summary) Topics in finite fields, 177\u2013191, Contemp. Math., 632, Amer. Math. Soc., Providence, RI (2015)","DOI":"10.1090\/conm\/632\/12628"},{"key":"705_CR22","doi-asserted-by":"publisher","first-page":"113","DOI":"10.1016\/j.ffa.2018.08.005","volume":"54","author":"X Hou","year":"2018","unstructured":"Hou, X.: Applications of the Hasse-Weil bound to permutation polynomials. Finite Fields Appl. 54, 113\u2013132 (2018)","journal-title":"Finite Fields Appl."},{"key":"705_CR23","doi-asserted-by":"publisher","DOI":"10.1090\/gsm\/190","volume-title":"Lectures on finite fields, Graduate Studies in Mathematics, 190","author":"X Hou","year":"2018","unstructured":"Hou, X.: Lectures on finite fields, Graduate Studies in Mathematics, 190. American Mathematical Society, Providence, RI (2018)"},{"key":"705_CR24","doi-asserted-by":"publisher","first-page":"819","DOI":"10.2307\/2372655","volume":"76","author":"S Lang","year":"1954","unstructured":"Lang, S., Weil, A.: Number of points of varieties in finite fields. Am. J. Math. 76, 819\u2013827 (1954)","journal-title":"Am. J. Math."},{"key":"705_CR25","doi-asserted-by":"publisher","first-page":"69","DOI":"10.1016\/j.ffa.2016.09.002","volume":"43","author":"K Li","year":"2017","unstructured":"Li, K., Qu, L., Chen, X.: New classes of permutation binomials and permutation trinomials over finite fields. Finite Fields Appl. 43, 69\u201385 (2017)","journal-title":"Finite Fields Appl."},{"key":"705_CR26","doi-asserted-by":"publisher","first-page":"2379","DOI":"10.1007\/s10623-017-0452-3","volume":"86","author":"K Li","year":"2018","unstructured":"Li, K., Qu, L., Wang, Q.: New constructions of permutation polynomials of the form $$x^rh(x^{q-1})$$ over $$\\mathbb{F} _{q^2}$$. Des. Codes Cryptogr. 86, 2379\u20132405 (2018)","journal-title":"Des. Codes Cryptogr."},{"key":"705_CR27","doi-asserted-by":"publisher","DOI":"10.1016\/j.ffa.2021.101831","volume":"72","author":"L Li","year":"2021","unstructured":"Li, L., Wang, Q., Xu, Y., Zeng, X.: Several classes of complete permutation polynomials with Niho exponents. Finite Fields Appl. 72, 101831 (2021)","journal-title":"Finite Fields Appl."},{"key":"705_CR28","volume-title":"Finite Fields","author":"R Lidl","year":"1997","unstructured":"Lidl, R., Niederreiter, H.: Finite Fields. Encyclopedia of Mathematics and its Applications), Cambridge University Press, Cambridge (1997)"},{"key":"705_CR29","doi-asserted-by":"publisher","DOI":"10.1201\/b15006","volume-title":"Handbook of Finite Fields, Discrete Mathematics and its Applications","author":"GL Mullen","year":"2013","unstructured":"Mullen, G.L., Panario, D.: Handbook of Finite Fields, Discrete Mathematics and its Applications. Boca Raton), CRC Press, Boca Raton, FL (2013)"},{"key":"705_CR30","doi-asserted-by":"crossref","unstructured":"\u00d6zbudak, F., G\u00fclmez Tem\u00fcr, B.: Classification of permutation polynomials of the form $$x^3g(x^{q-1})$$ of $$\\mathbb{F} _{q^2}$$ where $$g(x)=x^3+bx+c $$ and $$b,c \\in \\mathbb{F}_q^{*}$$. Des. Codes Cryptogr. 90, 1537\u20131556 (2022)","DOI":"10.1007\/s10623-022-01052-0"},{"key":"705_CR31","unstructured":"\u00d6zbudak, F., G\u00fclmez Tem\u00fcr, B., A survey on permutation polynomials over finite fields, to appear in Foundational principles of error-correcting codes and related concepts, Springer Lecture Notes in Mathematics"},{"key":"705_CR32","doi-asserted-by":"publisher","first-page":"67","DOI":"10.1017\/S0004972700019110","volume":"63","author":"YH Park","year":"2001","unstructured":"Park, Y.H., Lee, J.B.: Permutation polynomials and group permutation polynomials. Bull. Austral. Math. Soc. 63, 67\u201374 (2001)","journal-title":"Bull. Austral. Math. Soc."},{"key":"705_CR33","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-76878-4","volume-title":"Algebraic Function Fields and Codes (2nd edition), Graduate Texts in Mathematics, 254","author":"H Stichtenoth","year":"2009","unstructured":"Stichtenoth, H.: Algebraic Function Fields and Codes (2nd edition), Graduate Texts in Mathematics, 254. Springer, Berlin (2009)"},{"key":"705_CR34","doi-asserted-by":"publisher","first-page":"503","DOI":"10.1007\/s10801-013-0496-z","volume":"40","author":"K-U Schmidt","year":"2014","unstructured":"Schmidt, K.-U., Zhou, Y.: Planar functions over fields of characteristic two. J. Algebraic Combin. 40, 503\u2013526 (2014)","journal-title":"J. Algebraic Combin."},{"key":"705_CR35","doi-asserted-by":"publisher","first-page":"563","DOI":"10.1007\/s12095-018-0307-4","volume":"11","author":"Z Tu","year":"2019","unstructured":"Tu, Z., Zeng, X.: A class of permutation trinomials over finite fields of odd characteristic. Cryptogr. Commun. 11, 563\u2013583 (2019)","journal-title":"Cryptogr. Commun."},{"key":"705_CR36","doi-asserted-by":"publisher","first-page":"178","DOI":"10.1016\/j.ffa.2017.11.009","volume":"50","author":"Z Tu","year":"2018","unstructured":"Tu, Z., Zeng, X., Li, C., Helleseth, T.: A class of new permutation trinomials. Finite Fields Appl. 50, 178\u2013195 (2018)","journal-title":"Finite Fields Appl."},{"key":"705_CR37","doi-asserted-by":"publisher","first-page":"149","DOI":"10.1007\/BF01525801","volume":"112","author":"D Wan","year":"1991","unstructured":"Wan, D., Lidl, R.: Permutation polynomials of the form $$x^rf(x^{(q-1)\/d})$$ and their group structure. Monatshefte Math. 112, 149\u2013163 (1991)","journal-title":"Monatshefte Math."},{"key":"705_CR38","doi-asserted-by":"crossref","unstructured":"Wang, Q., Cyclotomic mapping permutation polynomials over finite fields, Sequences, subsequences, and consequences, Lecture Notes in Comput. Sci., 4893, Springer, Berlin, 119\u2013128, (2007)","DOI":"10.1007\/978-3-540-77404-4_11"},{"key":"705_CR39","doi-asserted-by":"crossref","unstructured":"Wang, Q., Polynomials over finite fields: an index approach, in: Combinatorics and Finite Fields, Difference Sets, Polynomials, Pseudorandomness and Applications, De Gruyter, 319\u2013348 (2019)","DOI":"10.1515\/9783110642094-015"},{"key":"705_CR40","doi-asserted-by":"publisher","first-page":"313","DOI":"10.1007\/s12095-018-0291-8","volume":"11","author":"G Wu","year":"2019","unstructured":"Wu, G., Li, N.: Several classes of permutation trinomials over $$\\mathbb{F} _{5^n}$$. Cryptogr. Commun. 11, 313\u2013324 (2019)","journal-title":"Cryptogr. Commun."},{"key":"705_CR41","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.ffa.2018.10.008","volume":"56","author":"D Zheng","year":"2019","unstructured":"Zheng, D., Yuan, Y., Yu, L.: Two types of permutation polynomials with special forms. Finite Fields Appl. 56, 1\u201316 (2019)","journal-title":"Finite Fields Appl."},{"key":"705_CR42","doi-asserted-by":"publisher","first-page":"2209","DOI":"10.1090\/S0002-9939-08-09767-0","volume":"137","author":"ME Zieve","year":"2009","unstructured":"Zieve, M.E.: On some permutation polynomials over $$\\mathbb{F} _q$$ of the form $$x^rh(x^{(q-1)\/d})$$. Proc. Amer. Math. Soc. 137, 2209\u20132216 (2009)","journal-title":"Proc. Amer. Math. Soc."}],"container-title":["Cryptography and Communications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12095-024-00705-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s12095-024-00705-2\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12095-024-00705-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,7,27]],"date-time":"2024-07-27T06:35:14Z","timestamp":1722062114000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s12095-024-00705-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,2,21]]},"references-count":42,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2024,7]]}},"alternative-id":["705"],"URL":"https:\/\/doi.org\/10.1007\/s12095-024-00705-2","relation":{},"ISSN":["1936-2447","1936-2455"],"issn-type":[{"type":"print","value":"1936-2447"},{"type":"electronic","value":"1936-2455"}],"subject":[],"published":{"date-parts":[[2024,2,21]]},"assertion":[{"value":"29 August 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 February 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 February 2024","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"Not applicable.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethics approval and consent to participate"}},{"value":"Not applicable.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Not applicable."}}]}}