{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,11]],"date-time":"2026-06-11T09:01:39Z","timestamp":1781168499543,"version":"3.54.1"},"reference-count":36,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2025,6,27]],"date-time":"2025-06-27T00:00:00Z","timestamp":1750982400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,6,27]],"date-time":"2025-06-27T00:00:00Z","timestamp":1750982400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["AAECC"],"published-print":{"date-parts":[[2026,7]]},"DOI":"10.1007\/s00200-025-00696-4","type":"journal-article","created":{"date-parts":[[2025,6,27]],"date-time":"2025-06-27T03:29:41Z","timestamp":1750994981000},"page":"943-968","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Several new classes of permutation polynomials over finite fields $$\\mathbb {F}_{5^{2m}}$$"],"prefix":"10.1007","volume":"37","author":[{"given":"Akshay Ankush","family":"Yadav","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Indivar","family":"Gupta","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Harshdeep","family":"Singh","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Arvind","family":"Yadav","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2025,6,27]]},"reference":[{"issue":"1","key":"696_CR1","doi-asserted-by":"publisher","DOI":"10.1155\/2007\/23408","volume":"2007","author":"A Akbary","year":"2007","unstructured":"Akbary, A., Wang, Q.: On polynomials of the form $$x^rf(x^{\\frac{q-1}{l}})$$. Int. J. Math. Math. Sci. 2007(1), 023408 (2007). https:\/\/doi.org\/10.1155\/2007\/23408","journal-title":"Int. J. Math. Math. Sci."},{"issue":"2","key":"696_CR2","doi-asserted-by":"publisher","first-page":"195","DOI":"10.1016\/j.ffa.2008.12.001","volume":"15","author":"A Akbary","year":"2009","unstructured":"Akbary, A., Ghioca, D., Wang, Q.: On permutation polynomials of prescribed shape. Finite Fields Appl. 15(2), 195\u2013206 (2009). https:\/\/doi.org\/10.1016\/j.ffa.2008.12.001","journal-title":"Finite Fields Appl."},{"issue":"1","key":"696_CR3","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 Appl. 17(1), 51\u201367 (2011). https:\/\/doi.org\/10.1016\/j.ffa.2010.10.002","journal-title":"Finite Fields Appl."},{"key":"696_CR4","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). https:\/\/doi.org\/10.1007\/s12095-017-0263-4","journal-title":"Cryptogr. Commun."},{"key":"696_CR5","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). https:\/\/doi.org\/10.1016\/j.ffa.2018.01.001","journal-title":"Finite Fields Appl."},{"issue":"9","key":"696_CR6","doi-asserted-by":"publisher","first-page":"5898","DOI":"10.1109\/TIT.2013.2260795","volume":"59","author":"C Ding","year":"2013","unstructured":"Ding, C., Helleseth, T.: Optimal ternary cyclic codes from monomials. IEEE Trans. Inf. Theory 59(9), 5898\u20135904 (2013). https:\/\/doi.org\/10.1109\/TIT.2013.2260795","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"7","key":"696_CR7","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 family of skew Hadamard difference sets. J. Comb. Theory Ser. A 113(7), 1526\u20131535 (2006). https:\/\/doi.org\/10.1016\/j.jcta.2005.10.006","journal-title":"J. Comb. Theory Ser. A"},{"issue":"1","key":"696_CR8","doi-asserted-by":"publisher","first-page":"79","DOI":"10.1137\/140960153","volume":"29","author":"C Ding","year":"2015","unstructured":"Ding, C., Qu, L., Wang, Q., Yuan, J., Yuan, P.: Permutation trinomials over finite fields with even characteristic. SIAM J. Discret. Math. 29(1), 79\u201392 (2015). https:\/\/doi.org\/10.1137\/140960153","journal-title":"SIAM J. Discret. Math."},{"issue":"4","key":"696_CR9","doi-asserted-by":"publisher","first-page":"825","DOI":"10.1007\/s12095-024-00705-2","volume":"16","author":"M Grassl","year":"2024","unstructured":"Grassl, M., \u00d6zbudak, F., \u00d6zkaya, B., Tem\u00fcr, B.G.: Complete characterization of a class of permutation trinomials in characteristic five. Cryptogr. Commun. Discrete Struct. Boolean Func. Seq. 16(4), 825\u2013841 (2024). https:\/\/doi.org\/10.1007\/s12095-024-00705-2","journal-title":"Cryptogr. Commun. Discrete Struct. Boolean Func. Seq."},{"issue":"08","key":"696_CR10","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. J. Algebra Appl. 22(08), 2350163 (2023). https:\/\/doi.org\/10.1142\/S0219498823501633","journal-title":"J. Algebra Appl."},{"key":"696_CR11","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.: Some new classes of permutation trinomials over finite fields with even characteristic. Finite Fields Appl. 41, 89\u201396 (2016). https:\/\/doi.org\/10.1016\/j.ffa.2016.05.004","journal-title":"Finite Fields Appl."},{"issue":"3","key":"696_CR12","doi-asserted-by":"publisher","first-page":"253","DOI":"10.4064\/aa166-3-3","volume":"166","author":"XD Hou","year":"2014","unstructured":"Hou, X.D.: Determination of a type of permutation trinomials over finite fields. Acta. Arith. 166(3), 253\u2013278 (2014)","journal-title":"Acta. Arith."},{"key":"696_CR13","doi-asserted-by":"publisher","first-page":"16","DOI":"10.1016\/j.ffa.2015.03.002","volume":"35","author":"XD Hou","year":"2015","unstructured":"Hou, X.D.: Determination of a type of permutation trinomials over finite fields. II. Finite Fields Appl. 35, 16\u201335 (2015). https:\/\/doi.org\/10.1016\/j.ffa.2015.03.002","journal-title":"II. Finite Fields Appl."},{"key":"696_CR14","doi-asserted-by":"publisher","unstructured":"Kyureghyan, G., Zieve, M.: Permutation polynomials of the form $$x + \\gamma \\operatorname{Tr}(x^k)$$. Contemporary developments in finite fields and applications, World Sci. Publ., Hackensack, pp. 178\u2013194, (2016). https:\/\/doi.org\/10.1142\/9789814719261_0011","DOI":"10.1142\/9789814719261_0011"},{"issue":"1","key":"696_CR15","doi-asserted-by":"publisher","first-page":"58","DOI":"10.1016\/j.ffa.2005.08.003","volume":"13","author":"Y Laigle-Chapuy","year":"2007","unstructured":"Laigle-Chapuy, Y.: Permutation polynomials and applications to coding theory. Finite Fields Appl. 13(1), 58\u201370 (2007). https:\/\/doi.org\/10.1016\/j.ffa.2005.08.003","journal-title":"Finite Fields Appl."},{"key":"696_CR16","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.ffa.2017.05.003","volume":"47","author":"N Li","year":"2017","unstructured":"Li, N.: On two conjectures about permutation trinomials over $$\\mathbb{F} _{3^{2k}}$$. Finite Fields Appl. 47, 1\u201310 (2017). https:\/\/doi.org\/10.1016\/j.ffa.2017.05.003","journal-title":"Finite Fields Appl."},{"key":"696_CR17","doi-asserted-by":"publisher","first-page":"693","DOI":"10.1007\/s12095-016-0210-9","volume":"9","author":"N Li","year":"2017","unstructured":"Li, N., Helleseth, T.: Several classes of permutation trinomials from niho exponents. Cryptogr. Commun. 9, 693\u2013705 (2017). https:\/\/doi.org\/10.1007\/s12095-016-0210-9","journal-title":"Cryptogr. Commun."},{"key":"696_CR18","doi-asserted-by":"publisher","first-page":"129","DOI":"10.1007\/s12095-018-0321-6","volume":"11","author":"N Li","year":"2019","unstructured":"Li, N., Helleseth, T.: New permutation trinomials from niho exponents over finite fields with even characteristic. Cryptogr. Commun. 11, 129\u2013136 (2019). https:\/\/doi.org\/10.1007\/s12095-018-0321-6","journal-title":"Cryptogr. Commun."},{"key":"696_CR19","doi-asserted-by":"publisher","first-page":"101","DOI":"10.4064\/aa8461-11-2017","volume":"183","author":"K Li","year":"2018","unstructured":"Li, K., Qu, L., Li, C., Fu, S.: New permutation trinomials constructed from fractional polynomials. Acta. Arith. 183, 101\u2013116 (2018)","journal-title":"Acta. Arith."},{"key":"696_CR20","doi-asserted-by":"publisher","first-page":"261","DOI":"10.1016\/j.ffa.2017.08.003","volume":"48","author":"J Ma","year":"2017","unstructured":"Ma, J., Ge, G.: A note on permutation polynomials over finite fields. Finite Fields Appl. 48, 261\u2013270 (2017). https:\/\/doi.org\/10.1016\/j.ffa.2017.08.003","journal-title":"Finite Fields Appl."},{"issue":"4","key":"696_CR21","doi-asserted-by":"publisher","first-page":"775","DOI":"10.1007\/s12095-023-00641-7","volume":"15","author":"F \u00d6zbudak","year":"2023","unstructured":"\u00d6zbudak, F., Tem\u00fcr, B.G.: Complete characterization of some permutation polynomials of the form $$x^r(1+ax^{s_1(q-1)}+bx^{s_2(q-1)})$$ over $$\\mathbb{F} _{q^2}$$. Cryptogr. Commun. 15(4), 775\u2013793 (2023). https:\/\/doi.org\/10.1007\/s12095-023-00641-7","journal-title":"Cryptogr. Commun."},{"issue":"1","key":"696_CR22","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. Aust. Math. Soc. 63(1), 67\u201374 (2001). https:\/\/doi.org\/10.1017\/S0004972700019110","journal-title":"Bull. Aust. Math. Soc."},{"issue":"2","key":"696_CR23","doi-asserted-by":"publisher","first-page":"120","DOI":"10.1145\/359340.359342","volume":"21","author":"RL Rivest","year":"1978","unstructured":"Rivest, R.L., Shamir, A., Adleman, L.: A method for obtaining digital signatures and public-key cryptosystems. Comm. ACM 21(2), 120\u2013126 (1978). https:\/\/doi.org\/10.1145\/359340.359342","journal-title":"Comm. ACM"},{"key":"696_CR24","doi-asserted-by":"publisher","first-page":"759","DOI":"10.1049\/el:19980569","volume":"34","author":"J Schwenk","year":"1998","unstructured":"Schwenk, J., Huber, K.: Public key encryption and digital signatures based on permutation polynomials. Electron. Lett. 34, 759\u2013760 (1998)","journal-title":"Electron. Lett."},{"key":"696_CR25","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 of $$f(x^{(q-1)\/d})$$ and their group structure. Monatshefte f\u00fcr Math. 112, 149\u2013163 (1991)","journal-title":"Monatshefte f\u00fcr Math."},{"key":"696_CR26","doi-asserted-by":"publisher","first-page":"119","DOI":"10.1007\/978-3-540-77404-4_11","volume":"4893","author":"Q Wang","year":"2007","unstructured":"Wang, Q.: Cyclotomic mapping permutation polynomials over finite fields. Seq. Subseq. Conseq. Springer Berl. Heidelb. 4893, 119\u2013128 (2007). https:\/\/doi.org\/10.1007\/978-3-540-77404-4_11","journal-title":"Seq. Subseq. Conseq. Springer Berl. Heidelb."},{"issue":"6","key":"696_CR27","doi-asserted-by":"publisher","first-page":"479","DOI":"10.1007\/s00200-018-0353-3","volume":"29","author":"Y Wang","year":"2018","unstructured":"Wang, Y., Zha, Z., Zhang, W.: Six new classes of permutation trinomials over $$\\mathbb{F} _{3^{3k}}$$. Appl. Algebra Eng. Commun. Comput. 29(6), 479\u2013499 (2018). https:\/\/doi.org\/10.1007\/s00200-018-0353-3","journal-title":"Appl. Algebra Eng. Commun. Comput."},{"issue":"3","key":"696_CR28","doi-asserted-by":"publisher","first-page":"1946","DOI":"10.1137\/17M1156666","volume":"32","author":"Y Wang","year":"2018","unstructured":"Wang, Y., Zhang, W., Zha, Z.: Six new classes of permutation trinomials over $$\\mathbb{F} _{2^{n}}$$. SIAM J. Discret. Math. 32(3), 1946\u20131961 (2018). https:\/\/doi.org\/10.1137\/17M1156666","journal-title":"SIAM J. Discret. Math."},{"key":"696_CR29","doi-asserted-by":"publisher","first-page":"1057","DOI":"10.1007\/s12095-019-0349-2","volume":"11","author":"L Wang","year":"2019","unstructured":"Wang, L., Wu, B., Yue, X., Zheng, Y.: Further results on permutation trinomials with niho exponents. Cryptogr. Commun. 11, 1057\u20131068 (2019). https:\/\/doi.org\/10.1007\/s12095-019-0349-2","journal-title":"Cryptogr. Commun."},{"issue":"2","key":"696_CR30","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 $$\\mathbb{F} _{5^{n}}$$ from niho exponents. Cryptogr. Commun. 11(2), 313\u2013324 (2019). https:\/\/doi.org\/10.1007\/s12095-018-0291-8","journal-title":"Cryptogr. Commun."},{"key":"696_CR31","doi-asserted-by":"publisher","first-page":"38","DOI":"10.1016\/j.ffa.2017.03.002","volume":"46","author":"D Wu","year":"2017","unstructured":"Wu, D., Yuan, P., Ding, C., Ma, Y.: Permutation trinomials over $$\\mathbb{F} _{2^m}$$. Finite Fields Appl. 46, 38\u201356 (2017)","journal-title":"Finite Fields Appl."},{"key":"696_CR32","doi-asserted-by":"publisher","DOI":"10.1016\/j.ffa.2024.102414","volume":"96","author":"AA Yadav","year":"2024","unstructured":"Yadav, A.A., Gupta, I., Singh, H., Yadav, A.: New classes of permutation trinomials of $$\\mathbb{F} _{2^{2m}}$$. Finite Fields Appl. 96, 102414 (2024). https:\/\/doi.org\/10.1016\/j.ffa.2024.102414","journal-title":"Finite Fields Appl."},{"key":"696_CR33","doi-asserted-by":"publisher","first-page":"43","DOI":"10.1016\/j.ffa.2016.11.011","volume":"45","author":"Z Zha","year":"2017","unstructured":"Zha, Z., Hu, L., Fan, S.: Further results on permutation trinomials over finite fields with even characteristic. Finite Fields Appl. 45, 43\u201352 (2017). https:\/\/doi.org\/10.1016\/j.ffa.2016.11.011","journal-title":"Finite Fields Appl."},{"key":"696_CR34","doi-asserted-by":"publisher","DOI":"10.1016\/j.ffa.2020.101754","volume":"68","author":"L Zheng","year":"2020","unstructured":"Zheng, L., Kan, H., Peng, J.: Two classes of permutation trinomials with niho exponents over finite fields with even characteristic. Finite Fields Appl. 68, 101754 (2020). https:\/\/doi.org\/10.1016\/j.ffa.2020.101754","journal-title":"Finite Fields Appl."},{"key":"696_CR35","unstructured":"Zieve, M.E.: Permutation polynomials on $${\\mathbb{F}}_q$$ induced from bijective redei functions on subgroups of the multiplicative group of $${\\mathbb{F}}_q$$, (2013). arXiv:1310.0776"},{"issue":"05","key":"696_CR36","doi-asserted-by":"publisher","first-page":"851","DOI":"10.1142\/s1793042108001717","volume":"04","author":"ME Zieve","year":"2008","unstructured":"Zieve, M.E.: Some families of permutation polynomials over finite fields. Int. J. Number Theory 04(05), 851\u2013857 (2008). https:\/\/doi.org\/10.1142\/s1793042108001717","journal-title":"Int. J. Number Theory"}],"container-title":["Applicable Algebra in Engineering, Communication and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00200-025-00696-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00200-025-00696-4","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00200-025-00696-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,6,11]],"date-time":"2026-06-11T08:43:39Z","timestamp":1781167419000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00200-025-00696-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,6,27]]},"references-count":36,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2026,7]]}},"alternative-id":["696"],"URL":"https:\/\/doi.org\/10.1007\/s00200-025-00696-4","relation":{},"ISSN":["0938-1279","1432-0622"],"issn-type":[{"value":"0938-1279","type":"print"},{"value":"1432-0622","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,6,27]]},"assertion":[{"value":"12 November 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 April 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"2 May 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 June 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no Conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}},{"value":"All the authors agree with the publication of this manuscript, if accepted.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Consent for publication"}}]}}