{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,18]],"date-time":"2026-01-18T05:43:04Z","timestamp":1768714984997,"version":"3.49.0"},"reference-count":40,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2025,1,14]],"date-time":"2025-01-14T00:00:00Z","timestamp":1736812800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,1,14]],"date-time":"2025-01-14T00:00:00Z","timestamp":1736812800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100001412","name":"Council of Scientific and Industrial Research, India","doi-asserted-by":"publisher","award":["09\/045(1751)\/2019-EMR-I"],"award-info":[{"award-number":["09\/045(1751)\/2019-EMR-I"]}],"id":[{"id":"10.13039\/501100001412","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Cryptogr. Commun."],"published-print":{"date-parts":[[2025,3]]},"DOI":"10.1007\/s12095-025-00775-w","type":"journal-article","created":{"date-parts":[[2025,1,14]],"date-time":"2025-01-14T04:40:11Z","timestamp":1736829611000},"page":"525-540","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A note on QM equivalence of known permutation polynomials"],"prefix":"10.1007","volume":"17","author":[{"given":"Akshay","family":"Ankush Yadav","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Harshdeep","family":"Singh","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Indivar","family":"Gupta","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,1,14]]},"reference":[{"key":"775_CR1","doi-asserted-by":"crossref","unstructured":"Akbary, A., Ghioca, D., Wang, Q.: On permutation polynomials of prescribed shape. Finite Fields Their Appl. 15(2), 195\u2013206 (2009). https:\/\/www.sciencedirect.com\/science\/article\/pii\/S1071579708000749","DOI":"10.1016\/j.ffa.2008.12.001"},{"issue":"1","key":"775_CR2","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 Their Appl. 17(1), 51\u201367 (2011). https:\/\/doi.org\/10.1016\/j.ffa.2010.10.002","journal-title":"Finite Fields Their Appl."},{"key":"775_CR3","doi-asserted-by":"crossref","unstructured":"Bai, T., Xia, Y.: A new class of permutation trinomials constructed from niho exponents. Cryptogr. Commun. 10, (2018)","DOI":"10.1007\/s12095-017-0263-4"},{"key":"775_CR4","doi-asserted-by":"crossref","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, 5898\u20135904 (2013)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"1","key":"775_CR5","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. Discr. Math. 29(1), 79\u201392 (2015). https:\/\/doi.org\/10.1137\/140960153","journal-title":"SIAM J. Discr. Math."},{"issue":"7","key":"775_CR6","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"},{"key":"775_CR7","doi-asserted-by":"crossref","unstructured":"Ding, Z., Zieve, M.E.: Determination of a class of permutation quadrinomials. Proc. Lond. Math. Soc. 127(2), 221\u2013260 (2023). https:\/\/londmathsoc.onlinelibrary.wiley.com\/doi\/abs\/10.1112\/plms.12540","DOI":"10.1112\/plms.12540"},{"key":"775_CR8","doi-asserted-by":"crossref","unstructured":"dong Hou, X.: A class of permutation binomials over finite fields. J. Number Theory 133(10), 3549\u20133558 (2013). https:\/\/www.sciencedirect.com\/science\/article\/pii\/S0022314X13001315","DOI":"10.1016\/j.jnt.2013.04.011"},{"key":"775_CR9","doi-asserted-by":"crossref","first-page":"253","DOI":"10.4064\/aa166-3-3","volume":"166","author":"X dong Hou","year":"2014","unstructured":"dong Hou, X.: Determination of a type of permutation trinomials over finite fields. Acta Arith. 166, 253\u2013278 (2014)","journal-title":"Acta Arith."},{"key":"775_CR10","doi-asserted-by":"crossref","first-page":"16","DOI":"10.1016\/j.ffa.2015.03.002","volume":"35","author":"X dong Hou","year":"2015","unstructured":"dong Hou, X.: Determination of a type of permutation trinomials over finite fields, ii. Finite Fields Their Appl. 35, 16\u201335 (2015). (https:\/\/www.sciencedirect.com\/science\/article\/pii\/S1071579715000337)","journal-title":"Finite Fields Their Appl."},{"key":"775_CR11","doi-asserted-by":"crossref","unstructured":"dong Hou, X., Lappano, S.D.: Determination of a type of permutation binomials over finite fields. J. Number Theory 147, 14\u201323 (2015). https:\/\/www.sciencedirect.com\/science\/article\/pii\/S0022314X14002273","DOI":"10.1016\/j.jnt.2014.06.021"},{"key":"775_CR12","doi-asserted-by":"crossref","unstructured":"dong Hou, X., Pallozzi Lavorante, V.: New results on permutation binomials of finite fields. Finite Fields Their Appl. 88, 102179 (2023). https:\/\/www.sciencedirect.com\/science\/article\/pii\/S1071579723000217","DOI":"10.1016\/j.ffa.2023.102179"},{"key":"775_CR13","doi-asserted-by":"publisher","first-page":"223","DOI":"10.1007\/s10623-019-00680-3","volume":"88","author":"R Gupta","year":"2020","unstructured":"Gupta, R.: Several new permutation quadrinomials over finite fields of odd characteristic. Des. Codes Cryptogr. 88, 223\u2013239 (2020). https:\/\/doi.org\/10.1007\/s10623-019-00680-3","journal-title":"Des. Codes Cryptogr."},{"key":"775_CR14","doi-asserted-by":"publisher","unstructured":"Gupta, R., Rai, A.: A note on a class of permutation trinomials. J. Algebra Appl. 22(08), (2023). https:\/\/doi.org\/10.1142\/S0219498823501633","DOI":"10.1142\/S0219498823501633"},{"key":"775_CR15","doi-asserted-by":"crossref","unstructured":"Gupta, R., Sharma, R.: Some new classes of permutation trinomials over finite fields with even characteristic. Finite Fields Their Appl. 41, 89\u201396 (2016). https:\/\/www.sciencedirect.com\/science\/article\/pii\/S1071579716300211","DOI":"10.1016\/j.ffa.2016.05.004"},{"key":"775_CR16","doi-asserted-by":"crossref","unstructured":"Kyureghyan, G., Zieve, M.: Permutation polynomials of the form $$x + \\gamma \\text{Tr}(x^k)$$. In: Contemporary Developments in Finite Fields and Applications, pp. 178\u2013194, (2016). https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/9789814719261_0011","DOI":"10.1142\/9789814719261_0011"},{"key":"775_CR17","doi-asserted-by":"crossref","unstructured":"Laigle-Chapuy, Y.: Permutation polynomials and applications to coding theory. Finite Fields Their Appl. 13(1), 58\u201370 (2007). https:\/\/www.sciencedirect.com\/science\/article\/pii\/S1071579705000705","DOI":"10.1016\/j.ffa.2005.08.003"},{"key":"775_CR18","doi-asserted-by":"crossref","unstructured":"Lappano, S.D.: A note regarding permutation binomials over fq2. Finite Fields Their Appl. 34, 153\u2013160 (2015). https:\/\/www.sciencedirect.com\/science\/article\/pii\/S1071579715000118","DOI":"10.1016\/j.ffa.2015.01.010"},{"key":"775_CR19","unstructured":"Li, K., Qu, L., Li, C., Fu, S.: New permutation trinomials constructed from fractional polynomials. arXiv:1605.06216, (2016)"},{"key":"775_CR20","doi-asserted-by":"crossref","unstructured":"Li, N.: On two conjectures about permutation trinomials over $$\\mathbb{F}_{3^{2k}}$$. Finite Fields Their Appl. 47, 1\u201310 (2017). https:\/\/www.sciencedirect.com\/science\/article\/pii\/S1071579717300515","DOI":"10.1016\/j.ffa.2017.05.003"},{"key":"775_CR21","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."},{"issue":"5","key":"775_CR22","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(5), 1057\u20131068 (2019). https:\/\/doi.org\/10.1007\/s12095-019-0349-2","journal-title":"Cryptogr. Commun."},{"key":"775_CR23","doi-asserted-by":"crossref","unstructured":"Oliveira, J.A., Brochero Mart\u00ednez, F.: Permutation binomials over finite fields. Discr. Math. 345(3), 112732 (2022). https:\/\/www.sciencedirect.com\/science\/article\/pii\/S0012365X21004453","DOI":"10.1016\/j.disc.2021.112732"},{"key":"775_CR24","doi-asserted-by":"crossref","first-page":"101906","DOI":"10.1016\/j.ffa.2021.101906","volume":"76","author":"T Pang","year":"2021","unstructured":"Pang, T., Xu, Y., Li, N., Zeng, X.: Permutation polynomials of the form $$x^d+l(x^s)$$ over $$\\mathbb{F} _{q^3}$$. Finite Fields Their Appl. 76, 101906 (2021). (https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/9789814719261_0011.)","journal-title":"Finite Fields Their Appl."},{"issue":"1","key":"775_CR25","doi-asserted-by":"crossref","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)","journal-title":"Bull. Aust. Math. Soc."},{"key":"775_CR26","doi-asserted-by":"crossref","unstructured":"Rivest, R.L., Shamir, A., Adelman, L.M.: A method for obtaining digital signatures and public-key cryptosystems. In: Comm. ACM 21, pp. 120\u2013126, (1978)","DOI":"10.1145\/359340.359342"},{"key":"775_CR27","doi-asserted-by":"crossref","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":"775_CR28","doi-asserted-by":"publisher","first-page":"304","DOI":"10.1016\/j.ffa.2017.11.013","volume":"50","author":"Z Tu","year":"2018","unstructured":"Tu, Z., Zeng, X., Helleseth, T.: New permutation quadrinomials over $$\\mathbb{F}_{2^{2m}}$$. Finite Fields Their Appl. 50, 304\u2013318 (2018). https:\/\/doi.org\/10.1016\/j.ffa.2017.11.013","journal-title":"Finite Fields Their Appl."},{"key":"775_CR29","doi-asserted-by":"crossref","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. Monatsh. f\u00fcr Math. 112, 149\u2013163 (1991)","journal-title":"Monatsh. f\u00fcr Math."},{"key":"775_CR30","doi-asserted-by":"publisher","unstructured":"Wang, Q.: Cyclotomic mapping permutation polynomials over finite fields. In: Golomb, S.W., Gong, G., Helleseth, T., Song, H.-Y. (Eds.), Sequences, Subsequences, and Consequences, pp. 119\u2013128, Berlin, Heidelberg, (2007). https:\/\/doi.org\/10.1007\/978-3-540-77404-4_11. Springer Berlin Heidelberg","DOI":"10.1007\/978-3-540-77404-4_11"},{"key":"775_CR31","doi-asserted-by":"crossref","unstructured":"Wang, Q.: Polynomials over finite fields: an index approach, pp. 319\u2013348. De Gruyter, Berlin, Boston, (2019)","DOI":"10.1515\/9783110642094-015"},{"key":"775_CR32","doi-asserted-by":"crossref","unstructured":"Wu, D., Yuan, P., Ding, C., Ma, Y.: Permutation trinomials over $$\\mathbb{F} _{2^m}$$. Finite Fields Their Appl. 46, 38\u201356 (2017). https:\/\/www.sciencedirect.com\/science\/article\/pii\/S107157971730014X","DOI":"10.1016\/j.ffa.2017.03.002"},{"issue":"2","key":"775_CR33","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":"775_CR34","doi-asserted-by":"publisher","first-page":"102414","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 Their Appl. 96, 102414 (2024). https:\/\/doi.org\/10.1016\/j.ffa.2024.102414","journal-title":"Finite Fields Their Appl."},{"key":"775_CR35","doi-asserted-by":"crossref","unstructured":"\u00d6zbudak, F., G\u00fclmez Tem\u00fcr, B.: Classification of some quadrinomials over finite fields of odd characteristic. Finite Fields Their Appl. 87, 102158 (2023). https:\/\/www.sciencedirect.com\/science\/article\/pii\/S1071579722001678","DOI":"10.1016\/j.ffa.2022.102158"},{"issue":"7","key":"775_CR36","doi-asserted-by":"publisher","first-page":"1537","DOI":"10.1007\/s10623-022-01052-0","volume":"90","author":"F \u00d6zbudak","year":"2022","unstructured":"\u00d6zbudak, F., Tem\u00fcr, B.G.: 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(7), 1537\u20131556 (2022). https:\/\/doi.org\/10.1007\/s10623-022-01052-0","journal-title":"Des. Codes Cryptogr."},{"issue":"4","key":"775_CR37","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."},{"key":"775_CR38","doi-asserted-by":"crossref","unstructured":"Zha, Z., Hu, L., Fan, S.: Further results on permutation trinomials over finite fields with even characteristic. Finite Fields Their Appl. 45, 43\u201352 (2017). https:\/\/www.sciencedirect.com\/science\/article\/pii\/S1071579716300880","DOI":"10.1016\/j.ffa.2016.11.011"},{"issue":"05","key":"775_CR39","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"},{"key":"775_CR40","unstructured":"Zieve, M.E.: Permutation polynomials on $$\\mathbb{F}_q$$ induced from bijective redei functions on subgroups of the multiplicative group of $$\\mathbb{F} _q$$. arXiv:1310.0776, (2013)"}],"container-title":["Cryptography and Communications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12095-025-00775-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s12095-025-00775-w\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12095-025-00775-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,3,20]],"date-time":"2025-03-20T13:13:05Z","timestamp":1742476385000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s12095-025-00775-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,1,14]]},"references-count":40,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2025,3]]}},"alternative-id":["775"],"URL":"https:\/\/doi.org\/10.1007\/s12095-025-00775-w","relation":{},"ISSN":["1936-2447","1936-2455"],"issn-type":[{"value":"1936-2447","type":"print"},{"value":"1936-2455","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,1,14]]},"assertion":[{"value":"7 November 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 January 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 January 2025","order":3,"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 competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}]}}