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In this paper we prove that the\n c<\/jats:italic>\n -Boomerang Uniformity,\n \n \n $$c \\ne 0$$<\/jats:tex-math>\n \n \n c<\/mml:mi>\n \u2260<\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , for all permutation monomials\n \n \n $$x^d$$<\/jats:tex-math>\n \n \n x<\/mml:mi>\n d<\/mml:mi>\n <\/mml:msup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , where\n \n \n $$d > 1$$<\/jats:tex-math>\n \n \n d<\/mml:mi>\n ><\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n and\n \n \n $$p \\not \\mid d$$<\/jats:tex-math>\n \n \n p<\/mml:mi>\n \u2224<\/mml:mo>\n d<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , is bounded by\n \n \n $$\\left\\{ \\begin{array}{ll} d^2, & c^2 \\ne 1, \\\\ d \\cdot (d - 1), & c = - 1, \\\\ d \\cdot (d - 2), & c = 1 \\end{array} \\right\\} .$$<\/jats:tex-math>\n \n \n \n \n \n \n \n \n \n d<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n ,<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mtd>\n \n \n \n c<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n \u2260<\/mml:mo>\n 1<\/mml:mn>\n ,<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mtd>\n <\/mml:mtr>\n \n \n \n \n d<\/mml:mi>\n \u00b7<\/mml:mo>\n (<\/mml:mo>\n d<\/mml:mi>\n -<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n ,<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mtd>\n \n \n c<\/mml:mi>\n =<\/mml:mo>\n -<\/mml:mo>\n 1<\/mml:mn>\n ,<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mtd>\n <\/mml:mtr>\n \n \n \n \n d<\/mml:mi>\n \u00b7<\/mml:mo>\n (<\/mml:mo>\n d<\/mml:mi>\n -<\/mml:mo>\n 2<\/mml:mn>\n )<\/mml:mo>\n ,<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mtd>\n \n \n c<\/mml:mi>\n =<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:mtd>\n <\/mml:mtr>\n <\/mml:mtable>\n <\/mml:mrow>\n <\/mml:mfenced>\n .<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n Further, we utilize this bound to estimate the\n c<\/jats:italic>\n -boomerang uniformity of a large class of generalized triangular dynamical systems, a polynomial-based approach to describe cryptographic permutations of\n \n \n $$\\mathbb{F}_{q}^{n}$$<\/jats:tex-math>\n \n \n F<\/mml:mi>\n \n q<\/mml:mi>\n <\/mml:mrow>\n n<\/mml:mi>\n <\/mml:msubsup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , including the well-known substitution\u2013permutation network.\n <\/jats:p>","DOI":"10.1007\/s00200-024-00670-6","type":"journal-article","created":{"date-parts":[[2024,10,21]],"date-time":"2024-10-21T08:02:39Z","timestamp":1729497759000},"page":"95-130","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A degree bound for the c-boomerang uniformity of permutation monomials"],"prefix":"10.1007","volume":"37","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5206-6579","authenticated-orcid":false,"given":"Matthias Johann","family":"Steiner","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2024,10,21]]},"reference":[{"key":"670_CR1","doi-asserted-by":"publisher","first-page":"151","DOI":"10.1007\/978-3-030-29962-0_8","volume-title":"ESORICS 2019: 24th European Symposium on Research in Computer Security, Part II. 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