{"title":"Inverses for Fourth-Degree Permutation Polynomials Modulo 32Ψ or 96Ψ, with Ψ as a Product of Different Prime Numbers Greater than Three","authors":"L. Trifina, D. Tarniceriu, Ana-Mirela Rotopanescu","doi":"10.3390/appliedmath4010020","DOIUrl":null,"url":null,"abstract":"In this paper, we address the inverse of a true fourth-degree permutation polynomial (4-PP), modulo a positive integer of the form 32kLΨ, where kL∈{1,3} and Ψ is a product of different prime numbers greater than three. Some constraints are considered for the 4-PPs to avoid some complicated coefficients’ conditions. With the fourth- and third-degree coefficients of the form k4,fΨ and k3,fΨ, respectively, we prove that the inverse PP is (I) a 4-PP when k4,f∈{1,3} and k3,f∈{1,3,5,7} or when k4,f=2 and (II) a 5-PP when k4,f∈{1,3} and k3,f∈{0,2,4,6}.","PeriodicalId":503400,"journal":{"name":"AppliedMath","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"AppliedMath","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.3390/appliedmath4010020","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper, we address the inverse of a true fourth-degree permutation polynomial (4-PP), modulo a positive integer of the form 32kLΨ, where kL∈{1,3} and Ψ is a product of different prime numbers greater than three. Some constraints are considered for the 4-PPs to avoid some complicated coefficients’ conditions. With the fourth- and third-degree coefficients of the form k4,fΨ and k3,fΨ, respectively, we prove that the inverse PP is (I) a 4-PP when k4,f∈{1,3} and k3,f∈{1,3,5,7} or when k4,f=2 and (II) a 5-PP when k4,f∈{1,3} and k3,f∈{0,2,4,6}.