{"title":"Characterization of weakly regular p-ary bent functions of $$\\ell $$ -form","authors":"Jong Yoon Hyun, Jungyun Lee, Yoonjin Lee","doi":"10.1007/s10623-024-01411-z","DOIUrl":null,"url":null,"abstract":"<p>We study the essential properties of weakly regular <i>p</i>-ary bent functions of <span>\\(\\ell \\)</span>-form, where a <i>p</i>-ary function is from <span>\\(\\mathbb {F}_{p^m}\\)</span> to <span>\\(\\mathbb {F}_p\\)</span>. We observe that most of studies on a weakly regular <i>p</i>-ary bent function <i>f</i> with <span>\\(f(0)=0\\)</span> of <span>\\(\\ell \\)</span>-form always assume the <i>gcd-condition</i>: <span>\\(\\gcd (\\ell -1,p-1)=1\\)</span>. We first show that whenever considering weakly regular <i>p</i>-ary bent functions <i>f</i> with <span>\\(f(0) = 0\\)</span> of <span>\\(\\ell \\)</span>-form, we can drop the gcd-condition; using the gcd-condition, we also obtain a characterization of a weakly regular bent function of <span>\\(\\ell \\)</span>-form. Furthermore, we find an additional characterization for weakly regular bent functions of <span>\\(\\ell \\)</span>-form; we consider two cases <i>m</i> being even or odd. Let <i>f</i> be a weakly regular bent function of <span>\\(\\ell \\)</span>-form preserving the zero element; then in the case that <i>m</i> is odd, we show that <i>f</i> satisfies <span>\\(\\gcd (\\ell ,p-1)=2\\)</span>. On the other hand, when <i>m</i> is even and <i>f</i> is also non-regular, we show that <i>f</i> satisfies <span>\\(\\gcd (\\ell ,p-1)=2\\)</span> as well. In addition, we present two explicit families of regular bent functions of <span>\\(\\ell \\)</span>-form in terms of the gcd-condition.</p>","PeriodicalId":11130,"journal":{"name":"Designs, Codes and Cryptography","volume":null,"pages":null},"PeriodicalIF":1.4000,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Designs, Codes and Cryptography","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10623-024-01411-z","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
引用次数: 0
Abstract
We study the essential properties of weakly regular p-ary bent functions of \(\ell \)-form, where a p-ary function is from \(\mathbb {F}_{p^m}\) to \(\mathbb {F}_p\). We observe that most of studies on a weakly regular p-ary bent function f with \(f(0)=0\) of \(\ell \)-form always assume the gcd-condition: \(\gcd (\ell -1,p-1)=1\). We first show that whenever considering weakly regular p-ary bent functions f with \(f(0) = 0\) of \(\ell \)-form, we can drop the gcd-condition; using the gcd-condition, we also obtain a characterization of a weakly regular bent function of \(\ell \)-form. Furthermore, we find an additional characterization for weakly regular bent functions of \(\ell \)-form; we consider two cases m being even or odd. Let f be a weakly regular bent function of \(\ell \)-form preserving the zero element; then in the case that m is odd, we show that f satisfies \(\gcd (\ell ,p-1)=2\). On the other hand, when m is even and f is also non-regular, we show that f satisfies \(\gcd (\ell ,p-1)=2\) as well. In addition, we present two explicit families of regular bent functions of \(\ell \)-form in terms of the gcd-condition.
期刊介绍:
Designs, Codes and Cryptography is an archival peer-reviewed technical journal publishing original research papers in the designated areas. There is a great deal of activity in design theory, coding theory and cryptography, including a substantial amount of research which brings together more than one of the subjects. While many journals exist for each of the individual areas, few encourage the interaction of the disciplines.
The journal was founded to meet the needs of mathematicians, engineers and computer scientists working in these areas, whose interests extend beyond the bounds of any one of the individual disciplines. The journal provides a forum for high quality research in its three areas, with papers touching more than one of the areas especially welcome.
The journal also considers high quality submissions in the closely related areas of finite fields and finite geometries, which provide important tools for both the construction and the actual application of designs, codes and cryptographic systems. In particular, it includes (mostly theoretical) papers on computational aspects of finite fields. It also considers topics in sequence design, which frequently admit equivalent formulations in the journal’s main areas.
Designs, Codes and Cryptography is mathematically oriented, emphasizing the algebraic and geometric aspects of the areas it covers. The journal considers high quality papers of both a theoretical and a practical nature, provided they contain a substantial amount of mathematics.