Finite Fields and Their Applications最新文献

{"title":"On the computation of r-th roots in finite fields","authors":"Gook Hwa Cho ,&nbsp;Soonhak Kwon","doi":"10.1016/j.ffa.2024.102479","DOIUrl":"10.1016/j.ffa.2024.102479","url":null,"abstract":"<div><p>Let <em>q</em> be a power of a prime such that <span><math><mi>q</mi><mo>≡</mo><mn>1</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mi>r</mi><mo>)</mo></math></span>. Let <em>c</em> be an <em>r</em>-th power residue over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>. In this paper, we present a new <em>r</em>-th root formula which generalizes G.H. Cho et al.'s cube root algorithm, and which provides a refinement of Williams' Cipolla-Lehmer based procedure. Our algorithm which is based on the recurrence relations arising from irreducible polynomial <span><math><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>r</mi></mrow></msup><mo>+</mo><msup><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>(</mo><mi>b</mi><mo>+</mo><msup><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mi>r</mi></mrow></msup><mi>r</mi><mo>)</mo><mo>(</mo><mi>x</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span> with <span><math><mi>b</mi><mo>=</mo><mi>c</mi><mo>+</mo><msup><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>r</mi></math></span> requires only <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>log</mi><mo>⁡</mo><mi>q</mi><mo>+</mo><msup><mrow><mi>r</mi></mrow><mrow><mn>4</mn></mrow></msup><mo>)</mo></math></span> multiplications for <span><math><mi>r</mi><mo>&gt;</mo><mn>1</mn></math></span>. The multiplications for computation of the main exponentiation of our algorithm are half of that of the Williams' Cipolla-Lehmer type algorithms.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141959399","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On linear representation, complexity and inversion of maps over finite fields","authors":"Ramachandran Ananthraman ,&nbsp;Virendra Sule","doi":"10.1016/j.ffa.2024.102475","DOIUrl":"10.1016/j.ffa.2024.102475","url":null,"abstract":"<div><p>This paper defines a linear representation for nonlinear maps <span><math><mi>F</mi><mo>:</mo><msup><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>→</mo><msup><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> where <span><math><mi>F</mi></math></span> is a finite field, in terms of matrices over <span><math><mi>F</mi></math></span>. This linear representation of the map <em>F</em> associates a unique number <em>N</em> and a unique matrix <em>M</em> in <span><math><msup><mrow><mi>F</mi></mrow><mrow><mi>N</mi><mo>×</mo><mi>N</mi></mrow></msup></math></span>, called the Linear Complexity and the Linear Representation of <em>F</em> respectively, and shows that the compositional powers <span><math><msup><mrow><mi>F</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msup></math></span> are represented by matrix powers <span><math><msup><mrow><mi>M</mi></mrow><mrow><mi>k</mi></mrow></msup></math></span>. It is shown that for a permutation map <em>F</em> with representation <em>M</em>, the inverse map has the linear representation <span><math><msup><mrow><mi>M</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span>. This framework of representation is extended to a parameterized family of maps <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>λ</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mi>F</mi><mo>→</mo><mi>F</mi></math></span>, defined in terms of a parameter <span><math><mi>λ</mi><mo>∈</mo><mi>F</mi></math></span>, leading to the definition of an analogous linear complexity of the map <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>λ</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math></span>, and a parameter-dependent matrix representation <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>λ</mi></mrow></msub></math></span> defined over the univariate polynomial ring <span><math><mi>F</mi><mo>[</mo><mi>λ</mi><mo>]</mo></math></span>. Such a representation leads to the construction of a parametric inverse of such maps where the condition for invertibility is expressed through the unimodularity of this matrix representation <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>λ</mi></mrow></msub></math></span>. Apart from computing the compositional inverses of permutation polynomials, this linear representation is also used to compute the cycle structures of the permutation map. Lastly, this representation is extended to a representation of the cyclic group generated by a permutation map <em>F</em>, and to the group generated by a finite number of permutation maps over <span><math><mi>F</mi></math></span>.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141959414","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Representations of group actions and their applications in cryptography","authors":"Giuseppe D'Alconzo,&nbsp;Antonio J. Di Scala","doi":"10.1016/j.ffa.2024.102476","DOIUrl":"10.1016/j.ffa.2024.102476","url":null,"abstract":"<div><p>Cryptographic group actions provide a flexible framework that allows the instantiation of several primitives, ranging from key exchange protocols to PRFs and digital signatures. The security of such constructions is based on the intractability of some computational problems. For example, given the group action <span><math><mo>(</mo><mi>G</mi><mo>,</mo><mi>X</mi><mo>,</mo><mo>⋆</mo><mo>)</mo></math></span>, the weak unpredictability assumption (Alamati et al. (2020) <span><span>[1]</span></span>) requires that, given random <span><math><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>'s in <em>X</em>, no probabilistic polynomial time algorithm can compute, on input <span><math><msub><mrow><mo>{</mo><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><mi>g</mi><mo>⋆</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo><mo>}</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>Q</mi></mrow></msub></math></span> and <em>y</em>, the set element <span><math><mi>g</mi><mo>⋆</mo><mi>y</mi></math></span>.</p><p>In this work, we study such assumptions, aided by the definition of <em>group action representations</em> and a new metric, the <em>q-linear dimension</em>, that estimates the “linearity” of a group action, or in other words, how much it is far from being linear. We show that under some hypotheses on the group action representation, and if the <em>q</em>-linear dimension is polynomial in the security parameter, then the weak unpredictability and other related assumptions cannot hold. This technique is applied to some actions from cryptography, like the ones arising from the equivalence of linear codes, as a result, we obtain the impossibility of using such actions for the instantiation of certain primitives.</p><p>As an additional result, some bounds on the <em>q</em>-linear dimension are given for classical groups, such as <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, <span><math><mi>GL</mi><mo>(</mo><msup><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span> and the cyclic group <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> acting on itself.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S1071579724001151/pdfft?md5=da2ac4d07e20b23f31147c448a4a4dc4&pid=1-s2.0-S1071579724001151-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141959413","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the stopping time of the Collatz map in F2[x]","authors":"Gil Alon ,&nbsp;Angelot Behajaina ,&nbsp;Elad Paran","doi":"10.1016/j.ffa.2024.102473","DOIUrl":"10.1016/j.ffa.2024.102473","url":null,"abstract":"<div><p>We study the stopping time of the Collatz map for a polynomial <span><math><mi>f</mi><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>[</mo><mi>x</mi><mo>]</mo></math></span>, and bound it by <span><math><mi>O</mi><mo>(</mo><mrow><mi>deg</mi></mrow><msup><mrow><mo>(</mo><mi>f</mi><mo>)</mo></mrow><mrow><mn>1.5</mn></mrow></msup><mo>)</mo></math></span>, improving upon the quadratic bound proven by Hicks, Mullen, Yucas and Zavislak. We also prove the existence of arithmetic sequences of unbounded length in the stopping times of certain sequences of polynomials, a phenomenon observed in the classical Collatz map.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141959410","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On algebraic degrees of inverted Kloosterman sums","authors":"Xin Lin ,&nbsp;Daqing Wan","doi":"10.1016/j.ffa.2024.102477","DOIUrl":"10.1016/j.ffa.2024.102477","url":null,"abstract":"<div><p>The study of <em>n</em>-dimensional inverted Kloosterman sums was suggested by Katz (1995) <span><span>[7]</span></span> who handled the case when <span><math><mi>n</mi><mo>=</mo><mn>1</mn></math></span> from complex point of view. For general <span><math><mi>n</mi><mo>≥</mo><mn>1</mn></math></span>, the <em>n</em>-dimensional inverted Kloosterman sums were studied from both complex and <em>p</em>-adic point of view in our previous paper (2024) <span><span>[10]</span></span>. In this note, we study the algebraic degree of the inverted <em>n</em>-dimensional Kloosterman sum as an algebraic integer.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141959411","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Two classes of LCD BCH codes over finite fields","authors":"Yuqing Fu ,&nbsp;Hongwei Liu","doi":"10.1016/j.ffa.2024.102478","DOIUrl":"10.1016/j.ffa.2024.102478","url":null,"abstract":"<div><p>BCH codes form a special subclass of cyclic codes and have been extensively studied in the past decades. Determining the parameters of BCH codes, however, has been an important but difficult problem. Recently, in order to further investigate the dual codes of BCH codes, the concept of dually-BCH codes was proposed. In this paper, we study BCH codes of lengths <span><math><mfrac><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>+</mo><mn>1</mn></mrow><mrow><mi>q</mi><mo>+</mo><mn>1</mn></mrow></mfrac></math></span> and <span><math><msup><mrow><mi>q</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>+</mo><mn>1</mn></math></span> over the finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>, both of which are LCD codes. The dimensions of narrow-sense BCH codes of length <span><math><mfrac><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>+</mo><mn>1</mn></mrow><mrow><mi>q</mi><mo>+</mo><mn>1</mn></mrow></mfrac></math></span> with designed distance <span><math><mi>δ</mi><mo>=</mo><mi>ℓ</mi><msup><mrow><mi>q</mi></mrow><mrow><mfrac><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo>+</mo><mn>1</mn></math></span> are determined, where <span><math><mi>q</mi><mo>&gt;</mo><mn>2</mn></math></span> and <span><math><mn>2</mn><mo>≤</mo><mi>ℓ</mi><mo>≤</mo><mi>q</mi><mo>−</mo><mn>1</mn></math></span>. Lower bounds on the minimum distances of the dual codes of narrow-sense BCH codes of length <span><math><msup><mrow><mi>q</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>+</mo><mn>1</mn></math></span> are developed for odd <em>q</em>, which are good in some cases. Moreover, sufficient and necessary conditions for the even-like subcodes of narrow-sense BCH codes of length <span><math><msup><mrow><mi>q</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>+</mo><mn>1</mn></math></span> being dually-BCH codes are presented, where <em>q</em> is odd and <span><math><mi>m</mi><mo>≢</mo><mn>0</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>4</mn><mo>)</mo></math></span>.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141950539","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"New results on PcN and APcN polynomials over finite fields","authors":"Zhengbang Zha ,&nbsp;Lei Hu","doi":"10.1016/j.ffa.2024.102471","DOIUrl":"10.1016/j.ffa.2024.102471","url":null,"abstract":"<div><p>Permutation polynomials with low <em>c</em>-differential uniformity have important applications in cryptography and combinatorial design. In this paper, we investigate perfect <em>c</em>-nonlinear (PcN) and almost perfect <em>c</em>-nonlinear (APcN) polynomials over finite fields. Based on some known permutation polynomials, we present several classes of PcN or APcN polynomials by using the Akbary-Ghioca-Wang criterion.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141959583","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"New results on n-to-1 mappings over finite fields","authors":"Xiaoer Qin ,&nbsp;Li Yan","doi":"10.1016/j.ffa.2024.102469","DOIUrl":"10.1016/j.ffa.2024.102469","url":null,"abstract":"<div><p><em>n</em>-to-1 mappings have many applications in cryptography, finite geometry, coding theory and combinatorial design. In this paper, we first use cyclotomic cosets to construct several kinds of <em>n</em>-to-1 mappings over <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup></math></span>. Then we characterize a new form of AGW-like criterion, and use it to present many classes of <em>n</em>-to-1 polynomials with the form <span><math><msup><mrow><mi>x</mi></mrow><mrow><mi>r</mi></mrow></msup><mi>h</mi><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>q</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></math></span> over <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mo>⁎</mo></mrow></msubsup></math></span>. Finally, by using monomials on the cosets of a subgroup of <span><math><msub><mrow><mi>μ</mi></mrow><mrow><mi>q</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> and another form of AGW-like criterion, we show some <em>n</em>-to-1 trinomials over <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mo>⁎</mo></mrow></msubsup></math></span>.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141638722","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Circularity in finite fields and solutions of the equations xm + ym − zm = 1","authors":"Wen-Fong Ke ,&nbsp;Hubert Kiechle","doi":"10.1016/j.ffa.2024.102467","DOIUrl":"10.1016/j.ffa.2024.102467","url":null,"abstract":"<div><p>An explicit formula for the number of solutions of the equation in the title is given when a certain condition, depending only on the exponent and the characteristic of the field, holds. This formula improves the one given by the authors in an earlier paper.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141623614","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Probabilistic Galois theory in function fields","authors":"Alexei Entin,&nbsp;Alexander Popov","doi":"10.1016/j.ffa.2024.102466","DOIUrl":"https://doi.org/10.1016/j.ffa.2024.102466","url":null,"abstract":"<div><p>We study the irreducibility and Galois group of random polynomials over function fields. We prove that a random polynomial <span><math><mi>f</mi><mo>=</mo><msup><mrow><mi>y</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>+</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msubsup><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><msup><mrow><mi>y</mi></mrow><mrow><mi>i</mi></mrow></msup><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><mi>x</mi><mo>]</mo><mo>[</mo><mi>y</mi><mo>]</mo></math></span> with i.i.d. coefficients <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> taking values in the set <span><math><mo>{</mo><mi>a</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><mi>x</mi><mo>]</mo><mo>:</mo><mi>deg</mi><mo>⁡</mo><mi>a</mi><mo>≤</mo><mi>d</mi><mo>}</mo></math></span> with uniform probability, is irreducible with probability tending to <span><math><mn>1</mn><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>d</mi></mrow></msup></mrow></mfrac></math></span> as <span><math><mi>n</mi><mo>→</mo><mo>∞</mo></math></span>, where <em>d</em> and <em>q</em> are fixed. We also prove that with the same probability, the Galois group of this random polynomial contains the alternating group <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. Moreover, we prove that if we assume a version of the polynomial Chowla conjecture over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><mi>x</mi><mo>]</mo></math></span>, then the Galois group of this polynomial is actually equal to the symmetric group <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> with probability tending to <span><math><mn>1</mn><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>d</mi></mrow></msup></mrow></mfrac></math></span>. We also study the other possible Galois groups occurring with positive limit probability. Finally, we study the same problems with <em>n</em> fixed and <span><math><mi>d</mi><mo>→</mo><mo>∞</mo></math></span>.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141606199","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}