Pub Date : 2024-08-26DOI: 10.1016/j.ffa.2024.102498
Maria Montanucci , Guilherme Tizziotti , Giovanni Zini
In this paper we compute the automorphism group of the curves and introduced in Tafazolian et al. [27] as new examples of maximal curves which cannot be covered by the Hermitian curve. They arise as subcovers of the first generalized GK curve (GGS curve). As a result, a new characterization of the GK curve, as a member of this family, is obtained.
{"title":"On the automorphism group of a family of maximal curves not covered by the Hermitian curve","authors":"Maria Montanucci , Guilherme Tizziotti , Giovanni Zini","doi":"10.1016/j.ffa.2024.102498","DOIUrl":"10.1016/j.ffa.2024.102498","url":null,"abstract":"<div><p>In this paper we compute the automorphism group of the curves <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>n</mi><mo>,</mo><mi>s</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>Y</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>s</mi></mrow></msub></math></span> introduced in Tafazolian et al. <span><span>[27]</span></span> as new examples of maximal curves which cannot be covered by the Hermitian curve. They arise as subcovers of the first generalized GK curve (GGS curve). As a result, a new characterization of the GK curve, as a member of this family, is obtained.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"99 ","pages":"Article 102498"},"PeriodicalIF":1.2,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142075829","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}
Pub Date : 2024-08-14DOI: 10.1016/j.ffa.2024.102492
Shamil Asgarli , Chi Hoi Yip
Blokhuis showed that all maximum cliques in Paley graphs of square order have a subfield structure. Recently, it has been shown that in Peisert-type graphs, all maximum cliques are affine subspaces, and yet some maximum cliques do not arise from a subfield. In this paper, we investigate the existence of a clique of size with a subspace structure in pseudo-Paley graphs of order q from unions of semi-primitive cyclotomic classes. We show that such a clique must have an equal contribution from each cyclotomic class and that most such pseudo-Paley graphs do not admit such cliques, suggesting that the Delsarte bound on the clique number can be improved in general. We also prove that generalized Peisert graphs are not isomorphic to Paley graphs or Peisert graphs, confirming a conjecture of Mullin.
{"title":"The subspace structure of maximum cliques in pseudo-Paley graphs from unions of cyclotomic classes","authors":"Shamil Asgarli , Chi Hoi Yip","doi":"10.1016/j.ffa.2024.102492","DOIUrl":"10.1016/j.ffa.2024.102492","url":null,"abstract":"<div><p>Blokhuis showed that all maximum cliques in Paley graphs of square order have a subfield structure. Recently, it has been shown that in Peisert-type graphs, all maximum cliques are affine subspaces, and yet some maximum cliques do not arise from a subfield. In this paper, we investigate the existence of a clique of size <span><math><msqrt><mrow><mi>q</mi></mrow></msqrt></math></span> with a subspace structure in pseudo-Paley graphs of order <em>q</em> from unions of semi-primitive cyclotomic classes. We show that such a clique must have an equal contribution from each cyclotomic class and that most such pseudo-Paley graphs do not admit such cliques, suggesting that the Delsarte bound <span><math><msqrt><mrow><mi>q</mi></mrow></msqrt></math></span> on the clique number can be improved in general. We also prove that generalized Peisert graphs are not isomorphic to Paley graphs or Peisert graphs, confirming a conjecture of Mullin.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"99 ","pages":"Article 102492"},"PeriodicalIF":1.2,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141985199","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}
Pub Date : 2024-08-14DOI: 10.1016/j.ffa.2024.102491
John Bamberg , Jesse Lansdown , Geertrui Van de Voorde
It is known that a Bruen chain of the three-dimensional projective space exists for every odd prime power q at most 37, except for . It was shown by Cardinali et al. (2005) that Bruen chains do not exist for . We develop a model, based on finite fields, which allows us to extend this result to , thereby adding more evidence to the conjecture that Bruen chains do not exist for . Furthermore, we show that Bruen chains can be realised precisely as the -cliques of a two related, yet distinct, undirected simple graphs.
{"title":"On Bruen chains","authors":"John Bamberg , Jesse Lansdown , Geertrui Van de Voorde","doi":"10.1016/j.ffa.2024.102491","DOIUrl":"10.1016/j.ffa.2024.102491","url":null,"abstract":"<div><p>It is known that a Bruen chain of the three-dimensional projective space <span><math><mrow><mi>PG</mi></mrow><mo>(</mo><mn>3</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span> exists for every odd prime power <em>q</em> at most 37, except for <span><math><mi>q</mi><mo>=</mo><mn>29</mn></math></span>. It was shown by Cardinali et al. (2005) that Bruen chains do not exist for <span><math><mn>41</mn><mo>⩽</mo><mi>q</mi><mo>⩽</mo><mn>49</mn></math></span>. We develop a model, based on finite fields, which allows us to extend this result to <span><math><mn>41</mn><mo>⩽</mo><mi>q</mi><mo>⩽</mo><mn>97</mn></math></span>, thereby adding more evidence to the conjecture that Bruen chains do not exist for <span><math><mi>q</mi><mo>></mo><mn>37</mn></math></span>. Furthermore, we show that Bruen chains can be realised precisely as the <span><math><mo>(</mo><mi>q</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>/</mo><mn>2</mn></math></span>-cliques of a two related, yet distinct, undirected simple graphs.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"99 ","pages":"Article 102491"},"PeriodicalIF":1.2,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S1071579724001308/pdfft?md5=731484f2ebf31e1586fb859e032c078c&pid=1-s2.0-S1071579724001308-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141985200","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}
Pub Date : 2024-08-13DOI: 10.1016/j.ffa.2024.102489
Rong Luo , Bingsheng Shen , Yang Yang , Zhengchun Zhou
A complete complementary code (CCC) consists of M sequence sets with size M. The sum of the auto-correlation functions of each sequence set is an impulse function, and the sum of cross-correlation functions of the different sequence sets is equal to zero. Thanks to their excellent correlation, CCCs received extensive use in engineering. In addition, they are strongly connected to orthogonal matrices. In some application scenarios, additional requirements are made for CCCs, such as recently proposed for concatenative CCC (CCCC) division multiple access (CCC-CDMA) technologies. In fact, CCCCs are a special kind of CCCs which requires that each sequence set in CCC be concatenated to form a zero-correlation-zone (ZCZ) sequence set. However, this requirement is challenging, and the literature is thin since there is only one construction in this context. We propose to go beyond the literature through this contribution to reduce the gap between their interest and our limited knowledge of CCCCs. This paper will employ novel methods for designing CCCCs and precisely derive two constructions of these objects. The first is based on perfect cross Z-complementary pair and Hadamard matrices, and the second relies on extended Boolean functions. Specifically, we highlight that optimal and asymptotic optimal CCCCs could be obtained through the proposed constructions. Besides, we shall present a comparison analysis with former structures in the literature and examples to illustrate our main results.
{"title":"Design of concatenative complete complementary codes for CCC-CDMA via specific sequences and extended Boolean functions","authors":"Rong Luo , Bingsheng Shen , Yang Yang , Zhengchun Zhou","doi":"10.1016/j.ffa.2024.102489","DOIUrl":"10.1016/j.ffa.2024.102489","url":null,"abstract":"<div><p>A complete complementary code (CCC) consists of <em>M</em> sequence sets with size <em>M</em>. The sum of the auto-correlation functions of each sequence set is an impulse function, and the sum of cross-correlation functions of the different sequence sets is equal to zero. Thanks to their excellent correlation, CCCs received extensive use in engineering. In addition, they are strongly connected to orthogonal matrices. In some application scenarios, additional requirements are made for CCCs, such as recently proposed for concatenative CCC (CCCC) division multiple access (CCC-CDMA) technologies. In fact, CCCCs are a special kind of CCCs which requires that each sequence set in CCC be concatenated to form a zero-correlation-zone (ZCZ) sequence set. However, this requirement is challenging, and the literature is thin since there is only one construction in this context. We propose to go beyond the literature through this contribution to reduce the gap between their interest and our limited knowledge of CCCCs. This paper will employ novel methods for designing CCCCs and precisely derive two constructions of these objects. The first is based on perfect cross Z-complementary pair and Hadamard matrices, and the second relies on extended Boolean functions. Specifically, we highlight that optimal and asymptotic optimal CCCCs could be obtained through the proposed constructions. Besides, we shall present a comparison analysis with former structures in the literature and examples to illustrate our main results.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"99 ","pages":"Article 102489"},"PeriodicalIF":1.2,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141978745","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}
The classification of the 2-designs with admitting a flag-transitive automorphism groups with socle is completed by settling the two open cases in [2]. The result is achieved by using conics and hyperovals of .
{"title":"Designs with a simple automorphism group","authors":"Alessandro Montinaro , Yanwei Zhao , Zhilin Zhang , Shenglin Zhou","doi":"10.1016/j.ffa.2024.102488","DOIUrl":"10.1016/j.ffa.2024.102488","url":null,"abstract":"<div><p>The classification of the 2-designs with <span><math><mi>λ</mi><mo>=</mo><mn>2</mn></math></span> admitting a flag-transitive automorphism groups with socle <span><math><mi>P</mi><mi>S</mi><mi>L</mi><mo>(</mo><mn>2</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span> is completed by settling the two open cases in <span><span>[2]</span></span>. The result is achieved by using conics and hyperovals of <span><math><mi>P</mi><mi>G</mi><mo>(</mo><mn>2</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span>.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"99 ","pages":"Article 102488"},"PeriodicalIF":1.2,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141963503","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}
Pub Date : 2024-08-08DOI: 10.1016/j.ffa.2024.102472
Paige Bright, Xinyu Fang, Barrett Heritage, Alex Iosevich, Tingsong Jiang, Hans Parshall, Maxwell Sun
In this paper, we generalize [6], [1], [5] and [3] by allowing the distance between two points in a finite field vector space to be defined by a general non-degenerate bilinear form or quadratic form. We prove the same bounds on the sizes of large subsets of for them to contain distance graphs with a given maximal vertex degree, under the more general notion of distance. We also prove the same results for embedding paths, trees and cycles in the general setting.
{"title":"Generalized point configurations in Fqd","authors":"Paige Bright, Xinyu Fang, Barrett Heritage, Alex Iosevich, Tingsong Jiang, Hans Parshall, Maxwell Sun","doi":"10.1016/j.ffa.2024.102472","DOIUrl":"10.1016/j.ffa.2024.102472","url":null,"abstract":"<div><p>In this paper, we generalize <span><span>[6]</span></span>, <span><span>[1]</span></span>, <span><span>[5]</span></span> and <span><span>[3]</span></span> by allowing the <em>distance</em> between two points in a finite field vector space to be defined by a general non-degenerate bilinear form or quadratic form. We prove the same bounds on the sizes of large subsets of <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>d</mi></mrow></msubsup></math></span> for them to contain distance graphs with a given maximal vertex degree, under the more general notion of distance. We also prove the same results for embedding paths, trees and cycles in the general setting.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"99 ","pages":"Article 102472"},"PeriodicalIF":1.2,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141952186","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}
Pub Date : 2024-08-08DOI: 10.1016/j.ffa.2024.102490
Hiroshi Onuki
In 2015, Abatzoglou, Silverberg, Sutherland, and Wong presented a framework for primality proving algorithms for special sequences of integers using an elliptic curve with complex multiplication. They applied their framework to obtain algorithms for elliptic curves with complex multiplication by imaginary quadratic field of class numbers one and two, but, they were not able to obtain primality proving algorithms in cases of higher class number. In this paper, we present a method to apply their framework to imaginary quadratic fields of class number three. In particular, our method provides a more efficient primality proving algorithm for special sequences of integers than the existing algorithms by using an imaginary quadratic field of class number three in which 2 splits. As an application, we give two special sequences of integers derived from and , which are all the imaginary quadratic fields of class number three in which 2 splits. Finally, we give a computational result for the primality of these sequences.
{"title":"Primality proving using elliptic curves with complex multiplication by imaginary quadratic fields of class number three","authors":"Hiroshi Onuki","doi":"10.1016/j.ffa.2024.102490","DOIUrl":"10.1016/j.ffa.2024.102490","url":null,"abstract":"<div><p>In 2015, Abatzoglou, Silverberg, Sutherland, and Wong presented a framework for primality proving algorithms for special sequences of integers using an elliptic curve with complex multiplication. They applied their framework to obtain algorithms for elliptic curves with complex multiplication by imaginary quadratic field of class numbers one and two, but, they were not able to obtain primality proving algorithms in cases of higher class number. In this paper, we present a method to apply their framework to imaginary quadratic fields of class number three. In particular, our method provides a more efficient primality proving algorithm for special sequences of integers than the existing algorithms by using an imaginary quadratic field of class number three in which 2 splits. As an application, we give two special sequences of integers derived from <span><math><mi>Q</mi><mo>(</mo><msqrt><mrow><mo>−</mo><mn>23</mn></mrow></msqrt><mo>)</mo></math></span> and <span><math><mi>Q</mi><mo>(</mo><msqrt><mrow><mo>−</mo><mn>31</mn></mrow></msqrt><mo>)</mo></math></span>, which are all the imaginary quadratic fields of class number three in which 2 splits. Finally, we give a computational result for the primality of these sequences.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"99 ","pages":"Article 102490"},"PeriodicalIF":1.2,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141952188","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}
Pub Date : 2024-08-08DOI: 10.1016/j.ffa.2024.102474
Tong Lin, Qiang Wang
Let q be a prime power. For , we construct stable polynomials of the form over by Capelli's lemma. Moreover, when and , we improve a lower bound for the number of stable quadratic polynomials over due to Goméz-Pérez and Nicolás [4]. When , we prove Ahmadi and Monsef-Shokri's conjecture [2] that is stable over .
{"title":"On the stable polynomials of degrees 2,3,4","authors":"Tong Lin, Qiang Wang","doi":"10.1016/j.ffa.2024.102474","DOIUrl":"10.1016/j.ffa.2024.102474","url":null,"abstract":"<div><p>Let <em>q</em> be a prime power. For <span><math><mi>m</mi><mo>=</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn></math></span>, we construct stable polynomials of the form <span><math><msup><mrow><mi>b</mi></mrow><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msup><msup><mrow><mo>(</mo><mi>x</mi><mo>+</mo><mi>a</mi><mo>)</mo></mrow><mrow><mi>m</mi></mrow></msup><mo>+</mo><mi>c</mi><mo>(</mo><mi>x</mi><mo>+</mo><mi>a</mi><mo>)</mo><mo>+</mo><mi>d</mi></math></span> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> by Capelli's lemma. Moreover, when <span><math><mi>m</mi><mo>=</mo><mn>2</mn></math></span> and <span><math><mi>q</mi><mo>≡</mo><mn>1</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>4</mn><mo>)</mo></math></span>, we improve a lower bound for the number of stable quadratic polynomials over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> due to Goméz-Pérez and Nicolás <span><span>[4]</span></span>. When <span><math><mi>m</mi><mo>=</mo><mn>3</mn></math></span>, we prove Ahmadi and Monsef-Shokri's conjecture <span><span>[2]</span></span> that <span><math><msup><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>+</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>1</mn></math></span> is stable over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"99 ","pages":"Article 102474"},"PeriodicalIF":1.2,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141952187","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}
Pub Date : 2024-08-05DOI: 10.1016/j.ffa.2024.102479
Gook Hwa Cho , Soonhak Kwon
Let q be a power of a prime such that . Let c be an r-th power residue over . In this paper, we present a new r-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 with requires only multiplications for . The multiplications for computation of the main exponentiation of our algorithm are half of that of the Williams' Cipolla-Lehmer type algorithms.
设 q 是一个质数的幂,使得 q≡1(modr)。设 c 是 Fq 上的 r 次幂残差。在本文中,我们提出了一个新的 r-th 根公式,它概括了 G.H. Cho 等人的立方根算法,并对 Williams 基于 Cipolla-Lehmer 的程序进行了改进。我们的算法基于不可还原多项式 h(x)=xr+(-1)r+1(b+(-1)rr)(x-1) 所产生的递推关系,其中 b=c+(-1)r+1r 对于 r>1 只需要 O(r2logq+r4) 次乘法。
{"title":"On the computation of r-th roots in finite fields","authors":"Gook Hwa Cho , 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>></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":"99 ","pages":"Article 102479"},"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}
Pub Date : 2024-08-05DOI: 10.1016/j.ffa.2024.102475
Ramachandran Ananthraman , Virendra Sule
This paper defines a linear representation for nonlinear maps where is a finite field, in terms of matrices over . This linear representation of the map F associates a unique number N and a unique matrix M in , called the Linear Complexity and the Linear Representation of F respectively, and shows that the compositional powers are represented by matrix powers . It is shown that for a permutation map F with representation M, the inverse map has the linear representation . This framework of representation is extended to a parameterized family of maps , defined in terms of a parameter , leading to the definition of an analogous linear complexity of the map , and a parameter-dependent matrix representation defined over the univariate polynomial ring . 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 . 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 F, and to the group generated by a finite number of permutation maps over .
本文定义了非线性映射 F:Fn→Fn 的线性表示,其中 F 是有限域,用 F 上的矩阵表示。映射 F 的这种线性表示关联了 FN×N 中唯一的数 N 和唯一的矩阵 M,分别称为 F 的线性复杂性和线性表示,并表明组成幂 F(k) 由矩阵幂 Mk 表示。这个表示框架被扩展到参数化的映射 Fλ(x):F→F 系列,以参数 λ∈F 定义,从而定义了映射 Fλ(x) 的类似线性复杂性,以及定义在单变量多项式环 F[λ] 上的与参数相关的矩阵表示 Mλ。通过这种表示,可以构建这种映射的参数逆,其中可逆性的条件是通过这种矩阵表示 Mλ 的单调性来表达的。除了计算置换多项式的组成逆之外,这种线性表示还用于计算置换映射的循环结构。最后,这一表示法被扩展为由置换映射 F 生成的循环群的表示法,以及由 F 上有限个置换映射生成的群的表示法。
{"title":"On linear representation, complexity and inversion of maps over finite fields","authors":"Ramachandran Ananthraman , 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":"99 ","pages":"Article 102475"},"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}