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On the Construction of Short Addition-Subtraction Chains and their Applications 短加减链的构造及其应用
Tatra Mountains Mathematical Publications Pub Date : 2023-02-01 DOI: 10.2478/tmmp-2023-0010
Moussa Ngom, A. Tall
{"title":"On the Construction of Short Addition-Subtraction Chains and their Applications","authors":"Moussa Ngom, A. Tall","doi":"10.2478/tmmp-2023-0010","DOIUrl":"https://doi.org/10.2478/tmmp-2023-0010","url":null,"abstract":"Abstract The problem of computing xn effciently, such that x and n are known to be very interesting, specially when n is very large. In order to find effcient methods to solve this problem, addition chains have been much studied, and generalized to addition-subtraction chains. These various chains have been useful in finding effcient exponentiation algorithms. In this paper, we present a new method to recover all existing exponentiation algorithms. It will be applied to design a new fast exponentiation method.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"83 1","pages":"131 - 144"},"PeriodicalIF":0.0,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43621143","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Irreducibility and Multiplicative Composition of Polynomials Over Finite Fields 有限域上多项式的不可约性和乘法复合
Tatra Mountains Mathematical Publications Pub Date : 2023-02-01 DOI: 10.2478/tmmp-2023-0001
L. Benferhat, O. Kihel, Jesse Larone, Rezki Ould Mohamed
{"title":"Irreducibility and Multiplicative Composition of Polynomials Over Finite Fields","authors":"L. Benferhat, O. Kihel, Jesse Larone, Rezki Ould Mohamed","doi":"10.2478/tmmp-2023-0001","DOIUrl":"https://doi.org/10.2478/tmmp-2023-0001","url":null,"abstract":"Abstract The aim of this paper is to provide integral polynomials irreducible over ℤ which are reducible over 𝔽p for every prime p. In particular, we show that certain composed products of integral polynomials are reducible modulo p for all primes p.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"83 1","pages":"1 - 10"},"PeriodicalIF":0.0,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46001987","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Tower Building Technique on Elliptic Curve with Embedding Degree 18 嵌入度为18的椭圆曲线建塔技术
Tatra Mountains Mathematical Publications Pub Date : 2023-02-01 DOI: 10.2478/tmmp-2023-0008
Ismail Assoujaa, Siham Ezzouak, Hakima Mouanis
{"title":"Tower Building Technique on Elliptic Curve with Embedding Degree 18","authors":"Ismail Assoujaa, Siham Ezzouak, Hakima Mouanis","doi":"10.2478/tmmp-2023-0008","DOIUrl":"https://doi.org/10.2478/tmmp-2023-0008","url":null,"abstract":"Abstract Pairing based cryptography is one of the best security solution that devote a lot of attention. So, to make pairing practical, secure and computationally effcient, we choose to work with extension finite field of the form 𝔽 p k with k ≥ 12. In this paper, we focus on the case of curves with embedding degree 18. We use the tower building technique, and study the case of degree 2 or 3 twist to carry out most arithmetics operations in 𝔽 p 2 , 𝔽 p 3 , 𝔽 p 6 , 𝔽 p 9 and 𝔽 p 18 , thus we speed up the computation in optimal ate pairing.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"219 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134942011","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A Generalization of Eisenstein-Schönemann’s Irreducibility Criterion Eisenstein-Schönemann不可约性准则的推广
Tatra Mountains Mathematical Publications Pub Date : 2023-02-01 DOI: 10.2478/tmmp-2023-0005
L. El Fadil
{"title":"A Generalization of Eisenstein-Schönemann’s Irreducibility Criterion","authors":"L. El Fadil","doi":"10.2478/tmmp-2023-0005","DOIUrl":"https://doi.org/10.2478/tmmp-2023-0005","url":null,"abstract":"Abstract The Eisenstein criterion is a particular case of the Schönemann’s irreducibility criterion stated in 1846. In 1906, based on Newton polygon techniques, Dumas gave a generalization of the Eisenstein criterion. In this paper, we extend this last generalization. Some applications on factorization of polynomials, and prime ideal factorization will be given, too.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"83 1","pages":"51 - 60"},"PeriodicalIF":0.0,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49641580","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Integral Bases and Monogenity of Pure Number Fields with Non-Square Free Parameters up to Degree 9 9次以下非平方自由参数纯数域的积分基与单性
Tatra Mountains Mathematical Publications Pub Date : 2023-02-01 DOI: 10.2478/tmmp-2023-0006
L. El Fadil, István Gaál
{"title":"Integral Bases and Monogenity of Pure Number Fields with Non-Square Free Parameters up to Degree 9","authors":"L. El Fadil, István Gaál","doi":"10.2478/tmmp-2023-0006","DOIUrl":"https://doi.org/10.2478/tmmp-2023-0006","url":null,"abstract":"Abstract Let K be a pure number field generated by a root α of a monic irreducible polynomial f (x)= xn − m with m a rational integer and 3 ≤ n ≤ 9 an integer. In this paper, we calculate an integral basis of ℤK , and we study the monogenity of K, extending former results to the case when m is not necessarily square-free. Collecting and completing the corresponding results in this more general case, our purpose is to provide a parallel to [Gaál, I.—Remete, L.: Power integral bases and monogenity of pure fields,J.Number Theory, 173 (2017), 129–146], where only square-free values of m were considered.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"83 1","pages":"61 - 86"},"PeriodicalIF":0.0,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43019836","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Automated Finite Element Solution of Diffusion Models for Image Denoising 图像去噪扩散模型的自动有限元解
Tatra Mountains Mathematical Publications Pub Date : 2023-02-01 DOI: 10.2478/tmmp-2023-0002
Abderrazzak Boufala, E. Kalmoun
{"title":"Automated Finite Element Solution of Diffusion Models for Image Denoising","authors":"Abderrazzak Boufala, E. Kalmoun","doi":"10.2478/tmmp-2023-0002","DOIUrl":"https://doi.org/10.2478/tmmp-2023-0002","url":null,"abstract":"Abstract We present in this paper a numerical solution of a generalized diffusion-based image denoising model, using the finite element computing platform FEniCS. The generalized model contains as special cases three classical denoising techniques: linear isotropic diffusion, total variation, and Perona-Malik method. The numerical simulation using four classical grayscale images demonstrates the superior performance of the finite element method over the finite difference method in terms of both the denoising quality and the computational work.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"83 1","pages":"11 - 24"},"PeriodicalIF":0.0,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46394081","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On The Geometric Determination of Extensions of Non-Archimedean Absolute Values 关于非阿基米德绝对值扩张的几何判定
Tatra Mountains Mathematical Publications Pub Date : 2023-02-01 DOI: 10.2478/tmmp-2023-0007
Mohamed Faris, L. El Fadil
{"title":"On The Geometric Determination of Extensions of Non-Archimedean Absolute Values","authors":"Mohamed Faris, L. El Fadil","doi":"10.2478/tmmp-2023-0007","DOIUrl":"https://doi.org/10.2478/tmmp-2023-0007","url":null,"abstract":"Abstract Let | | be a discrete non-archimedean absolute value of a field K with valuation ring 𝒪, maximal ideal 𝓜 and residue field 𝔽 = 𝒪/𝓜. Let L be a simple finite extension of K generated by a root α of a monic irreducible polynomial F ∈ O[x]. Assume that F¯=ϕ¯l$overline F = overline varphi ^l$ in 𝔽[x] for some monic polynomial φ ∈ O[x] whose reduction modulo 𝓜 is irreducible, the φ-Newton polygon Nφ¯(F)$Noverline phi left( F right)$ has a single side of negative slope λ, and the residual polynomial Rλ(F )(y) has no multiple factors in 𝔽φ[y]. In this paper, we describe all absolute values of L extending | |. The problem is classical but our approach uses new ideas. Some useful remarks and computational examples are given to highlight some improvements due to our results.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"83 1","pages":"87 - 102"},"PeriodicalIF":0.0,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42616597","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Two Disjoint and Infinite Sets of Solutions for An Elliptic Equation with Critical Hardy-Sobolev-Maz’ya Term and Concave-Convex Nonlinearities 一类具有临界Hardy-Sobolev-Maz 'ya项和凹凸非线性的椭圆方程的两个不相交无穷解集
Tatra Mountains Mathematical Publications Pub Date : 2023-02-01 DOI: 10.2478/tmmp-2023-0003
R. Echarghaoui, Zakaria Zaimi
{"title":"Two Disjoint and Infinite Sets of Solutions for An Elliptic Equation with Critical Hardy-Sobolev-Maz’ya Term and Concave-Convex Nonlinearities","authors":"R. Echarghaoui, Zakaria Zaimi","doi":"10.2478/tmmp-2023-0003","DOIUrl":"https://doi.org/10.2478/tmmp-2023-0003","url":null,"abstract":"Abstract In this paper, we consider the following critical Hardy-Sobolev-Maz’ya problem {−Δu=|u|2∗(t)−2u|y|t+μ|u|q−2u in Ω,u=0 on ∂Ω, begin{cases}-Delta u=frac{|u|^{2^*(t)-2} u}{|y|^t}+mu|u|^{q-2} u & text { in } Omega, u=0 & text { on } partial Omega,end{cases} where Ω is an open bounded domain in ℝN , which contains some points (0,z*), μ>0,10,1<q<2,2^*(t)=frac{2(N-t)}{N-2}, 0 ≤ t < 2, x = (y, z) ∈ ℝk × ℝN−k, 2 ≤ k ≤ N. We prove that if N>2q+1q−1+t$N > 2{{q + 1} over {q - 1}} + t$, then the above problem has two disjoint and infinite sets of solutions. Here, we give a positive answer to one open problem proposed by Ambrosetti, Brezis and Cerami in [1] for the case of the critical Hardy-Sobolev-Maz’ya problem.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"83 1","pages":"25 - 42"},"PeriodicalIF":0.0,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46219455","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Twisted Edwards Curve Over the Ring 扭曲的爱德华兹曲线在环上
Tatra Mountains Mathematical Publications Pub Date : 2023-02-01 DOI: 10.2478/tmmp-2023-0004
Moha Ben Taleb El Hamam, A. Chillali, L. El Fadil
{"title":"Twisted Edwards Curve Over the Ring","authors":"Moha Ben Taleb El Hamam, A. Chillali, L. El Fadil","doi":"10.2478/tmmp-2023-0004","DOIUrl":"https://doi.org/10.2478/tmmp-2023-0004","url":null,"abstract":"Abstract Let 𝔽q be a finite field of q elements, where q is a power of an odd prime number. In this paper, we study the twisted Edwards curves denoted EEa,d over the local ring 𝔽q[e], where e2 = 0. In the first time, we study the arithmetic of the ring 𝔽q[e], e2 = 0. After that we define the twisted Edwards curves EEa,d over this ring and we give essential properties and we define the group EEa,d , these properties. Precisely, we give a bijection between the groups EEa,d and EEa,d0 × Fq,where EEa,d0 is the twisted Edwards curves over the finite field 𝔽q.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"83 1","pages":"43 - 50"},"PeriodicalIF":0.0,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46753686","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Tower Building Technique on Elliptic Curve with Embedding Degree 18 嵌入度为18的椭圆曲线上的塔楼施工技术
Tatra Mountains Mathematical Publications Pub Date : 2022-12-27 DOI: 10.37394/23205.2022.21.39
Ismail Assoujaa, Siham Ezzouak, H. Mouanis
{"title":"Tower Building Technique on Elliptic Curve with Embedding Degree 18","authors":"Ismail Assoujaa, Siham Ezzouak, H. Mouanis","doi":"10.37394/23205.2022.21.39","DOIUrl":"https://doi.org/10.37394/23205.2022.21.39","url":null,"abstract":"Abstract Pairing based cryptography is one of the best security solution that devote a lot of attention. So, to make pairing practical, secure and computationally effcient, we choose to work with extension finite field of the form 𝔽pk with k ≥ 12. In this paper, we focus on the case of curves with embedding degree 18. We use the tower building technique, and study the case of degree 2 or 3 twist to carry out most arithmetics operations in 𝔽p2 , 𝔽p3, 𝔽p6, 𝔽p9 and 𝔽p18, thus we speed up the computation in optimal ate pairing.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"83 1","pages":"103 - 118"},"PeriodicalIF":0.0,"publicationDate":"2022-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41445202","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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