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.
{"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}
L. Benferhat, O. Kihel, Jesse Larone, Rezki Ould Mohamed
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.
{"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}
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.
{"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}
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.
{"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}
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.
{"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}
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.
{"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}
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.
摘要:设| |是域K的离散非阿基米德绝对值，其值环为，极大理想为剩余域为 = / 。设L是由一元不可约多项式F∈O[x]的根α生成的K的简单有限扩展。假设F¯= φ¯1$overline F = overline varphi ^l$ 对于某一元多项式φ∈O[x]，其约化模是不可约的，在n [x]中，φ-牛顿多边形Nφ¯(F)$Noverline phi left( F right)$ 单侧斜率为负λ，残差多项式Rλ(F)(y)在𝔽φ[y]中没有多因子。本文描述了扩展| |的L的所有绝对值。这个问题很经典，但我们的方法采用了新思路。给出了一些有用的评论和计算实例，以突出我们的结果所带来的一些改进。
{"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}
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>0,10,12q+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.
文摘中,我们考虑以下关键Hardy-Sobolev-Maz大家问题{−Δu = | | 2∗(t)−2 u y | | t +μ| | q−2 uΩ,在∂u = 0Ω,开始{病例}-δu = 压裂{| u | ^ {2 ^ * (t) 2} u} {y | | ^ t} + uμ| | ^ {q2} u & 文本的{}ω u = 0 & 文本上{}部分ω,结束{病例}Ω是一个开放的有限域在ℝN,其中包含一些点(0,z *),μ> 0,10,12 + 1 q−1 + t $ N > 2 {{q + 1} / {q - 1}} +新台币,然后上面的问题有两个不相交的无限集的解决方案。对于临界Hardy-Sobolev-Maz 'ya问题，我们给出了Ambrosetti、Brezis和Cerami在1996年提出的一个开放问题的肯定答案。
{"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}
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.
{"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}
Pub Date : 2022-12-27DOI: 10.37394/23205.2022.21.39
Ismail Assoujaa, Siham Ezzouak, H. Mouanis
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.
{"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}
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