{"title":"A special scalar multiplication algorithm on Jacobi quartic curves","authors":"Jiang Weng, Aiwang Chen, Tao Huang, Weifeng Ji","doi":"10.1007/s00200-023-00633-3","DOIUrl":null,"url":null,"abstract":"<p>At present, GLV/GLS scalar multiplication mainly focuses on elliptic curves in Weierstrass form, attempting to find and construct more and more efficiently computable endomorphism. In this paper, we investigate the application of the GLV/GLS scalar multiplication technique to Jacobi Quartic curves. Firstly, we present a concrete construction of efficiently computable endomorphisms for this type of curves over prime fields by exploiting birational equivalence between curves, and obtain a 2-dimensional GLV method. Secondly, we consider the quadratic twists of elliptic curves. By using birational equivalence and Frobenius mapping between curves, we present methods to construct efficiently computable endomorphisms for this type of curves over the quadratic extension field, and obtain a 2-dimensional GLS method. Finally, we obtain a 4-dimensional GLV method on elliptic curves with <i>j</i>-invariant 0 or 1728 by using higher degree twists. The experimental results show that the speedups of the 2-dimensional GLV method and 4-dimensional GLV method compared to 5-NAF method exceed <span>\\(37.2\\%\\)</span> and <span>\\(109.4\\%\\)</span> for Jacobi Quartic curves, respectively. At the same time, when utilizing one of the proposed methods, the scalar multiplication on Jacobi Quartic curves is consistently more efficient than on elliptic curves in Weierstrass form.</p>","PeriodicalId":50742,"journal":{"name":"Applicable Algebra in Engineering Communication and Computing","volume":"80 1","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2023-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Applicable Algebra in Engineering Communication and Computing","FirstCategoryId":"5","ListUrlMain":"https://doi.org/10.1007/s00200-023-00633-3","RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
引用次数: 0
Abstract
At present, GLV/GLS scalar multiplication mainly focuses on elliptic curves in Weierstrass form, attempting to find and construct more and more efficiently computable endomorphism. In this paper, we investigate the application of the GLV/GLS scalar multiplication technique to Jacobi Quartic curves. Firstly, we present a concrete construction of efficiently computable endomorphisms for this type of curves over prime fields by exploiting birational equivalence between curves, and obtain a 2-dimensional GLV method. Secondly, we consider the quadratic twists of elliptic curves. By using birational equivalence and Frobenius mapping between curves, we present methods to construct efficiently computable endomorphisms for this type of curves over the quadratic extension field, and obtain a 2-dimensional GLS method. Finally, we obtain a 4-dimensional GLV method on elliptic curves with j-invariant 0 or 1728 by using higher degree twists. The experimental results show that the speedups of the 2-dimensional GLV method and 4-dimensional GLV method compared to 5-NAF method exceed \(37.2\%\) and \(109.4\%\) for Jacobi Quartic curves, respectively. At the same time, when utilizing one of the proposed methods, the scalar multiplication on Jacobi Quartic curves is consistently more efficient than on elliptic curves in Weierstrass form.
期刊介绍：
Algebra is a common language for many scientific domains. In developing this language mathematicians prove theorems and design methods which demonstrate the applicability of algebra. Using this language scientists in many fields find algebra indispensable to create methods, techniques and tools to solve their specific problems.
Applicable Algebra in Engineering, Communication and Computing will publish mathematically rigorous, original research papers reporting on algebraic methods and techniques relevant to all domains concerned with computers, intelligent systems and communications. Its scope includes, but is not limited to, vision, robotics, system design, fault tolerance and dependability of systems, VLSI technology, signal processing, signal theory, coding, error control techniques, cryptography, protocol specification, networks, software engineering, arithmetics, algorithms, complexity, computer algebra, programming languages, logic and functional programming, algebraic specification, term rewriting systems, theorem proving, graphics, modeling, knowledge engineering, expert systems, and artificial intelligence methodology.
Purely theoretical papers will not primarily be sought, but papers dealing with problems in such domains as commutative or non-commutative algebra, group theory, field theory, or real algebraic geometry, which are of interest for applications in the above mentioned fields are relevant for this journal.
On the practical side, technology and know-how transfer papers from engineering which either stimulate or illustrate research in applicable algebra are within the scope of the journal.
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