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Добірка наукової літератури з теми "Elliptic curve points group"

Автор: Grafiati

Опубліковано: 30 травня 2022

Оновлено: 31 липня 2025

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Статті в журналах з теми "Elliptic curve points group"

Shchur, Nataliia, Oleksandra Pokotylo, and Yelyzaveta Bailiuk. "ELLIPTIC CURVE CRYPTOGRAPHY AND ITS PRACTICAL APPLICATION." Cybersecurity: Education, Science, Technique 1, no. 21 (2023): 48–64. http://dx.doi.org/10.28925/2663-4023.2023.21.4864.

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Elliptic curves are one of the most promising tools for constructing modern cryptographic algorithms. The security of elliptic curve cryptography is based on the complexity of solving the discrete logarithm problem in the group of points of the elliptic curve over a finite field. Elliptic curve cryptography enables two parties communicating over public channel using elliptic curve encryption and signing algorithms. Elliptic curves allow to achieve the same level of security with small key sizes than other asymmetric cryptographic algorithms. The article describes the mathematical apparatus of

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Skuratovskii, Ruslan. "SUPERSINGULAR EDWARDS CURVES AND EDWARDS CURVE POINTS COUNTING METHOD OVER FINITE FIELD." Journal of Numerical and Applied Mathematics, no. 1 (133) (2020): 68–88. http://dx.doi.org/10.17721/2706-9699.2020.1.06.

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We consider problem of order counting of algebraic affine and projective curves of Edwards [2, 8] over the finite field $F_{p^n}$. The complexity of the discrete logarithm problem in the group of points of an elliptic curve depends on the order of this curve (ECDLP) [4, 20] depends on the order of this curve [10]. We research Edwards algebraic curves over a finite field, which are one of the most promising supports of sets of points which are used for fast group operations [1]. We construct a new method for counting the order of an Edwards curve over a finite field. It should be noted that thi

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Wohlfahrt, K. "Macbeath's Curve and the Modular Group." Glasgow Mathematical Journal 28, no. 2 (1986): 241. http://dx.doi.org/10.1017/s0017089500006583.

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Анотація:

On p. 244 of Glasgow Math. J.27 (1985) on the right hand side of one of the 6 equations characterizing the 4 fixed points of the involution v a sign error has occurred.The relevant equation should ready0y3y5y6=–1,or the points would not lie on the curve.Correcting the error unfortunately invalidates the model of an elliptic curve given in §6, which therefore has to be re-evaluated. First we find, in the notation of the paper,2 f (x) = ((r + 1)/(R + 2))2.

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Fan, Jing, Xuejun Fan, Ningning Song, and Long Wang. "Hyperelliptic Covers of Different Degree for Elliptic Curves." Mathematical Problems in Engineering 2022 (July 4, 2022): 1–11. http://dx.doi.org/10.1155/2022/9833393.

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In elliptic curve cryptography (ECC) and hyperelliptic curve cryptography (HECC), the size of cipher-text space defined by the cardinality of Jacobian is a significant factor to measure the security level. Counting problems on Jacobians of elliptic curve can be solved in polynomial time by Schoof–Elkies–Atkin (SEA) algorithm. However, counting problems on Jacobians of hyperelliptic curves are solved less satisfactorily than those on elliptic curves. So, we consider the construction of the cover map from the hyperelliptic curves to the elliptic curves to convert point counting problems on hyper

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CONCEIÇÃO, RICARDO. "ON INTEGRAL POINTS ON ISOTRIVIAL ELLIPTIC CURVES OVER FUNCTION FIELDS." Bulletin of the Australian Mathematical Society 102, no. 2 (2020): 177–85. http://dx.doi.org/10.1017/s0004972720000155.

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Let $k$ be a finite field and $L$ be the function field of a curve $C/k$ of genus $g\geq 1$. In the first part of this note we show that the number of separable $S$-integral points on a constant elliptic curve $E/L$ is bounded solely in terms of $g$ and the size of $S$. In the second part we assume that $L$ is the function field of a hyperelliptic curve $C_{A}:s^{2}=A(t)$, where $A(t)$ is a square-free $k$-polynomial of odd degree. If $\infty$ is the place of $L$ associated to the point at infinity of $C_{A}$, then we prove that the set of separable $\{\infty \}$-points can be bounded solely i

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Tama, Yanuar Bhakti Wira, and Muhammad Firdhausi Fahmi. "Sistem Kriptografi Klasik Dengan Memanfaatkan Orde Dari Grup Titik Pada Kurva Eliptik Bentuk Montgomery." Euler : Jurnal Ilmiah Matematika, Sains dan Teknologi 11, no. 2 (2023): 361–71. http://dx.doi.org/10.37905/euler.v11i2.23009.

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Elliptic curve cryptography is one of the application fields of algebra and number theory concepts. One form of elliptic curve cryptography is Montgomery elliptic curve cryptography. In this paper, a method for a classical cryptographic system be formulated, consisting of encryption and decryption involving twenty-six alphabetical letters which are mapped to points on an elliptic curve by utilizing the order of the point group on the Montgomery elliptic curve. Several examples of implementation in simple cases are given to verify the results.

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Jayanti, Sravani, K. Chittibabu, and Chandra Sekhar Akkapeddi. "A Cryptosystem of Skewed Affine Cipher of Multiple Keys." ECS Transactions 107, no. 1 (2022): 15071–80. http://dx.doi.org/10.1149/10701.15071ecst.

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Анотація:

In this era, where communication over technology has become vital, the reliability of the same is of utmost need. Cryptography ensures confidentiality, user authentication, and integrity of data. One of the techniques is the Elliptic Curve Cryptography (ECC). Several classical ciphers are designed based on mathematical backgrounds. In this paper, we focus on combining Affine Cipher and ECC to magnify the security provided by an Affine cipher. Hence a skewed Affine cipher that uses multiple keys over Elliptic curves is proposed. The keys chosen are derived from the points on the specified Ellip

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Dalkılıç, Şeyda, and Ercan Altınışık. "Mestre's Finite Field Method for Searching Elliptic Curves with High Ranks." Journal of New Theory, no. 47 (June 30, 2024): 20–27. http://dx.doi.org/10.53570/jnt.1467401.

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Анотація:

The theory of elliptic curves is one of the popular topics of recent times with its unsolved problems and interesting conjectures. In 1922, Mordell proved that the group of $\mathbb{Q}$-rational points on an elliptic curve is finitely generated. However, the rank of this group, signifying the number of independent generators, can be arbitrarily high for certain curves, a fact yet to be definitively proven. This study leverages the computer algebra system Magma to investigate curves with potentially high ranks using a technique developed by Mestre.

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Kamthawee, Krissanee, and Bhichate Chiewthanakul. "The Construction of ElGamal over Koblitz Curve." Advanced Materials Research 931-932 (May 2014): 1441–46. http://dx.doi.org/10.4028/www.scientific.net/amr.931-932.1441.

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Анотація:

Recently elliptic curve cryptosystems are widely accepted for security applications key generation, signature and verification. Cryptographic mechanisms based on elliptic curves depend on arithmetic involving the points of the curve. it is possible to use smaller primes, or smaller finite fields, with elliptic curves and achieve a level of security comparable to that for much larger integers. Koblitz curves, also known as anomalous binary curves, are elliptic curves defined over F2. The primary advantage of these curves is that point multiplication algorithms can be devised that do not use any

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Carita, Sa'aadah Sajjana, and Herman Kabetta. "MODIFICATION OF POLLARD RHO ALGORITHM USING NEGATION MAPPING." BAREKENG: Jurnal Ilmu Matematika dan Terapan 16, no. 4 (2022): 1159–66. http://dx.doi.org/10.30598/barekengvol16iss4pp1159-1166.

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El Gamal encryption was introduced in 1985 and is still commonly used today. Its hardness is based on a discrete logarithm problem defined over the finite abelian cyclic group group chosen in the original paper was but later it was proven that using the group of Elliptic Curve points could significantly reduce the key size required. The modified El Gamal encryption is dubbed its analog version. This analog encryption bases its hardness on Elliptic Curve Discrete Logarithm Problem (ECDLP). One of the fastest attacks in cracking ECDLP is the Pollard Rho algorithm, with the expected number of ite

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Дисертації з теми "Elliptic curve points group"

McGee, John J. "René Schoof's Algorithm for Determining the Order of the Group of Points on an Elliptic Curve over a Finite Field." Thesis, Virginia Tech, 2006. http://hdl.handle.net/10919/31911.

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Анотація:

Elliptic curves have a rich mathematical history dating back to Diophantus (c. 250 C.E.), who used a form of these cubic equations to find right triangles of integer area with rational sides. In more recent times the deep mathematics of elliptic curves was used by Andrew Wiles et. al., to construct a proof of Fermat's last theorem, a problem which challenged mathematicians for more than 300 years. In addition, elliptic curves over finite fields find practical application in the areas of cryptography and coding theory. For such problems, knowing the order of the group of points satisfying t

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Hişil, Hüseyin. "Elliptic curves, group law, and efficient computation." Thesis, Queensland University of Technology, 2010. https://eprints.qut.edu.au/33233/1/H%C3%BCseyin_Hi%C5%9Fil_Thesis.pdf.

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Анотація:

This thesis is about the derivation of the addition law on an arbitrary elliptic curve and efficiently adding points on this elliptic curve using the derived addition law. The outcomes of this research guarantee practical speedups in higher level operations which depend on point additions. In particular, the contributions immediately find applications in cryptology. Mastered by the 19th century mathematicians, the study of the theory of elliptic curves has been active for decades. Elliptic curves over finite fields made their way into public key cryptography in late 1980’s with independent pro

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Li, Chao. "2-Selmer groups and Heegner points on elliptic curves." Thesis, Harvard University, 2015. http://nrs.harvard.edu/urn-3:HUL.InstRepos:17464036.

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This thesis studies several aspects of the arithmetic of elliptic curves. In particular, we explore the prediction of the Birch and Swinnerton-Dyer conjecture when the 2-Selmer group has rank one. For certain elliptic curves $E/\mathbb{Q}: y^2=F(x)$ with additive reduction at 2, we determine their 2-Selmer ranks in terms of the 2-rank of the class group of the cubic field $L=\mathbb{Q}[x]/F(x)$. We then interpret this result as a mod 2 congruence between the Hasse-Weil $L$-function of $E$ and a degree two Artin $L$-function associated to the cubic field $L$. When the class number of $L$ is o

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Kouchaki, Barzi Behnaz. "Points of High Order on Elliptic Curves : ECDSA." Thesis, Linnéuniversitetet, Institutionen för matematik (MA), 2016. http://urn.kb.se/resolve?urn=urn:nbn:se:lnu:diva-58449.

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This master thesis is about Elliptic Curve Digital Signature Algorithm or ECDSA and two of the known attacks on this security system. The purpose of this thesis is to find points that are likely to be points of high order on an elliptic curve. If we have a point P of high order and if Q = mP, then we have a large set of possible values of m. Therefore it is hard to solve the Elliptic Curve Discrete Logarithm Problem or ECDLP. We have investigated on the time of finding the solution of ECDLP for a certain amount of elliptic curves based on the order of the point which is used to create the digi

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DeLorme, Cheryl Lynn 1969. "On the Shafarevich-Tate group of an elliptic curve." Diss., The University of Arizona, 1997. http://hdl.handle.net/10150/288706.

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This dissertation work concentrates on finding non-trivial elements in the Shafarevich-Tate group of an elliptic curve. The set of K-rational points on an elliptic curve, E, are known to form a finitely generated abelian group. My results are of interest when trying to find the rank of this group, which in general is a hard problem. The Selmer group of E,S(E/K), can be used to give a bound on this rank, and the obstruction to using this to find the exact rank is the Shafarevich-Tate group, scIII(E/K). There is a pairing on scIII(E/K), called the Cassels-Tate pairing, which is non-degenerate mo

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Ito, Koki. "Elliptic hypergeometric functions associated to the configuration space of three-points on an affine elliptic curve." 京都大学 (Kyoto University), 2006. http://hdl.handle.net/2433/144133.

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Kyoto University (京都大学) 0048 新制・課程博士 博士(理学) 甲第12064号 理博第2958号 新制||理||1443(附属図書館) 23900 UT51-2006-J59 京都大学大学院理学研究科数学・数理解析専攻 (主査)教授深谷賢治, 教授河野明, 助教授梅田亨 学位規則第4条第1項該当

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Lester, Jeremy W. "The Elliptic Curve Group Over Finite Fields: Applications in Cryptography." Youngstown State University / OhioLINK, 2012. http://rave.ohiolink.edu/etdc/view?acc_num=ysu1348847698.

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Weimerskirch, Andre. "The Application of the Mordell-Weil Group to Cryptographic Systems." Digital WPI, 2001. https://digitalcommons.wpi.edu/etd-theses/321.

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This thesis examines the Mordell-Weil group for application in cryptography. This approach has recently been proposed by Gerhard Frey. The use of the Mordell-Weil group for discrete logarithm schemes is a variant of elliptic curve cryptosystems. We extended the original idea by Frey with the goal of a performance improvement. The arithmetic complexity using the Mordell-Weil group will be compared to ordinary elliptic curve cryptosystems. The main goals of this thesis are (1) to investigate the algorithmic complexity of Mordell-Weil cryptosystems relative to elliptic curve cryptosystems; (2) th

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Wade, Darryl Gene. "The tropical Jacobian of an elliptic curve is the group S¹(Q) /." Diss., CLICK HERE for online access, 2008. http://contentdm.lib.byu.edu/ETD/image/etd2521.pdf.

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Wilcox, Nicholas. "A Computational Introduction to Elliptic and Hyperelliptic Curve Cryptography." Oberlin College Honors Theses / OhioLINK, 2018. http://rave.ohiolink.edu/etdc/view?acc_num=oberlin1528649455201473.

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Книги з теми "Elliptic curve points group"

Fourier expansions for Eisenstein series twisted by modular symbols and the distribution of multiples of real points on an elliptic curve. [publisher not identified], 2019.

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Gaitsgory, Dennis, and Jacob Lurie. Weil's Conjecture for Function Fields. Princeton University Press, 2019. http://dx.doi.org/10.23943/princeton/9780691182148.001.0001.

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A central concern of number theory is the study of local-to-global principles, which describe the behavior of a global field K in terms of the behavior of various completions of K. This book looks at a specific example of a local-to-global principle: Weil's conjecture on the Tamagawa number of a semisimple algebraic group G over K. In the case where K is the function field of an algebraic curve X, this conjecture counts the number of G-bundles on X (global information) in terms of the reduction of G at the points of X (local information). The goal of this book is to give a conceptual proof of

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Частини книг з теми "Elliptic curve points group"

Silverman, Joseph H., and John Tate. "The Group of Rational Points." In Rational Points on Elliptic Curves. Springer New York, 1992. http://dx.doi.org/10.1007/978-1-4757-4252-7_4.

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Silverman, Joseph H., and John T. Tate. "The Group of Rational Points." In Rational Points on Elliptic Curves. Springer International Publishing, 2015. http://dx.doi.org/10.1007/978-3-319-18588-0_3.

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Lenstra, H. W., and J. Pila. "Does the Set of Points of an Elliptic Curve Determine the Group?" In Computational Algebra and Number Theory. Springer Netherlands, 1995. http://dx.doi.org/10.1007/978-94-017-1108-1_8.

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Schoof, René. "The exponents of the groups of points on the reductions of an elliptic curve." In Arithmetic Algebraic Geometry. Birkhäuser Boston, 1991. http://dx.doi.org/10.1007/978-1-4612-0457-2_15.

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Menezes, Alfred. "Counting Points on Elliptic Curves Over F2m." In Elliptic Curve Public Key Cryptosystems. Springer US, 1993. http://dx.doi.org/10.1007/978-1-4615-3198-2_7.

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Silverman, Joseph H. "The Formal Group of an Elliptic Curve." In The Arithmetic of Elliptic Curves. Springer New York, 2009. http://dx.doi.org/10.1007/978-0-387-09494-6_4.

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Silverman, Joseph H. "The Formal Group of an Elliptic Curve." In The Arithmetic of Elliptic Curves. Springer New York, 1986. http://dx.doi.org/10.1007/978-1-4757-1920-8_5.

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Husemöller, Dale. "Elementary Properties of the Chord-Tangent Group Law on a Cubic Curve." In Elliptic Curves. Springer New York, 1987. http://dx.doi.org/10.1007/978-1-4757-5119-2_2.

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Chakraborty, Debopam. "On Class Number Divisibility of Number Fields and Points on Elliptic Curves." In Class Groups of Number Fields and Related Topics. Springer Singapore, 2020. http://dx.doi.org/10.1007/978-981-15-1514-9_10.

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Mérai, László. "On Pseudorandom Properties of Certain Sequences of Points on Elliptic Curve." In Arithmetic of Finite Fields. Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-55227-9_4.

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Тези доповідей конференцій з теми "Elliptic curve points group"

Rahman, Md Sazedur, and Kalyan Kumar Halder. "Efficient Implementation of Group Operations for Elliptic Curve Cryptography." In 2024 IEEE International Conference on Signal Processing, Information, Communication and Systems (SPICSCON). IEEE, 2024. https://doi.org/10.1109/spicscon64195.2024.10940954.

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TAKEUCHI, RYOUHEI. "ON DISTRIBUTION OF THE GROUP OF RATIONAL POINTS OF REDUCTIONS OF AN ELLIPTIC CURVE." In Proceedings of the First International Congress of Mathematical Software. WORLD SCIENTIFIC, 2002. http://dx.doi.org/10.1142/9789812777171_0027.

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Baanen, Anne, Alex J. Best, Nirvana Coppola, and Sander R. Dahmen. "Formalized Class Group Computations and Integral Points on Mordell Elliptic Curves." In CPP '23: 12th ACM SIGPLAN International Conference on Certified Programs and Proofs. ACM, 2023. http://dx.doi.org/10.1145/3573105.3575682.

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Burhanuddin, Iftikhar A., and Ming-Deh A. Huang. "Elliptic curve torsion points and division polynomials." In Computational Aspects of Algebraic Curves. WORLD SCIENTIFIC, 2005. http://dx.doi.org/10.1142/9789812701640_0002.

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Faz-Hernández, Armando, and Julio López. "Generation of Elliptic Curve Points in Tandem." In Simpósio Brasileiro de Segurança da Informação e de Sistemas Computacionais. Sociedade Brasileira de Computação - SBC, 2020. http://dx.doi.org/10.5753/sbseg.2020.19230.

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A hash to curve function H, mapping bit strings to points on an elliptic curve, is often required in cryptographic schemes based on elliptic curves. Its construction is based on a deterministic encoding and a cryptographic hash function, which complementarily dominate its execution time. To improve the performance of H, we propose a parallel strategy where two units execute in tandem the internal operations of H. We instantiate this approach with a parallel software implementation of a hash to curve function that outputs points on a twisted Edwards curve. A performance benchmark on Haswell and

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Zhao, Jianhong, and Lixing Yang. "Elliptic Curve Integral Points on y2=x3+19x-46." In 2nd International Forum on Management, Education and Information Technology Application (IFMEITA 2017). Atlantis Press, 2018. http://dx.doi.org/10.2991/ifmeita-17.2018.108.

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Du, Xiancun, Jianhong Zhao, and Shuangqing Lv. "On the integral points of a special elliptic curve." In 2023 2nd International Conference on Applied Statistics, Computational Mathematics and Software Engineering (ASCMSE 2023), edited by Paulo Batista and Yudong Zhang. SPIE, 2023. http://dx.doi.org/10.1117/12.2692839.

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Sakarindr, Pitipatana, and Nirwan Ansari. "Elliptic Curve Cryptosystem-Based Group Key Management for Secure Group Communications." In MILCOM 2007 - IEEE Military Communications Conference. IEEE, 2007. http://dx.doi.org/10.1109/milcom.2007.4455002.

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Guangming Yang, Qi Wang, Qian Zhu, Jian Xu, Jin Chang, and Long Yan. "An Improved Group Key Agreement Based on Elliptic Curve." In 2010 Fourth International Conference on Genetic and Evolutionary Computing (ICGEC 2010). IEEE, 2010. http://dx.doi.org/10.1109/icgec.2010.114.

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Luhaib, Qasim Mohsin, and Ruma Kareem K. Ajeena. "Elliptic curve group ring for hybrid public key cryptosystem." In INTERNATIONAL RESEARCH CONFERENCE ON ENGINEERING AND APPLIED SCIENCES 2023: IRCEAS2023. AIP Publishing, 2025. https://doi.org/10.1063/5.0257049.

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Звіти організацій з теми "Elliptic curve points group"

Wang, Yao, Jeehee Lim, Rodrigo Salgado, Monica Prezzi, and Jeremy Hunter. Pile Stability Analysis in Soft or Loose Soils: Guidance on Foundation Design Assumptions with Respect to Loose or Soft Soil Effects on Pile Lateral Capacity and Stability. Purdue University, 2022. http://dx.doi.org/10.5703/1288284317387.

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The design of laterally loaded piles is often done in practice using the p-y method with API p-y curves representing the behavior of soil at discretized points along the pile length. To account for pile-soil-pile interaction in pile groups, AASHTO (2020) proposes the use of p-multipliers to modify the p-y curves. In this research, we explored, in depth, the design of lateral loaded piles and pile groups using both the Finite Element (FE) method and the p-y method to determine under what conditions pile stability problems were likely to occur. The analyses considered a wide range of design scen

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