Solving Polynomial Systems

Ding, Jíntai; Petzoldt, Albrecht; Schmidt, Dieter

doi:10.1007/978-1-0716-0987-3_8

Cited by 4 publications

(7 citation statements)

References 14 publications

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“…From a root of ( 11), all other quantities can be determined. However, Ryley does not compute any root of (11) or values of a, b, and c for the given l, m, and n.…”

Section: A Renewed Interest In Colonel Titus' Problemmentioning

confidence: 99%

“…lxxi-lxxii. By interchanging the variables, Equation ( 13) can be derived from (11). For l = 16, m = 17, and n = 18 we obtain the equation…”

Section: First Reformulation and Then Eliminationmentioning

confidence: 99%

“…They arise in robotics, coding theory, optimization, mathematical biology, computer vision, game theory, statistics, machine learning, control theory, and numerous other areas [10]. Systems of quadratic polynomial equations appear in nearly every crypto-system [11] and in robotics [12].…”

mentioning

confidence: 99%

“…For a modern treatment of numerical methods for roots of polynomials, see [13,14] and references therein. For solving systems of polynomial equations, see [10,11] and references therein.…”

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confidence: 99%

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A Story of Computational Science: Colonel Titus’ Problem from the 17th Century

Steihaug

2022

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Experimentation and the evaluation of algorithms have a long history in algebra. In this paper we follow a single test example over more than 250 years. In 1685, John Wallis published A treatise of algebra, both historical and practical, containing a solution of Colonel Titus’ problem that was proposed to him around 1650. The Colonel Titus problem consists of three algebraic quadratic equations in three unknowns, which Wallis transformed into the problem of finding the roots of a fourth-order (quartic) polynomial. When Joseph Raphson published his method in 1690, he demonstrated the method on 32 algebraic equations and one of the examples was this quartic equation. Edmund Halley later used the same polynomial as an example for his new methods in 1694. Although Wallis used the method of Vietè, which is a digit–by–digit method, the more efficient methods of Halley and Raphson are clearly demonstrated in the works by Raphson and Halley. For more than 250 years the quartic equation has been used as an example in a wide range of solution methods for nonlinear equations. This paper provides an overview of the Colonel Titus problem and the equation first derived by Wallis. The quartic equation has four positive roots and the equation has been found to be very useful for analyzing the number of roots and finding intervals for the individual roots, in the Cardan–Ferrari direct approach for solving quartic equations, and in Sturm’s method of determining the number of real roots of an algebraic equation. The quartic equation, together with two other algebraic equations, have likely been the first set of test examples used to compare different iteration methods of solving algebraic equations.

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“…From a root of ( 11), all other quantities can be determined. However, Ryley does not compute any root of (11) or values of a, b, and c for the given l, m, and n.…”

Section: A Renewed Interest In Colonel Titus' Problemmentioning

confidence: 99%

“…lxxi-lxxii. By interchanging the variables, Equation ( 13) can be derived from (11). For l = 16, m = 17, and n = 18 we obtain the equation…”

Section: First Reformulation and Then Eliminationmentioning

confidence: 99%

mentioning

confidence: 99%

“…For a modern treatment of numerical methods for roots of polynomials, see [13,14] and references therein. For solving systems of polynomial equations, see [10,11] and references therein.…”

mentioning

confidence: 99%

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A Story of Computational Science: Colonel Titus’ Problem from the 17th Century

Steihaug

2022

Axioms

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“…Алгоритмы такого типа разрабатываются и исследуются более 35 лет многочисленными исследователями из разных стран [14,15]. Широкий интерес к ним связан с тем, что их стойкость базируется на вычислительной трудности решения систем многих степенных уравнений с многими неизвестными [16], т. е. на задаче, для решения которой квантовый компьютер не является эффективным. Последнее означает, что стойкость этих алгоритмов к атакам с использованием обычных компьютеров обеспечивает стойкость и к квантовым атакам.…”

Section: Introductionunclassified

Algebraic Signature Algorithms With Complete Signature Randomization

2024

Cybersecurity Issues

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Цель работы: устранение потенциального снижения стойкости алгоритмов ЭЦП на некоммутативных алгебрах с увеличением числа подписанных электронных документов.Метод исследования: обеспечение полной рандомизации подписи путем включения в формулу генерации подгоночного элемента подписи случайного обратимого вектора как одного из множителей. Использование двух проверочных уравнений с вхождение одного и того же подгоночного элемента подписи. Формирование открытого ключа в виде набора векторов, вычисляемых в зависимости от векторов, содержащихся в скрытой (секретной) коммутативной группе конечной некоммутативной ассоциативной алгебры, используемой в качестве алгебраического носителя алгоритма ЭЦП.Результаты исследования: показана ограниченность рандомизации подписи, приводящая к снижению стойкости при увеличении числа подписанных документов, в ранее предложенных алгебраических алгоритмах ЭЦП со скрытой группой, стойкость которых основана на вычислительной трудности решения большой системы степенных уравнений. Разработан способ обеспечения полной рандомизации подписи в алгебраических алгоритмах указанного типа. Показано, что результаты изучения строения конечных некоммутативных алгебр (с точки зрения декомпозиции на множество коммутативных подалгебр), используемых в качестве алгебраического носителя, имеют существенное значение как для выбора параметров разрабатываемого алгоритма ЭЦП, так и для оценки его стойкости. Разработан новый алгебраический алгоритм ЭЦП, представляющий интерес как практичная постквантовая криптосхема благодаря достаточно малым размерам открытого ключа и подписи.Научная и практическая значимость результатов статьи состоит в разработке и апробации способа обеспечения полной рандомизации подписи и обоснования необходимости реализации последней в алгоритмах ЭЦП на некоммутативных ассоциативных алгебрах. Разработанный новый алгоритм ЭЦП является достаточно практичным и представляет интерес как прототип для разработки постквантовых алгоритмов ЭЦП, ориентированных на применение в условиях ограниченности доступных вычислительных ресурсов.

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Post-Quantum Public-Key Cryptoschemes on Finite Algebras

Moldovyan,

Moldovyan

2024

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One direction in the development of practical post-quantum public-key cryptographic algorithms is the use of finite algebras as their algebraic carrier. Two approaches in this direction are considered: 1) construction of electronic digital signature algorithms with a hidden group on non-commutative associative algebras and 2) construction of multidimensional cryptography algorithms using the exponential operation in a vector finite field (in a commutative algebra, which is a finite field) to specify a nonlinear mapping with a secret trapdoor. The first approach involves the development of two types of cryptoschemes: those based on the computational difficulty of a) the hidden discrete logarithm problem and b) solving a large system of quadratic equations. For the second type, problems arise in ensuring complete randomization of the digital signature and specifying non-commutative associative algebras of large dimension. Ways to solve these problems are discussed. The importance of studying the structure of finite non-commutative algebras from the point of view of decomposition into a set of commutative subalgebras is shown. Another direction is aimed at a significant (10 or more times) reduction in the size of the public key in multivariate-cryptography algorithms and is associated with the problem of developing formalized, parameterizable, unified methods for specifying vector finite fields of large dimensions (from 5 to 130) with a sufficiently large number of potentially implementable types and modifications each type (up to 2500 or more). Variants of such methods and topologies of nonlinear mappings on finite vector fields of various dimensions are proposed. It is shown that the use of mappings that specify the exponential operation in vector finite fields potentially eliminates the main drawback of known multivariate-cryptography algorithms, which is associated with the large size of the public key.

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Solving Polynomial Systems

Cited by 4 publications

References 14 publications

A Story of Computational Science: Colonel Titus’ Problem from the 17th Century

A Story of Computational Science: Colonel Titus’ Problem from the 17th Century

Algebraic Signature Algorithms With Complete Signature Randomization

Post-Quantum Public-Key Cryptoschemes on Finite Algebras

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