Modeling of 3- and 5-Isogenies of Supersingular Edwards Curves

An analysis is made of the properties and conditions for the existence of 3- and 5-isogenies of complete and quadratic supersingular Edwards curves. For the encapsulation of keys based on the SIDH algorithm, it is proposed to use isogeny of minimal odd 3 and 5 degrees, which allows bypassing the problem of singular points of the 2nd and 4th orders, characteristic of 2-isogenies. A review of the main properties of the classes of complete, quadratic and twisted Edwards curves over a simple field is given. Formulas for the isogeny of odd degrees are reduced to a form adapted to curves in Weierstrass form. To do this, the modified law of addition of curve points in the generalized Edwards form is used, which preserves the horizontal symmetry of the curve’s return points. Examples of the calculation of 3- and 5-isogenies of complete Edwards supersingular curves over small simple fields are given, and the properties of the isogeny composition for computing isogenies with large-order kernels are discussed. Formulas of upper bounds for the complexity of computing isogeny of odd degrees 3 and 5 in the classes of complete and quadratic Edwards curves in projective coordinates are obtained. Algorithms for calculating 3- and 5-isogenies of Edwards curves with complexity and 12M+5S, respectively, are constructed. The conditions for the existence of supersingular complete and quadratic Edwards curves of the order 4·3m·5n and 8·3m·5n are found. Some parameters of the cryptosystem were determined during the implementation of the SIDH algorithm at the quantum security level of 128 bits

3- and 5-Isogenies of Supersingular Edwards Curves

An analysis is made of the properties and conditions for the existence of 3- and 5-isogenies of complete and quadratic supersingular Edwards curves. For the encapsulation of keys based on the SIDH algorithm, it is proposed to use isogeny of minimal odd degrees 3 and 5, which allows bypassing the problem of singular points of the 2nd and 4th orders, characteristic of 2-isogenies. A review of the main properties of the classes of complete, quadratic, and twisted Edwards curves over a simple field is given. Equations for the isogeny of odd degrees are reduced to a form adapted to curves in the form of Weierstrass. To do this, use the modified law of addition of curve points in the generalized Edwards form, which preserves the horizontal symmetry of the curve return points. Examples of the calculation of 3- and 5-isogenies of complete Edwards supersingular curves over small simple fields are given, and the properties of the isogeny composition for their calculation with large-order kernels are discussed. Equations are obtained for upper complexity estimates for computing isogeny of odd degrees 3 and 5 in the classes of complete and quadratic Edwards curves in projective coordinates; algorithms are constructed for calculating 3- and 5-isogenies of Edwards curves with complexity 6M + 4S and 12M + 5S, respectively. The conditions for the existence of supersingular complete and quadratic Edwards curves of order 4x3mx5n and 8x3mx5n are found. Some parameters of the cryptosystem are determined when implementing the SIDH algorithm at the level of quantum security of 128 bits

Nanoeconomics: A Statistical Model of Company Profit Influenced by Individual Interests of Managers

A concept and mathematical model of modern economics are formulated in which a company profit is defined when taking into account possible interests of individual decision makers rather than based exclusively on benefits of either the company (ground of microeconomics) or the whole society (ground of macroeconomics). We call this approach “nanoeconomics” as a terminological step down from macro- and micro- economics. The growing gap between interests of the business owner and decision makers in large corporations is obviously detrimental for the company performance. Here we formulate a statistical model to describe this situation quantitatively. In this model, the company profit is an accumulative effect of statistical contributions of a variety of decisions of individual decision-makers, rather than looking at the company as a whole (microeconomics). The model demonstrates the decrease of profit of the company compared to the microeconomic estimation. We study a relative contribution to the profit decrease from different human factors considered in the model, “careerism” and “expertise” of the decision-making managers, their “loyalty” to the company, and the lack of strong company management (“decentralization”)

A shift in the paradigm of energy cooperation between Russia and Northeast Asia countries facing new global and regional challenges: from predominant sale of energy resources to innovation and technology cooperation

The paper is concerned with the specific features of Russia’s economic development in a new economic environment that caused the need to revise the priorities of energy policy. The research presents the initial conditions, targets and strategic directions of energy development in the East of the country. The focus is made on the priority lines of innovation and technology cooperation between Russia and Northeast Asian countries in the field of energy, and recommendations on necessary conditions and initiatives for their successful implementation are given

Energy cooperation between Mongolia and Russia: past, present, and future

The paper showcases the experience of energy cooperation between Mongolia and Russia and it gives a brief description of the current state and outlines strategic directions of energy cooperation between the two countries from the perspective of current energy cooperation priorities in the Northeast Asian countries, and active mutually beneficial participation of Mongolia and Russia in this process. Prerequisites for cooperation and conditions for successful intercountry projects in individual industries of the energy sector are considered in the study

Computing of Odd Degree Isogenies on Supersingular Twisted Edwards Curves

An overview of the properties of three classes of curves in generalized Edwards form Ea,d with two parameters is given. The known formulas for the odd degree isogenies on curves Ed with one parameter are generalized to all classes of curves in Edwards form, and Theorem 1 on the isogenic mapping of the points of these curves is proved. The analysis of the known effective method for computing isogenies in Farashahi-Hosseini w-coordinates, justified for the curve Ed, is given. Theorem 2 proves the applicability of this method to the class of twisted Edwards curves. Examples of 3- and 5-isogenies of twisted Edwards curves are given. Methods for bypassing the exceptional points of such curves in PQC cryptosystems like CSIDH are proposed