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Number Theory and Cryptographic Hardness Assumptions
Published in Jonathan Katz, Yehuda Lindell, Introduction to Modern Cryptography, 2020
A useful way to think about E(ℤp) is to look at the graph of Equation (9.2) over the reals (i.e., the equation y2 = x3 +Ax+B without reduction modulo p) as in Figure 9.2. This figure does not correspond exactly to E(ℤp) because, for example, E(ℤp) has a finite number of points (ℤp is, after all, a finite set) while there are an infinite number of solutions to the same equation if we allow x and y to range over all real numbers. Nevertheless, the picture provides useful intuition. In such a figure, one can think of the “point at infinity” O as sitting at the top of the y-axis and lying on every vertical line.
Functions of a Complex Variable
Published in Vladimir Eiderman, An Introduction to Complex Analysis and the Laplace Transform, 2021
Now we bring into consideration an additional, special, point not shown in the diagram: this is called the point at infinity and is denoted by z=∞. This imaginable point z=∞ will be added to the plane ℂ, and we place it in correspondence with the point P∈S. This combination of ℂ with the point at infinity is called the extended complex plane and is denoted by ℂ¯ (this notation is used because in fact ℂ¯ has some of the properties of a closed domain). Each point z∈ℂ¯ corresponds to a unique point Z∈S, and conversely. The sphere S is called the Riemann sphere.2
The pixel's visual territory
Published in Linda Matthews, Design Strategies for Reimagining the City, 2022
One example concerns the conceptualisation of the line and its trajectory. In linear perspective geometry, parallel lines intersect at the point of infinity rather than abiding by the Euclidean proposition that they never intersect. However, although reliant upon the same perceptual principle, paradoxically, digital lines can cross without intersecting because they are composed of pixel sequences that intersect in segments. There is no common pixel at the intersection point. Put simply, while these two geometries are perceptually alike, they are structurally radically dissimilar, and it is precisely this structural dissimilarity that describes the digital image's many new representational opportunities.
Entropy generation analysis in MHD hybrid nanofluid flow: Effect of thermal radiation and chemical reaction
Published in Numerical Heat Transfer, Part B: Fundamentals, 2023
For computing the results of the flow problem modeled on the geometry of the disk, the dimensionless ODE’s system (10)–(15) along with boundary conditions (16) is solved by a numerical technique called the BVP Mid rich scheme. The results are obtained using Maple software. The BVP cannot be solved over an infinite interval, and even solving it for a sizeable finite interval would be impractical. The infinite boundary condition at a finite point at infinity in this research is 4 and 7 for velocity and temperature field, respectively. Also, in order to verify the accuracy of the current analysis, the numerical results of and have been compared with those of Rashidi et al. [47] for the limiting case when and at various values of M (Table 3). The table shows that the numerical values of the physical entity derived by us and Rashidi et al. [47] exhibit excellent agreement. The good agreement is an encouragement for further study of the effects of other parameters on the flow.
An Improved 2-Factor Authentication Scheme for WSN Based on ECC
Published in IETE Technical Review, 2023
Bhanu Chander, Gopalakrishnan Kumaravelan
Elliptic Curve Cryptography (ECC) is a special kind of public-key cryptography built on mathematically programmed elliptic curves that require smaller key sizes. Thus, it could be an appropriate target for resource-constrained situations. Famous well-known researchers Victor Miller projected ECC in 1885, then Neal Koblitz in 1985. An elliptic curve over a finite field demarcated as the set of every (x, y) ∈ Fp × Fp such that = + ax + b, here a, b ∈ Fp and mod p ≠ 0, and a prominent point at infinity which O. symbolizes. It should assume that the Gateway node is more computationally efficient than sensor nodes and holds a secured database where the list of registered sensor nodes is stored. In addition, every sensor node and Gateway node hoard their corresponding individualities in memory formerly than network placement.
Transcritical bifurcation at infinity in planar piecewise polynomial differential systems with two zones
Published in Dynamical Systems, 2022
Denis de Carvalho Braga, Jaume Llibre, Luis Fernando Mello
By means of another projection, for instance, the gnomonic projection such as in [11], we can study the vector field obtained by the projection of onto , where is the Poincaré disc. It follows that there exists a one-to-one correspondence between points placed at infinity of G and points on of . In this sense, we say is a singular point at infinity of the vector field G, if . When G has no singular points at infinity we say G has a periodic orbit at infinity which is identified with .