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Discrete Logarithm Problem
Published in Khaleel Ahmad, M. N. Doja, Nur Izura Udzir, Manu Pratap Singh, Emerging Security Algorithms and Techniques, 2019
Khaleel Ahmad, Afsar Kamal, Khairol Amali Bin Ahmad
The term “logarithm” is nothing but just an exponent in the sense of real number. For example, 32 = 9 (where 2 is called exponent). Now translating into the language of Logarithm it becomes Log3 9 = 2 (where Log3 9 is the exponent equal to 2 and 3 is the base of the logarithm). Basically, the exponent is used to calculate the product as x = 32 = 9, while Logarithm is to compute the exponent as x = log3 9 = 2. This prompts to the definition of Logarithm as Logxy = z, or xz = y, assuming x and y as positive besides x ≠ 1 in algebraic form. Indeed, Logxy is generally not an integer number. For example, the solution of logarithm Log2 3 = ? ↔ 2? = 3 gives the approximation result? ≈ 1.58496 as an exponent because of 21.58496 ≈ 2.99999 that is exactly 3. The word “discrete” in discrete logarithm refers to the aspect in discrete group {1, …, P − 1} only integer numbers rather than fractions (real numbers) (Pomerance, 2008). Now coming to the point of DLP.
Frequency Analysis
Published in Hector J. Rabal, Roberto A. Braga, Dynamic Laser Speckle and Applications, 2018
Lucía Isabel Passoni, Gonzalo Hernán, Gonzalo Hernán Sendra, Constancio Miguel Arizmendi
Selecting the mother wavelet and a discrete group of parameters aj = 2 j and bj = 2 jk with j, k ∈ Z (integer set), the family () ψj,k(t)=2−j/2ψ(2−jt−k),j,k,∈Z
Weighted estimates for commutators of anisotropic Calderón-Zygmund operators
Published in Applicable Analysis, 2022
For the anisotropy property, we can express it by a general discrete group of dilations with A being an matrix and all its eigenvalues λ satisfying in mathematics. For the function spaces, they also can be extended from classical Hardy spaces to anisotropic, parabolic and homogeneous Hardy spaces; see [8–10]. Moreover, we can also extend them to the weighted cases associated to general Muckenhoupt weights; see [11,12]. The theory of weighted Hardy space on was established by García-Cuerva [11] and Strömberg and Torchinsky [12], which generalized classical Hardy space but still retained the basic properties.