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Securing Cloud Data
Published in Olivier Terzo, Lorenzo Mossucca, Cloud Computing with e-Science Applications, 2017
We assume that file F (potentially encoded using Reed-Solomon codes) is divided into n blocks m1, m2, … , mn, where mi ∊ ℤq and q is a large prime. Let e: G × G → GT be a bilinear map and H: {0, 1}* → G be a hash function that converts binary strings to elements of G and is viewed as a random oracle. Let g be the generator of G.
Nonlinear Systems Analysis and Modeling
Published in Naim A. Kheir, Systems Modeling and Computer Simulation, 2018
Definition 4 Lie Algebra.A Lie algebra over the real field ℜ or the complex field l is a vector space G for which a bilinear map (X, Y) → [X, Y] is defined from G × G → G such that
Master-slave Robots, Optimal Control, Quantum Mechanics and Informationlds
Published in Harish Parthasarathy, Electromagnetics, Control and Robotics, 2023
Then, it is clear that span (Range(ϕ)) = W. To see this simply consider x = ei, y = fj where ei is the n × 1 vector with a one in its ith position and zero at all the other positions and likewise, ϕj is the m × 1 vector with a one in its jth position and zero at all the other positions. Then, ϕ(ei, fj) is the nm × 1 vector with a one in its (m(i –1) + j) th position and a zero at all the other positions. Further m(i–1) + j varies over the entire set {1, 2,...,nm} as i varies over 1, 2,...,n and j over 1, 2,...,m and hence any z ∈ W can be expressed as a linear combination of the vectors {ϕ(ei, fj): 1 ≤ i ≤ n, 1 ≤ j ≤ m}. In fact, this argument shows that {ϕ(ei, fj): 1 ≤ i ≤ n, 1 ≤ j ≤ m} is a permutation of the standard ordered basis for W = ℂnm. The second property is verified as follows: Let X be any vector space and T: U × V → W be a bilinear map. Then, T(x,y)=T(∑xiei,∑yjfj)=∑i,jxiyjT(ei,fj)
Lightweight MapReduce Application Service Integrity Auditing on the Cloud
Published in IETE Journal of Research, 2022
Panpan Li, Zhengxia Xie, Zengkai Wang, Zhigang Zhou
In the remainder of this paper, we will use the following notations. represents a unique identifier for the auditing list which is a list of auditing points on one worker. is the symmetric encryption of under key . is the one-way weakly collision hash function of . is the symmetric message authentication code under the key of X., represents the concatenation of with .Bilinear Map: Let ,, and be multiplicative cyclic groups of prime order . Let and be generators of and , respectively. A bilinear map is a map such that for all ,, and , . This bilinearity implies that for any , , . There is an efficiently computable algorithm for computing and the map should be non-trivial, i.e. is non-degenerate: .