China
Africa
Explore chapters and articles related to this topic
Algebraic Structures I (Matrices, Groups, Rings, and Fields)
Published in R. Balakrishnan, Sriraman Sridharan, Discrete Mathematics, 2019
R. Balakrishnan, Sriraman Sridharan
A group G is non-AbelianGroup!nonabelian if it is not Abelian, that is, there exists a pair of elements x, y in G with xy ≠ yx.
The Group Theory
Published in Mikhail G. Brik, Chong-Geng Ma, Theoretical Spectroscopy of Transition Metal and Rare Earth Ions, 2019
Mikhail G. Brik, Chong-Geng Ma
As seen from this table, all four basic properties of a group are fulfilled. It is also easy to find the inverse elements for each member of this group: The E, A, B, C elements are inverse for themselves, whereas the F and D elements are inverse for each other. This is an example of a finite non-Abelian group.
Elements of topology and homology
Published in Rodrigo Rojas Moraleda, Nektarios A. Valous, Wei Xiong, Niels Halama, Computational Topology for Biomedical Image and Data Analysis, 2019
Rodrigo Rojas Moraleda, Nektarios A. Valous, Wei Xiong, Niels Halama
Non-Abelian group A non-Abelian group is a group (G, *) in which there are at least two elements a, b ∈ G such that a*b ≠ b*a.
An improved anti-quantum MST3 public key encryption scheme for remote sensing images
Published in Enterprise Information Systems, 2021
Xianmin Wang, Jing Li, Hongyang Yan
To address this problem, some encryption algorithms are employed for remote sensing images, in which the public key encryption (PKE) schemes are the most representative schemes (Huang et al. 2019b,a; Li et al. 2019). However, with the great progress in quantum algorithms, most PKE schemes using the integer factoring (IFP) and discrete logarithm (DLP) problems have been threatened due to the emergence of quantum algorithms, among which Shor’s algorithm (Shor 1999; Li et al. 2019) is famous and efficient for solving the above problems. Thus, it is important to design secure PKE schemes based on new hard assumptions that can resist quantum attacks (Riad and Ke. n.d., 2018). In the previous quantum cryptography studies, some difficult problems in group theory were given much attention, and construction of the corresponding cryptosystems was attempted. In 1984, an approach to construct public key cryptosystems with the undecidable word problem over groups and semigroups was presented by Wagner et al. (Wagner and Magyarik 1984). In 2000, Ko et al. (Ko et al. 2000) introduced the theory of braid-based cryptography, where the security depends on the hardness of the conjugator search problem (CSP) for braid groups. In 2005, Thompson’s group was suggested to be a good platform for designing cryptosystems (Shpilrain and Ushakov 2005). Subsequently, Kahrobaei et al. used matrices over group rings to construct a key exchange protocol. At the same time, another new non-commutative cryptosystem on the basis of the factorisation problem over non-abelian groups (GFP) produced promising results. Magliveras et al. designed two public key cryptosystems, MST1 and MST2, based on random covers and logarithmic signatures for non-abelian groups (Magliveras et al. 2002) (here, random covers act as public keys in encryption (Li et al. 2018a), and logarithmic signature serves as the private key in decryption). In 2009, Magliveras et al. constructed a new public key MST3 cryptosystem, achieving higher security (Lempken et al. 2009). Certainly, there are some interesting works studying attacks on MST1, MST2 and MST3. Svaba and Van Trung (2010) presented an improved MST3 cryptosystem by introducing a homomorphism as a private key (Svaba and Van Trung 2010). However, the existing MST series PKE schemes are insecure under malleable attacks for ciphertexts.