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NoC Topology
Published in Marcello Coppola, Miltos D. Grammatikakis, Riccardo Locatelli, Giuseppe Maruccia, Lorenzo Pieralisi, Design of Cost-Efficient Interconnect Processing Units, 2020
Marcello Coppola, Miltos D. Grammatikakis, Riccardo Locatelli, Giuseppe Maruccia, Lorenzo Pieralisi
Most constant degree topologies discussed in this Section can be derived from the well known family of Cayley graphs. This rich family of graphs can be used to generate small degree, low diameter networks. Cayley graphs are based on algebraic group theory, e.g. the N-node permutation group with composition operator, or integer with add/multiply operator. Nodes of the Cayley graph are all group elements, while edges (and therefore routing) are based on applying the group operator function (called generator, ⊗), i.e. node x connects to node y, if and only if x ⊗ γi, =y for some γi, ∈ S. Cayley graphs share many nice topological properties. For example, all Cayley graphs are vertex symmetric, and many are also edge-symmetric if all operator pairs are related through a group automorphism. Moreover, almost all Cayley graphs are Hamiltonian, and many are hierarchically recursive and optimally fault tolerant.
Very-Large-Scale Integration Implementations of Cryptographic Algorithms
Published in Tomasz Wojcicki, Krzysztof Iniewski, VLSI: Circuits for Emerging Applications, 2017
Elliptic curves were introduced to cryptography by Koblitz [2] and Miller [20] independently in the 1980s. Elliptic curves are described by the Weierstrass equation given below: E:y2+a1xy+a3y=x3+a2x2+a4x+a6 with ai,i = 1, 2, 3, 4, 6; x; y ∈ K, where K is a field. A point P = (x,y) is a valid point on the elliptic curve if it satisfies the Weierstrass equation. The basic operations performed on the elliptic curve are point addition and point doubling. Using these operations, a scalar multiplication Q = kP can be defined on an elliptic curve. The set of points on an elliptic curve together with a point at infinity ∞ with point addition as an operation makes the set of points an algebraic group.
Minimal control effort and time Lie-group synchronisation design based on proportional-derivative control
Published in International Journal of Control, 2022
A smooth manifold that is also an algebraic group is termed a Lie group. An algebraic group is a structure () where m : is an associative binary operation (such as a multiplication), is an identity element with respect to m and i : is the function which represents the inversion with respect to the operation m, so that for each . A left translation on a Lie group is denoted by and is defined as .