Accelerating Galois Field Arithmetic for Reed-Solomon Erasure Codes in Storage Applications

2011 ◽  
Author(s):  
Sebastian Kalcher ◽  
Volker Lindenstruth
Keyword(s):  
Galois Field ◽  
Erasure Codes

scholarly journals Private and rateless adaptive coded matrix-vector multiplication

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Rawad Bitar ◽  
Yuxuan Xing ◽  
Yasaman Keshtkarjahromi ◽  
Venkat Dasari ◽  
Salim El Rouayheb ◽  
...  
Keyword(s):  
Computing Methods ◽  
Erasure Codes ◽  
Time Varying ◽  
New Paradigm ◽  
Matrix Vector Multiplication ◽  
Processing Data ◽  
Computationally Intensive ◽  
Matrix Vector ◽  
Monitoring Devices ◽  
Theoretical Results

AbstractEdge computing is emerging as a new paradigm to allow processing data near the edge of the network, where the data is typically generated and collected. This enables critical computations at the edge in applications such as Internet of Things (IoT), in which an increasing number of devices (sensors, cameras, health monitoring devices, etc.) collect data that needs to be processed through computationally intensive algorithms with stringent reliability, security and latency constraints. Our key tool is the theory of coded computation, which advocates mixing data in computationally intensive tasks by employing erasure codes and offloading these tasks to other devices for computation. Coded computation is recently gaining interest, thanks to its higher reliability, smaller delay, and lower communication costs. In this paper, we develop a private and rateless adaptive coded computation (PRAC) algorithm for distributed matrix-vector multiplication by taking into account (1) the privacy requirements of IoT applications and devices, and (2) the heterogeneous and time-varying resources of edge devices. We show that PRAC outperforms known secure coded computing methods when resources are heterogeneous. We provide theoretical guarantees on the performance of PRAC and its comparison to baselines. Moreover, we confirm our theoretical results through simulations and implementations on Android-based smartphones.

scholarly journals Modular invariant models of leptons at level 7

2020 ◽  
Vol 2020 (8) ◽  
Author(s):  
Gui-Jun Ding ◽  
Stephen F. King ◽  
Cai-Chang Li ◽  
Ye-Ling Zhou
Keyword(s):  
Modular Forms ◽  
Galois Field ◽  
Special Linear Group ◽  
Type I ◽  
Time Level ◽  
Projective Special Linear Group ◽  
Modular Invariant ◽  
Seesaw Mechanism ◽  
Linearly Independent ◽  
First Time

Abstract We consider for the first time level 7 modular invariant flavour models where the lepton mixing originates from the breaking of modular symmetry and couplings responsible for lepton masses are modular forms. The latter are decomposed into irreducible multiplets of the finite modular group Γ7, which is isomorphic to PSL(2, Z7), the projective special linear group of two dimensional matrices over the finite Galois field of seven elements, containing 168 elements, sometimes written as PSL2(7) or Σ(168). At weight 2, there are 26 linearly independent modular forms, organised into a triplet, a septet and two octets of Γ7. A full list of modular forms up to weight 8 are provided. Assuming the absence of flavons, the simplest modular-invariant models based on Γ7 are constructed, in which neutrinos gain masses via either the Weinberg operator or the type-I seesaw mechanism, and their predictions compared to experiment.

scholarly journals A Technique for Securing Multiple Digital Images Based on 2D Linear Congruential Generator, Silver Ratio, and Galois Field

IEEE Access ◽  
2021 ◽  
pp. 1-1
Author(s):  
Abdul Gaffar ◽  
Anand B. Joshi ◽  
Sonali Singh ◽  
Vishnu Narayan Mishra ◽  
Hamurabi Gamboa Rosales ◽  
...  
Keyword(s):  
Digital Images ◽  
Galois Field ◽  
Congruential Generator ◽  
Linear Congruential Generator ◽  
Linear Congruential
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