# Hardware-software Scalable Architectures for Gaussian Elimination over GF(2) and Higher Galois Fields

### Prateek Saxena, Vinay B. Y. Kumar, Dilawar Singh, H Narayanan, Sachin B. Patkar

#### Abstract

Solving a System of Linear Equations over Finite Fields finds one of the most important practical applications, for instance, in problems arising in cryptanalysis and network coding among others. However, other than software-only approaches to acceleration, the amount of focus particularly towards hardware or hardware/software based solutions is small, in comparison to that towards general linear equation solvers. We present scalable architectures for Gaussian elimination with pivoting over GF(2) and higher fields, both as custom extensions to commodity processors or as dedicated hardware for larger problems. In particular, we present: 1) Designs of components---Matrix Multiplication and `Basis search and Inversion'---for Gaussian elimination over GF(2), prototyped as custom instruction extensions to Nios-II on DE2-70(DE2, 2008), which even with a 50MHz clock perform at $\approx$30~GOPS (billion GF(2) operations); and also report results for GF($2^8$) or higher order matrix multiplication with about 20~GOPS performance at 200MBps. 2) A scalable extension of a previous design [Bogdanov et. al, 2006] for multiple FPGAs and with $\approx$2.5 TrillionOPS performance at ~5GBps bandwidth on a Virtex-5 FPGA

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#### Paper Citation

#### in Harvard Style

Saxena P., B. Y. Kumar V., Singh D., Narayanan H. and B. Patkar S. (2013). **Hardware-software Scalable Architectures for Gaussian Elimination over GF(2) and Higher Galois Fields** . In *Proceedings of the 3rd International Conference on Pervasive Embedded Computing and Communication Systems - Volume 1: PECCS,* ISBN 978-989-8565-43-3, pages 195-201. DOI: 10.5220/0004313201950201

#### in Bibtex Style

@conference{peccs13,

author={Prateek Saxena and Vinay B. Y. Kumar and Dilawar Singh and H Narayanan and Sachin B. Patkar},

title={Hardware-software Scalable Architectures for Gaussian Elimination over GF(2) and Higher Galois Fields},

booktitle={Proceedings of the 3rd International Conference on Pervasive Embedded Computing and Communication Systems - Volume 1: PECCS,},

year={2013},

pages={195-201},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0004313201950201},

isbn={978-989-8565-43-3},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the 3rd International Conference on Pervasive Embedded Computing and Communication Systems - Volume 1: PECCS,

TI - Hardware-software Scalable Architectures for Gaussian Elimination over GF(2) and Higher Galois Fields

SN - 978-989-8565-43-3

AU - Saxena P.

AU - B. Y. Kumar V.

AU - Singh D.

AU - Narayanan H.

AU - B. Patkar S.

PY - 2013

SP - 195

EP - 201

DO - 10.5220/0004313201950201