Area-efficient two-dimensional architectures for finite field inversion and division

Many high-throughput two-dimensional architectures for finite field inversion and division are based on reformulations of the extended Euclidean algorithm (EEA). These reformulated EEAs usually keep track of two pairs of data polynomials (or registers), and the operations of the reformulated EEAs are only within each pair of polynomials. In this paper, we propose a new reformulated EEA wherein the operations within the two pairs of polynomials are identical. Hence, the two pairs of polynomials in our new reformulated EEA can be concatenated into one pair. By utilizing some inherent properties of the EEA, we further reduce the computational complexity of our reformulated EEA by 25%. Based on our reformulated EEA, we propose new two-dimensional inversion and division architectures. How much hardware saving the reduced computational complexity translates into depends on how control mechanisms are implemented. Regardless of the implementation of control signals, our new architectures require smaller numbers of gates and latches while achieving comparable or better throughput, latency, and critical path delay in comparison to the best architectures in the literature.

[1]  Dorothy Bollman,et al.  Fixed Points in Discrete Models for Regulatory Genetic Networks , 2007, EURASIP J. Bioinform. Syst. Biol..

[2]  V.K. Bhargava,et al.  A Modified Massey-Omura Parallel Multiplier for a Class of Finite Fields , 1993, IEEE Trans. Computers.

[3]  George Theodoridis,et al.  An efficient reconfigurable multiplier architecture for Galois field GF(2m) , 2003, Microelectron. J..

[4]  Kazuyoshi Takagi,et al.  A VLSI Algorithm for Division in GF(2m) Based on Extended Binary GCD Algorithm , 2002, IEICE Trans. Fundam. Electron. Commun. Comput. Sci..

[5]  Berk Sunar,et al.  Constructing Composite Field Representations for Efficient Conversion , 2003, IEEE Trans. Computers.

[6]  Máire O'Neill,et al.  FPGA Montgomery multiplier architectures - a comparison , 2004, 12th Annual IEEE Symposium on Field-Programmable Custom Computing Machines.

[7]  Humberto Ortiz-Zuazaga,et al.  Error Correction and Clustering Gene Expression Data Using Majority Logic Decoding , 2007, BIOCOMP.

[8]  Ke Wang,et al.  The VLSI Implementation of a Reed—Solomon Encoder Using Berlekamp's Bit-Serial Multiplier Algorithm , 1984, IEEE Transactions on Computers.

[9]  Zhiyuan Yan,et al.  New Systolic Architectures for Inversion and Division in GF(2^m) , 2003, IEEE Trans. Computers.

[10]  J. Pollard,et al.  The fast Fourier transform in a finite field , 1971 .

[11]  Patrik D'haeseleer,et al.  Genetic network inference: from co-expression clustering to reverse engineering , 2000, Bioinform..

[12]  Hidde de Jong,et al.  Modeling and Simulation of Genetic Regulatory Networks , 2003, POSTA.

[13]  Richard A. Spinello,et al.  Property Rights in Genetic Information , 2004, Ethics and Information Technology.

[14]  Edward R. Dougherty,et al.  Probabilistic Boolean networks: a rule-based uncertainty model for gene regulatory networks , 2002, Bioinform..

[15]  Keshab K. Parhi,et al.  VLSI digital signal processing systems , 1999 .

[16]  Christof Paar,et al.  Breaking Ciphers with COPACOBANA - A Cost-Optimized Parallel Code Breaker , 2006, CHES.

[17]  Werner Dubitzky,et al.  Modeling gene-regulatory networks using evolutionary algorithms and distributed computing , 2005, CCGrid 2005. IEEE International Symposium on Cluster Computing and the Grid, 2005..

[18]  Ilya Shmulevich,et al.  On Learning Gene Regulatory Networks Under the Boolean Network Model , 2003, Machine Learning.

[19]  James H. Cross,et al.  Reverse engineering and design recovery: a taxonomy , 1990, IEEE Software.

[20]  M. Anwar Hasan,et al.  Low complexity bit parallel architectures for polynomial basis multiplication over GF(2m) , 2004, IEEE Transactions on Computers.

[21]  Guillermo Morales-Luna,et al.  A Parallel Version of the Itoh-Tsujii Multiplicative Inversion Algorithm , 2007, ARC.

[22]  J. D. Lipson,et al.  Chinese remainder and interpolation algorithms , 1971, SYMSAC '71.

[23]  Uwe Meyer-Baese,et al.  Digital Signal Processing with Field Programmable Gate Arrays , 2001 .

[24]  A. F. Tenca,et al.  Novel Multiplier Architectures for GF ( p ) and GF ( 2 n ) , 2003 .

[25]  Reinhard Laubenbacher,et al.  Equivalence Relations on Finite Dynamical Systems , 2001, Adv. Appl. Math..

[26]  Tarek A. El-Ghazawi,et al.  Is High-Performance, Reconfigurable Computing the Next Supercomputing Paradigm? , 2006, ACM/IEEE SC 2006 Conference (SC'06).

[27]  Elwyn R. Berlekamp,et al.  Bit-serial Reed - Solomon encoders , 1982, IEEE Transactions on Information Theory.

[28]  Andrew A. Quong,et al.  Linear fuzzy gene network models obtained from microarray data by exhaustive search , 2004, BMC Bioinformatics.

[29]  H. Ortiz-Zuazaga,et al.  A finite field deterministic genetic network model , 2002 .

[30]  Brent E. Nelson,et al.  Optimal Finite Field Multipliers for FPGAs , 1999, FPL.

[31]  Tarek A. El-Ghazawi,et al.  Guest Editors' Introduction: High-Performance Reconfigurable Computing , 2007, Computer.

[32]  M. Anwar Hasan,et al.  A class of unidirectional bit serial systolic architectures for multiplicative inversion and division over GF(2/sup m/) , 2005, IEEE Transactions on Computers.

[33]  Nir Friedman,et al.  Inferring subnetworks from perturbed expression profiles , 2001, ISMB.

[34]  Min Zou,et al.  A new dynamic Bayesian network (DBN) approach for identifying gene regulatory networks from time course microarray data , 2005, Bioinform..

[35]  Alexander Vardy,et al.  Correcting errors beyond the Guruswami-Sudan radius in polynomial time , 2005, 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05).

[36]  Nikola Kasabov,et al.  Knowledge-based neural networks for gene expression data analysis, modelling and profile discovery , 2004 .

[37]  Ting Chen,et al.  Modeling Gene Expression with Differential Equations , 1998, Pacific Symposium on Biocomputing.

[38]  Thomas C. Bartee,et al.  Computation with Finite Fields , 1963, Inf. Control..

[39]  Çetin Kaya Koç,et al.  Mastrovito Multiplier for General Irreducible Polynomials , 1999, IEEE Trans. Computers.

[40]  Dennis Dalessandro,et al.  Accelerated FPGA Based Encryption * , 2005 .

[41]  M. Anwar Hasan,et al.  Look-Up Table-Based Large Finite Field Multiplication in Memory Constrained Cryptosystems , 2000, IEEE Trans. Computers.

[42]  V. Thorsson,et al.  Discovery of regulatory interactions through perturbation: inference and experimental design. , 1999, Pacific Symposium on Biocomputing. Pacific Symposium on Biocomputing.

[43]  Ashish Choudhury,et al.  Control approaches for probabilistic gene regulatory networks - What approaches have been developed for addreassinig the issue of intervention? , 2007, IEEE Signal Processing Magazine.

[44]  Chien-Ming Wu,et al.  Systolic VLSI realization of a novel iterative division algorithm over GF(2/sup m/): a high-speed, low-complexity design , 2001, ISCAS 2001. The 2001 IEEE International Symposium on Circuits and Systems (Cat. No.01CH37196).

[45]  M. Anwar Hasan,et al.  Low complexity word-level sequential normal basis multipliers , 2005, IEEE Transactions on Computers.

[46]  Oscar Moreno,et al.  Toward a Solution of the Reverse Engineering Problem Using FPGAs , 2006, Euro-Par Workshops.

[47]  Chaoyang Zhang,et al.  Comparison of probabilistic Boolean network and dynamic Bayesian network approaches for inferring gene regulatory networks , 2007, BMC Bioinformatics.

[48]  S. Kauffman.  Homeostasis and Differentiation in Random Genetic Control Networks , 1969, Nature.

[49]  Sung-Ming Yen.  IMPROVED NORMAL BASIS INVERSION IN GF(2M) , 1997 .

[50]  Vincent Frouin,et al.  Evolutionary approaches for the reverse-engineering of gene regulatory networks: A study on a biologically realistic dataset , 2008, BMC Bioinformatics.

[51]  Edusmildo Orozco,et al.  Reconfigurable Computing. Accelerating Computation with Field-Programmable Gate Arrays , 2007, Scalable Comput. Pract. Exp..

[52]  R. Laubenbacher,et al.  A computational algebra approach to the reverse engineering of gene regulatory networks. , 2003, Journal of theoretical biology.

[53]  Suteaki Shioya,et al.  Clustering gene expression pattern and extracting relationship in gene network based on artificial neural networks. , 2003, Journal of bioscience and bioengineering.

[54]  Michal Linial,et al.  Using Bayesian Networks to Analyze Expression Data , 2000, J. Comput. Biol..

[55]  Ziv Bar-Joseph,et al.  Analyzing time series gene expression data , 2004, Bioinform..

[56]  J. Zukas.  Introduction to the Modern Theory of Dynamical Systems , 1998 .

[57]  Brandilyn Suzanne Stigler,et al.  An Algebraic Approach to Reverse Engineering with an Application to Biochemical Networks , 2005 .

[58]  Klaus Huber.  Some comments on Zech's logarithms , 1990, IEEE Trans. Inf. Theory.

[59]  Tommi S. Jaakkola,et al.  Predictive Discretization during Model Selection , 2004, AISTATS.

[60]  Jesper Tegnér,et al.  Reverse engineering gene networks using singular value decomposition and robust regression , 2002, Proceedings of the National Academy of Sciences of the United States of America.

[61]  Le Yu,et al.  Inference of a Probabilistic Boolean Network from a Single Observed Temporal Sequence , 2007, EURASIP J. Bioinform. Syst. Biol..

[62]  Alfonso Valencia,et al.  A hierarchical unsupervised growing neural network for clustering gene expression patterns , 2001, Bioinform..

[63]  S. Kung,et al.  VLSI Array processors , 1985, IEEE ASSP Magazine.

[64]  Michael K. Ng,et al.  Simulation study in Probabilistic Boolean Network models for genetic regulatory networks , 2007, Int. J. Data Min. Bioinform..

[65]  H. Ortiz-Zuazaga.  Abstract of "Analysis of Gene Regulation Networks Using Finite-Field Models" , 2005 .

[66]  Trieu-Kien Truong,et al.  A Comparison of VLSI Architecture of Finite Field Multipliers Using Dual, Normal, or Standard Bases , 1988, IEEE Trans. Computers.

[67]  Kazuyoshi Takagi,et al.  A Fast Algorithm for Multiplicative Inversion in GF(2m) Using Normal Basis , 2001, IEEE Trans. Computers.

[68]  J. Hasty,et al.  Reverse engineering gene networks: Integrating genetic perturbations with dynamical modeling , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[69]  Hadi Shahriar Shahhoseini,et al.  The Best Irreducible Pentanomials For A Mastrovito GF Multiplier , 2006, IEEE International Conference on Computer Systems and Applications, 2006..

[70]  Zhiyuan Yan,et al.  High-speed systolic architectures for finite field inversion , 2005, Integr..

[71]  Zhiyuan Yan,et al.  High-speed systolic architectures for finite field inversion and division , 2004, GLSVLSI '04.

[72]  Berk Sunar,et al.  Achieving efficient polynomial multiplication in fermat fields using the fast Fourier transform , 2006, ACM-SE 44.

[73]  Zeljko Zilic,et al.  A Deterministic Multivariate Interpolation Algorithm for Small Finite Fields , 2002, IEEE Trans. Computers.

[74]  Soonhak Kwon,et al.  A Compact and Fast Division Architecture for a Finite Field , 2003, ICCSA.

[75]  Keshab K. Parhi,et al.  Low-complexity modified Mastrovito multipliers over finite fields GF(2/sup M/) , 1999, ISCAS'99. Proceedings of the 1999 IEEE International Symposium on Circuits and Systems VLSI (Cat. No.99CH36349).

[76]  Guillermo Morales-Luna,et al.  Parallel Itoh–Tsujii multiplicative inversion algorithm for a special class of trinomials , 2007, Des. Codes Cryptogr..

[77]  Francisco Rodríguez-Henríquez,et al.  Parallel Multipliers Based on Special Irreducible Pentanomials , 2003, IEEE Trans. Computers.

[78]  Davide De Caro,et al.  A Novel Architecture for Galois Fields GF(2^m) Multipliers Based on Mastrovito Scheme , 2007, IEEE Transactions on Computers.

[79]  Satoru Miyano,et al.  Identification of Genetic Networks from a Small Number of Gene Expression Patterns Under the Boolean Network Model , 1998, Pacific Symposium on Biocomputing.

[80]  Berk Sunar,et al.  Mastrovito Multiplier for All Trinomials , 1999, IEEE Trans. Computers.

[81]  Tommi S. Jaakkola,et al.  Using Graphical Models and Genomic Expression Data to Statistically Validate Models of Genetic Regulatory Networks , 2000, Pacific Symposium on Biocomputing.

[82]  Dieter Gollmann.  Equally Spaced Polynomials, Dual Bases, and Multiplication in F2n , 2002, IEEE Trans. Computers.

[83]  R. Laubenbacher,et al.  Boolean Monomial Dynamical Systems , 2004, math/0403166.

[84]  M. Anwar Hasan,et al.  A New Construction of Massey-Omura Parallel Multiplier over GF(2m) , 2002, IEEE Trans. Computers.

[85]  Bernd Sturmfels,et al.  Monomial Dynamical Systems over Finite Fields , 2006, Complex Syst..

[86]  P. L. Montgomery.  Modular multiplication without trial division , 1985 .

[87]  Hidde de Jong,et al.  Modeling and Simulation of Genetic Regulatory Systems: A Literature Review , 2002, J. Comput. Biol..

[88]  G. Amdhal,et al.  Validity of the single processor approach to achieving large scale computing capabilities , 1967, AFIPS '67 (Spring).

[89]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[90]  T. Itoh,et al.  A Fast Algorithm for Computing Multiplicative Inverses in GF(2^m) Using Normal Bases , 1988, Inf. Comput..

[91]  Francisco Rodríguez-Henríquez,et al.  A fast implementation of multiplicative inversion over GF(2/sup m/) , 2005, International Conference on Information Technology: Coding and Computing (ITCC'05) - Volume II.

[92]  Martin C. Herbordt,et al.  Achieving High Performance with FPGA-Based Computing , 2007, Computer.

[93]  M. A. Aviño-Diaz,et al.  FINITE DYNAMICAL SYSTEMS, LINEAR AUTOMATA, AND FINITE FIELDS , 2006, math/0604534.

[94]  Trieu-Kien Truong,et al.  VLSI Architectures for Computing Multiplications and Inverses in GF(2m) , 1983, IEEE Transactions on Computers.

[95]  Jean-Pierre Deschamps,et al.  Hardware Implementation of Finite-Field Division , 2006 .

[96]  Graham H. Norton.  Extending the Binary GCD Algorithm , 1985, AAECC.

[97]  Christof Paar,et al.  Itoh-Tsujii Inversion in Standard Basis and Its Application in Cryptography and Codes , 2002, Des. Codes Cryptogr..

[98]  Erkay Savas,et al.  Efficient Methods for Composite Field Arithmetic , 1999 .

[99]  S Fuhrman,et al.  Reveal, a general reverse engineering algorithm for inference of genetic network architectures. , 1998, Pacific Symposium on Biocomputing. Pacific Symposium on Biocomputing.

[100]  Guillermo Morales-Luna,et al.  FPGA implementation of an efficient multiplier over finite fields GF(2/sup m/) , 2005, 2005 International Conference on Reconfigurable Computing and FPGAs (ReConFig'05).

[101]  R. May.  Uses and Abuses of Mathematics in Biology , 2004, Science.

[102]  Dirk Repsilber,et al.  Reverse engineering of regulatory networks: simulation studies on a genetic algorithm approach for ranking hypotheses. , 2002, Bio Systems.

[103]  S. Kauffman.  Metabolic stability and epigenesis in randomly constructed genetic nets. , 1969, Journal of theoretical biology.

[104]  Victor Mitrana,et al.  A Formal Language-Based Approach in Biology , 2004, Comparative and functional genomics.

[105]  Oscar Moreno,et al.  A Parallel Solution to Reverse Engineering Genetic Networks , 2004, ICCSA.

[106]  Satoru Miyano,et al.  Identification of genetic networks by strategic gene disruptions and gene overexpressions under a boolean model , 2003, Theor. Comput. Sci..

[107]  Roger E Bumgarner,et al.  Integrated genomic and proteomic analyses of a systematically perturbed metabolic network. , 2001, Science.

[108]  M Wahde,et al.  Coarse-grained reverse engineering of genetic regulatory networks. , 2000, Bio Systems.

[109]  Mark G. Karpovsky,et al.  Fault Detection in Combinational Networks by Reed-Muller Transforms , 1989, IEEE Trans. Computers.

[110]  Linda Sellie,et al.  Learning Sparse Multivariate Polynomials over a Field with Queries and Counterexamples , 1996, J. Comput. Syst. Sci..

[111]  C.-L. Wang,et al.  Hardware-efficient systolic architecture for inversion and division in GF(2m) , 1998 .

[112]  Kiyomichi Araki,et al.  Fast Inverters over Finite Field Based on Euclid's Algorithm , 1989 .

[113]  Jean-Jacques Quisquater,et al.  Iterative Modular Division over GF(2m): Novel Algorithm and Implementations on FPGA , 2006, ARC.

[114]  Harald Niederreiter,et al.  Introduction to finite fields and their applications: List of Symbols , 1986 .

[115]  Soonhak Kwon,et al.  Efficient bit-serial systolic array for division over GF(2/sup m/) [elliptic curve cryptosystem applications] , 2003, Proceedings of the 2003 International Symposium on Circuits and Systems, 2003. ISCAS '03..

[116]  Christophe Soulé,et al.  Mathematical approaches to differentiation and gene regulation. , 2005, Comptes rendus biologies.

[117]  Michael N. Vrahatis,et al.  Aitken and Neville Inverse Interpolation Methods over Finite Fields , 2005 .

[118]  Mike Bond,et al.  Experience Using a Low-Cost FPGA Design to Crack DES Keys , 2002, CHES.

[119]  Christophe Bobda,et al.  Introduction to reconfigurable computing - architectures, algorithms, and applications , 2010 .

[120]  Alfred Menezes,et al.  Field inversion and point halving revisited , 2004, IEEE Transactions on Computers.

[121]  Tet Hin Yeap,et al.  Efficient FPGA implementation of FFT based multipliers , 2005, Canadian Conference on Electrical and Computer Engineering, 2005..

[122]  Ruth Chadwick,et al.  How the Role of Computing is Driving New Genetics' Public Policy , 2004, Ethics and Information Technology.

[123]  Hitoshi Iba,et al.  Inference of a gene regulatory network by means of interactive evolutionary computing , 2002, Inf. Sci..

[124]  加来 道雄,et al.  Visions : how science will revolutionize the 21st century , 1997 .