A Graph-Based Unified Technique for Computing and Representing Coefficients over Finite Fields

This paper presents the generalized theory and an efficient graph-based technique for the calculation and representation of coefficients of multivariate canonic polynomials over arbitrary finite fields in any polarity. The technique presented for computing coefficients is unlike polynomial interpolation or matrix-based techniques and takes into consideration efficient graph-based forms which can be available as an existing resource during synthesis, verification, or simulation of digital systems. Techniques for optimization of the graph-based forms for representing the coefficients are also presented. The efficiency of the algorithm increases for larger fields. As a test case, the proposed technique has been applied to benchmark circuits over GF(2m). The experimental results show that the proposed technique can significantly speed up execution time.

[1]  Claudio Moraga,et al.  Efficient derivation of Reed-Muller expansions in multiple-valued logic systems , 1992, [1992] Proceedings The Twenty-Second International Symposium on Multiple-Valued Logic.

[2]  Zeljko Zilic,et al.  A Multiple-Valued Reed-Muller Transform for Incompletely Specified Functions , 1995, IEEE Trans. Computers.

[3]  Dhiraj K. Pradhan,et al.  Mathematical framework for representing discrete functions as word-level polynomials , 2003, Eighth IEEE International High-Level Design Validation and Test Workshop.

[4]  Tsutomu Sasao,et al.  Exact Minimization of FPRMs Using Multi-Terminal Exor TDDs , 1996 .

[5]  R. Bryant,et al.  Verification of Arithmetic Functions with Binary Moment Diagrams , 1994 .

[6]  Dhiraj K. Pradhan,et al.  MODD: a new decision diagram and representation for multiple output binary functions , 2004, Proceedings Design, Automation and Test in Europe Conference and Exhibition.

[7]  Zhihong Zeng,et al.  Taylor expansion diagrams: a compact, canonical representation with applications to symbolic verification , 2002, Proceedings 2002 Design, Automation and Test in Europe Conference and Exhibition.

[8]  S. Wicker.  Error Control Systems for Digital Communication and Storage , 1994 .

[9]  Malgorzata Marek-Sadowska,et al.  Boolean Matching Using Generalized Reed-Muller Forms , 1994, 31st Design Automation Conference.

[10]  Claudio Moraga,et al.  Spectral Transform Decision Diagrams , 1996 .

[11]  Christof Paar,et al.  Fast Arithmetic for Public-Key Algorithms in Galois Fields with Composite Exponents , 1999, IEEE Trans. Computers.

[12]  Marek Karpinski,et al.  On Zero-Testing and Interpolation of k-Sparse Multivariate Polynomials Over Finite Fields , 1991, Theor. Comput. Sci..

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

[14]  Radomir Stanković.  Unified View of Decision Diagrams for Representation of Discrete Functions , 2002 .

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

[16]  Rolf Drechsler,et al.  Circuit design from Kronecker Galois field decision diagrams for multiple-valued functions , 1997, Proceedings 1997 27th International Symposium on Multiple- Valued Logic.

[17]  Dhiraj K. Pradhan,et al.  Reed-Muller Like Canonic Forms for Multivalued Functions , 1975, IEEE Transactions on Computers.

[18]  Claudio Moraga,et al.  Optimization of GF(4) expressions using the extended dual polarity property , 2003, 33rd International Symposium on Multiple-Valued Logic, 2003. Proceedings..

[19]  Suman Purwar.  An Efficient Method of Computing Generalized Reed-Muller Expansions from Binary Decision Diagram , 1991, IEEE Trans. Computers.

[20]  Shu Lin,et al.  Error control coding : fundamentals and applications , 1983 .

[21]  D. Green.  Families of Reed-Muller canonical forms , 1991 .

[22]  Radomir S. Stankovic.  Functional decision diagrams for multiple-valued functions , 1995, Proceedings 25th International Symposium on Multiple-Valued Logic.

[23]  Chien-Ming Wu,et al.  High-Speed, Low-Complexity Systolic Designs of Novel Iterative Division Algorithms in GF(2^m) , 2004, IEEE Trans. Computers.

[24]  Rolf Drechsler,et al.  On the construction of multiple-valued decision diagrams , 2002, Proceedings 32nd IEEE International Symposium on Multiple-Valued Logic.

[25]  Dhiraj K. Pradhan.  A Theory of Galois Switching Functions , 1978, IEEE Transactions on Computers.

[26]  Marek Karpinski,et al.  Fast Parallel Algorithms for Sparse Multivariate Polynomial Interpolation over Finite Fields , 1988, SIAM J. Comput..

[27]  Michael Ben-Or,et al.  A deterministic algorithm for sparse multivariate polynomial interpolation , 1988, STOC '88.

[28]  Elena Dubrova,et al.  Generalized Reed-Muller Canonical Form for a Multiple-Valued Algebra , 1996 .

[29]  Katarzyna Radecka,et al.  Design verification by test vectors and arithmetic transform universal test set , 2004, IEEE Transactions on Computers.

[30]  Zeljko Zilic,et al.  Current-mode CMOS Galois field circuits , 1993, [1993] Proceedings of the Twenty-Third International Symposium on Multiple-Valued Logic.

[31]  Johannes Grabmeier,et al.  Applying Coding Theory to Sparse Interpolation , 1993, SIAM J. Comput..

[32]  Tsutomu Sasao.  And-Exor Expressions and their Optimization , 1993 .

[33]  Randal E. Bryant,et al.  Graph-Based Algorithms for Boolean Function Manipulation , 1986, IEEE Transactions on Computers.

[34]  Randal E. Bryant,et al.  Verification of Arithmetic Circuits with Binary Moment Diagrams , 1995, 32nd Design Automation Conference.

[35]  Allan O. Steinhardt,et al.  Fast algorithms for digital signal processing , 1986, Proceedings of the IEEE.

[36]  Tiziano Villa,et al.  Multi-valued decision diagrams: theory and applications , 1998 .