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This paper presents a novel low-complexity cross parity code, with a wide range of multiple bit error correction capability at a lower overhead, for improving the reliability in circuits over GF (2^m) . For an m input circuit, the proposed scheme can correct m\le D_{\textit {w}}\le 3^{{m}/{2}}-1 multiple error combinations out of all the possible 2^{m}-1$ errors, which is superior to many existing approaches. From the mathematical and practical evaluations, the best case error correction is {m}/{2} bit errors. Tests on 80-bit parallel and, for the first time, on 163-bit Federal Information Processing Standard/National Institute of Standards and Technology (FIPS/NIST) standard word-level Galois field (GF) multipliers, suggest that it requires only 106% and 170% area overheads, respectively, which is lower than the existing approaches, while error injection-based behavioral analysis demonstrates its wider error correction capability. [ABSTRACT FROM AUTHOR]

This paper presents an efficient technique for synthesis and optimization of the polynomials over GF(2m), where m is a nonzero positive integer. The technique is based on a graph-based decomposition and factorization of the polynomials, followed by efficient network factorization and optimization. A technique for efficiently computing the coefficients of the polynomials over GF(pm), where p is a prime number, is first presented. The coefficients are stored as polynomial graphs over GF(pm). The synthesis and optimization is initiated from this graph-based representation. The technique has been applied to minimize multipliers over the fields GF(2κ ), where k = 2,..., 8, generated with all the 51 primitive polynomials in the 0.18-/~m CMOS technology with the help of the Synopsys design compiler. It has also been applied to minimize combinational exponentiation circuits, parallel integer adders and multipliers, and other multivariate bit- as well as word-level polynomials. The experimental results suggest that the proposed technique can reduce area, delay, and power by significant amounts. We also observed that the technique is capable of producing 100% testable circuits for stuck-at faults. [ABSTRACT FROM AUTHOR]

This paper presents a technique for representing multiple-output binary and word-level functions in CF(N) (where N = pm, p is a prime number, and m is a nonzero positive integer) based on decision diagrams (DDs). The presented DD is canonical and can be made minimal with respect to a given variable order. The DD has been tested on benchmarks, including integer multiplier circuits, and the results show that it can produce better node compression (more than an order of magnitude in some eases) compared to shared binary DDs (BDDs). The benchmark results also reflect the effect of varying the input and output field sizes on the number of nodes. Methods of graph-based representation of characteristic and encoded characteristic functions in GF(N) are also presented. Performance of the proposed representations has been studied in terms of average path lengths and the actual evaluation times with 50,000 randomly generated patterns on many benchmark circuits. All of these results reflect that the proposed technique can outperform existing techniques. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]