Galois field arithmetic is a critical component in communication and security-related hardware, requiring dedicated arithmetic circuit architectures for greater performance. In many Galois field applications, such as cryptography, the datapath size in the circuits can be very large. Formal verification of such circuits is beyond the capabilities of contemporary verification techniques. This paper addresses formal verification of combinational arithmetic circuits over Galois fields of the type F 2 k using a computer-algebra/algebraic-geometry based approach.The verification problem is formulated as membership testing of a given specification polynomial in a corresponding ideal generated by the circuit constraints. Ideal membership testing requires the computation of a Gröbner basis, which is computationally very expensive. To overcome this limitation, we analyze the circuit topology and derive a term order to represent the polynomials. Subsequently, using the theory Gröbner bases over F 2 k , we show that this term order renders the set of polynomials itself a minimal Gröbner basis of this ideal. Consequently, the verification test reduces to a much simpler case of Gröbner basis reduction via polynomial division, significantly enhancing verification efficiency.To further improve our approach, we exploit the concepts presented in the F 4 algorithm for Gröbner basis, and show that our verification test can be formulated as Gaussian elimination on a matrix representation of the problem. Finally, we demonstrate the ability of our approach to verify the correctness of, and detect bugs in, up to 163-bit circuits in F 2 163 -whereas verification utilizing contemporary techniques proves infeasible.

In urban areas, signalized intersections are hot spots of emissions and have significant negative environmental and health impacts. Ecodriving is a strategy that aims to reduce fuel consumption and emissions through the modification or optimization of driver behaviors. By the use of information on the signal phases and the queue discharge time, ecodriving could optimize the speed trajectories for a vehicle approaching an intersection to reduce fuel consumption and emissions. This research developed an optimization model to determine the optimal ecodriving trajectory (the speed profile) at a signalized intersection. The model aimed to achieve the minimization of a linear combination of emissions and travel time. The Motor Vehicle Emissions Simulator was used to estimate the emissions (nitrogen oxide), and the genetic algorithm was selected to solve the optimization problem that was developed. A sensitivity analysis was conducted to analyze and compare the performance of the optimal solution in various scenarios. The results of the case study showed that ecodriving could achieve satisfactory reductions in emissions by more than 50% and in travel time by about 7% compared with the emissions and travel times obtained by use of a normal driving strategy.

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Most existing intersection signals are timed based on delay minimization. However, minimizing delay does not necessarily lead to the minimization of emissions at an intersection. No study has focused on the difference or the trade-off between delay based and emissions based signal optimization. Delay-based optimization typically uses macroscopic flow conditions such as traffic demands, saturation flow rates, and average delay. However, the latest emission model, MOVES (Motor Vehicle Emission Simulation), requires second-by-second individual vehicle speed profiles, which makes the model difficult to formulate directly in an emission-based signal optimization problem. This study first develops a methodology to derive vehicle profiles given macroscopic inputs so that MOVES can be applied to estimating emissions. Then an optimization methodology of signal timing is developed and solved with a genetic algorithm. The objective function of the optimization problem considers both delay and emissions, with the signal timing elements being the decision variables. Through a case study, the air quality benefit by reducing vehicle emissions through intersection signal control is demonstrated, and thetrade-off between operational and emission performance measures is investigated. Furthermore, the air quality benefit from intersection signal control is discussed under different scenarios of cycle lengths, percentages of turning vehicles, and traffic demands on major/minor roads.