Computing a Group of Polynomials over a Galois Field in FPGA Architecture

Abstract

For the most extensive range of tasks, such as real-time data processing in intelligent transport systems, etc., advanced computer-based techniques are required. They include field-programmable gate arrays (FPGAs). This paper proposes a method of pre-calculating the hardware complexity of computing a group of polynomial functions depending on the number of input variables of the said functions, based on the microchips of FPGAs. These assessments are reduced for a group of polynomial functions due to computing the common values of elementary polynomials. Implementation is performed using similar software IP-cores adapted to the architecture of user-programmable logic arrays. The architecture of FPGAs includes lookup tables and D flip-flops. This circumstance ensures that the pipelined data processing provides the highest operating speed of a device, which implements the group of polynomial functions defined over a Galois field, independently of the number of variables of the said functions. A group of polynomial functions is computed based on common variables. Therefore, the input/output blocks of FPGAs are not a significant limiting factor for the hardware complexity estimates. Estimates obtained in using the method proposed allow evaluating the amount of the reconfigurable resources of FPGAs, required for implementing a group of polynomial functions defined over a Galois field. This refers to both the existing FPGAs and promising ones that have not yet been implemented.