This chapter deals with the synthesis of circuits implementing the main finite field operations: addition, subtraction, product, exponentiation, and inversion. The reason why these operations should be implemented in hardware, instead of just being programmed for some target microprocessor, is the reduction of the computation time. This is particularly true in the case of computer and communications systems including the execution of cryptographic algorithms for security purposes: they use very long operands so that their software-only execution time could become prohibitively long for some real-time applications. An efficient solution is the development of specific circuits (coprocessors) executing the most time-consuming operations.

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