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Multiplication is one of the most important operations in finite field arithmetic. It is used in cryptographic and coding applications, such as elliptic curve cryptography and Reed-Solomon codes. In this paper, we consider the finite field multiplication used in elliptic curve cryptography and design concurrent error detection circuits. It is shown in the literature that the Montgomery multiplication can be used in cryptography to accelerate the scalar multiplication. Here, we use a parity-based concurrent error detection approach to increase the reliability of different Montgomery multipliers available in the literature. First, we consider bit-serial Montgomery multiplication and propose an error detection circuit. Then, we apply the same technique on the digit-serial Montgomery multiplication. Finally, we consider low time-complexity bit-parallel Montgomery multiplication and design the required components to implement the concurrent error detection circuits. ASIC implementations have been completed to analyze the time and area overheads of the proposed schemes. Also, the error detection capability is investigated by software simulations. We show that our approach results in efficient error detection schemes with small time and area overheads.