We have proposed a flexible coprocessor key-authentication architecture for 80/112-bit security-related applications over $GF\left(2^m\right)$ field by employing Elliptic-curve Diffie Hellman (ECDH) protocol. Towards flexibility, a serial input/output interface is used to load/produce secret, public, and shared keys sequentially. Moreover, to reduce the hardware resources and to achieve a reasonable time for cryptographic computations, we have proposed a finite field digit-serial multiplier architecture using combined shift and accumulate techniques. Furthermore, two finite-state-machine controllers are used to perform efficient control functionalities. The proposed coprocessor architecture over $GF\left(2^{163}\right)$ and $GF\left(2^{233}\right)$ is programmed using Verilog and then implemented on Xilinx Virtex-7 FPGA (field-programmable-gate-array) device. For $GF\left(2^{163}\right)$ and $GF\left(2^{233}\right)$, the proposed flexible coprocessor use 1351 and 1789 slices, the achieved clock frequency is 250 and 235 MHz, time for one public key computation is 40.50 and 79.20 μs and time for one shared key generation is 81.00 and 158.40 μs. Similarly, the consumed power over $GF\left(2^{163}\right)$ and $GF\left(2^{233}\right)$ is 0.91 and 1.37 mW, respectively. The proposed coprocessor architecture outperforms state-of-the-art ECDH designs in terms of hardware resources.