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The high level of security and the fast hardware and software implementations of the Advanced Encryption Standard have made it the first choice for many critical applications. Nevertheless, the transient and permanent internal faults or malicious faults aiming at revealing the secret key may reduce its reliability. In this paper, we present a concurrent fault detection scheme for the S-box and the inverse S-box as the only two nonlinear operations within the Advanced Encryption Standard. The proposed parity-based fault detection approach is based on the low-cost composite field implementations of the S-box and the inverse S-box. We divide the structures of these operations into three blocks and find the predicted parities of these blocks. Our simulations show that except for the redundant units approach which has the hardware and time overheads of close to 100 percent, the fault detection capabilities of the proposed scheme for the burst and random multiple faults are higher than the previously reported ones. Finally, through ASIC implementations, it is shown that for the maximum target frequency, the proposed fault detection S-box and inverse S-box in this paper have the least areas, critical path delays, and power consumptions compared to their counterparts with similar fault detection capabilities.