Naturally occurring and maliciously injected faults reduce the reliability of Advanced Encryption Standard (AES) and may leak confidential information. We developed an invariance-based concurrent error detection (CED) scheme which is independent of the implementation of AES encryption/decryption. Additionally, we improve the security of our scheme with Randomized CED Round Insertion and adaptive checking. Experimental results show that the invariance-based CED scheme detects all single-bit, all single-byte fault, and 99.99999997% of burst faults. The area and delay overheads of this scheme are compared with those of previously reported CED schemes on two Xilinx Virtex FPGAs. The hardware overhead is in the 13.2-27.3% range and the throughput is between 1.8-42.2Gbps depending on the AES architecture, FPGA family, and the detection latency. One can implement our scheme in many ways; designers can trade off performance, reliability, and security according to the available resources.