In cryptographic engineering, extensive attention has been devoted to ameliorating the performance and security of the algorithms within. Nonetheless, in the state-of-the-art, the approaches for increasing the reliability of the efficient hash functions ECHO and Fugue have not been presented to date. We propose efficient fault detection schemes by presenting closed formulations for the predicted signatures of different transformations in these algorithms. These signatures are derived to achieve low overhead for the specific transformations and can be tailored to include byte/word-wide predicted signatures. Through simulations, we show that the proposed fault detection schemes are highly-capable of detecting natural hardware failures and are capable of deteriorating the effectiveness of malicious fault attacks. The proposed reliable hardware architectures are implemented on the application-specific integrated circuit (ASIC) platform using a 65-nm standard technology to benchmark their hardware and timing characteristics. The results of our simulations and implementations show very high error coverage with acceptable overhead for the proposed schemes.