This paper presents an efficient algebraic fault analysis on all four modes of SHA-3 under relaxed fault models. This is the first work to apply algebraic techniques on fault analysis of SHA-3. Results show that algebraic fault analysis on SHA-3 is very efficient and effective due to the clear algebraic properties of Keccak operations. Comparing with previous work on differential fault analysis of SHA-3, algebraic fault analysis can identify the injected faults with much higher rates, and recover an entire internal state of the penultimate round with much fewer fault injections.