Secret sharing is a useful method which divides a secret message into several shares for security. During the recovery procedure, only when sufficient shares are obtained, the secret message can be recovered. This paper proposes two novel reversible data hiding schemes in encrypted image via secret sharing over Galois fields <inline-formula> <tex-math notation="LaTeX">{{GF}}{({p})} </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">{{GF}}{(}{{2}^{{8}}}{)} </tex-math></inline-formula>. The content owner first applies a specific encryption method through block and pixel permutation and Shamir's secret sharing. Then, the theoretical demonstration is introduced to explain that the generated shares are suitable for data embedding over <inline-formula> <tex-math notation="LaTeX">{{GF}}{({p})} </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">{{GF}}{(}{{2}^{{8}}}{)} </tex-math></inline-formula>. Finally, two embedding algorithms over <inline-formula> <tex-math notation="LaTeX">{{GF}}{({p})} </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">{{GF}}{(}{{2}^{{8}}}{)} </tex-math></inline-formula> are presented, and on the receiver side, with different keys, additional data can be extracted correctly and original image can be recovered losslessly. Experimental results show that our schemes can achieve better rate-distortion performance than some state-of-the-art schemes.