To reduce the transmission cost of blockchain confidential transactions, we propose SymmeProof, a novel communication efficient non-interactive zero-knowledge range proof protocol without a trusted setup. We design and integrate two new techniques in SymmeProof, namely vector compression and inner-product range proof. The proposed vector compression is able to reduce the communication cost to $\log (n)$ for $n$-size vectors. The proposed inner-product range proof converts a range proof relation into an inner-product form, which can further reduce the range proof size with the vector compression technique. Based on these two techniques, SymmeProof can eventually achieve a $\log (n)$-size range proof. The proposed SymmeProof can be used in many important applications such as blockchain confidential transactions as well as arguments for arithmetic circuit satisfiability. We evaluate the performance of SymmeProof. The results show that SymmeProof substantially outperforms representative methods such as Bulletproofs in the proof size without a trusted setup.