Logic locking has demonstrated its potential to protect the intellectual property of integrated circuits (ICs). The security strength of logic locking is typically evaluated through functional and structural analysis-based attacks. There is limited work analyzing logic locking techniques' resilience against power-based side-channel attacks. To fill this gap, we propose an attack flow for the correlation power analysis (CPA) attack on the circuits encrypted with transistor-level logic locking. Our case studies indicate that CPA attacks outperform DPA attacks in terms of key recovery rate (KRR). To improve the CPA attack resilience of an existing transistor-level logic locking technique, we propose a logic-cone conjunction (LCC) method to enlarge the key space and reduce the correlation between the locking key and the power consumption of locked circuits. The experimental results show that the LCC method successfully reduces the KRR from 100% to 0% by using cyclic logic structures. The FPGA emulation indicates that the proposed method incurs 2.6% more delay and 1.5% more power consumption than the baseline.