Successive approximation register analog-to-digital converter (SAR ADC) is widely adopted in the Internet of Things (IoT) systems due to its simple structure and high energy efficiency. Unfortunately, SAR ADC dissipates various and unique power features when it converts different input signals, leading to severe vulnerability to power side-channel attack (PSA). The adversary can accurately derive the input signal by only measuring the power information from the analog supply pin (AVDD), digital supply pin (DVDD), and/or reference pin (Ref) which feed to the trained machine learning models. This paper first presents the detailed mathematical analysis of power side-channel attack (PSA) to SAR ADC, concluding that the power information from AVDD is the most vulnerable to PSA compared with the other supply pin. Then, an LSB-reused protection technique is proposed, which utilizes the characteristic of LSB from the SAR ADC itself to protect against PSA. Lastly, this technique is verified in a 12-bit 5 MS/s secure SAR ADC implemented in 65nm technology. By using the current waveform from AVDD, the adopted convolutional neural network (CNN) algorithms can achieve >99% prediction accuracy from LSB to MSB in the SAR ADC without protection. With the proposed protection, the bit-wise accuracy drops to around 50%.