One classical example of this is a side-channel attack on RSA. In brief, the central computation is to raise a number to the power of a secret key. This is naively accomplished through a square-and-multiply algorithm -- where multiplications happen for every 1 of the binary representation of the secret key. By watching the power consumption of a device as it encodes/decodes a message, you can read off the secret key with relative ease.

https://en.wikipedia.org/wiki/Power_analysis