First of all, you should do a good job, leave on good terms, and wrap up all loose ends. Make it as easy as possible for whoever is taking over your responsibilities to pick up where you left off.
Cutting your notice short doesn't sound appropriate in your situation. I can think of only two situations where it really would be reasonable.
Circumstances beyond your control force you to leave earlier
"The new company wants me to start sooner," is not an example of this. "My spouse's job relocation got pushed up and we're leaving now" or "I need to drop everything and care for a sick family member on the other side of the country" are what I'm talking about here.
Your work situation becomes untenable
This is more than just your supervisor not treating you as well as they did before. It's expected that things will be a little awkward and that they won't be happy you're leaving, especially after so long. But if the supervisor is verbally abusive or is "punishing" you in some way, it's reasonable to move your end date up.
If you do decide you want out early, despite all the advice to the contrary, you really should tell your boss this up-front. Doing nothing in order to make them shorten your notice period is even more unprofessional than shortening it yourself.
As a note for the future, a month is a lot of notice (in the US, at least). Assuming you work somewhere where you choose how much notice to give, it's worth considering the next time you're in this position. Some employers are thrilled to have long notice periods and will treat people well when they're leaving, because it results in a smoother transition. Others make your last days miserable, or even walk you out when you give notice. In your next job, pay attention to how people who leave are treated. That will help you decide how much notice to give. (This is all based on Alison Green's excellent advice at Ask a Manager.) Why you're quitting is another useful thing to consider. If you're leaving a job because you hate it, giving more notice than the two-week minimum just seems masochistic.