Don't worry about people stealing your idea - very few people have the time or talent to sit down and write a book.

Explain your plot to a few people and see how they receive (just bare bones, nothing more). Try to write a chapter (any part) and see if it is any good. Send it to someone for criticism.

If you still feel it has legs after that, go for it. Some people can write a book in few months, for others it takes years...