Are we to trusting?

We’ve had a slightly nerdy debate going on in work (well it is a computing department) for the last few days about public /private key encryption (this all started when I was having a heated debate with Peter about how Public/Private Keys were designed to alleviate the lack of authentication in some systems, Stuart has picked this up and is like a dog with a bone now). The principle of the debate is, does anyone understand how it works ? Stuart cannot grasp how the math works (and he is very clever) and neither can I, i just keep saying, It must work because loads of people use it. a bit trusting i admit. anyway the best (easy) explanation i have found so far is from the Hitchhikers Guide although that leaves some math questions unanswered, and unfortunately that’s the bit Stuart is struggling with.