There’s been an increasing call in recent weeks and months for encryption to have government ‘backdoors’ put into them. This is a bad idea. No really, it’s an incredibly bad idea. Even if we took the assumption that it is a push that’s made with only the purest of intentions, and the government universal key is kept 100% safe and secure and never leaked or misused, it’s still a really, unbelievably, stupid idea.
No, it’s a Fifth Amendment issue. The Wikipedia article I linked to discusses it. Being compelled to provide a password runs into some of the same problems that compelling self-incriminating testimony does.
You’re confusing the Fourth Amendment – which deals with searches – and the Fifth Amendment. You’re right that it’s not an issue of protection against illegal searches, which is what one might assume to be the case, but not correct as to the actual rationale that it runs into.