According to the 4th Amendment to the U.S. Constitution, the People have the right "to be secure in their persons, houses, papers, and effects, against unreasonable searches and seizures". That is an explicit legal right to privacy, even if it doesn't use the literal word "privacy" in the text.
Forget the word "privacy" then. The Constitution does explicitly give you the right to be secure against unreasonable searches, and this is definitely an unreasonable search.
Yes, the federal constitution gives zero rights to people. It provides for federal government to do various things, and puts limits on that. It explicitly reserves everything else to the People and the States.
According to the 4th Amendment to the U.S. Constitution, the People have the right "to be secure in their persons, houses, papers, and effects, against unreasonable searches and seizures". That is an explicit legal right to privacy, even if it doesn't use the literal word "privacy" in the text.