"The right of the people to be secure in their persons, houses, papers, and effects, against unreasonable searches and seizures, shall not be violated, and no Warrants shall issue, but upon probable cause, supported by Oath or affirmation, and particularly describing the place to be searched, and the persons or things to be seized."
There is no right to public payments. The drug test would not be used to prosecute for a crime.
The recipient voluntarily seeks public funds. As a condition to receive those funds, the recipient must meet certain criteria and thus qualify to receive those funds. Passing a drug test is simply another criterion required to qualify for the program.
Many banks require a thumbprint to cash a check. One does not have to supply a thumbprint but the bank is not required to negotiate the check. Similarly, one does not have to submit to a vision or driving test. However, one may not qualify for a driver's license. Try obtaining a passport without submitting a photo meeting the required standards. However, seeking a passport is a voluntary act.
The Fourth Amendment protection does not apply to assistance payments. Florida has passed a law earlier this year requiring drug testing for welfare recipients and a federal judge has issued a temporary injunction. However, the case will likely end up before the Supreme Court.
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