Restoring Important Voter Eligibility Requirements to States Act of 2013 - Amends the National Voter Registration Act of 1993 to require an applicant for voter registration for federal elections to state affirmatively in the application form provided along with an application for a motor vehicle driver's license (as well as in other voter registration forms) that the applicant meets the eligibility requirements for voting in such elections as a condition of the application's acceptance.
Requires all such forms to include a statement specifying the penalty for perjury in an application for voter registration.
Prohibits states from registering an individual to vote in federal elections unless they verify that the information provided in the individual's application is correct.
Requires the head of an office of the federal government, upon request, to enter into an agreement with the chief state election official to enable the official to verify the eligibility of such an applicant.