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We solve a long-standing challenge to the integrity of votes cast without the supervision of a voting booth: ``{\it improper influence},'' which typically refers to any combination of vote buying and voter coercion. Our approach allows each voter, or their trusted agents (which we call ``{\it hedgehogs}''), to {\it ``nullify''} (effectively cancel) their vote in a way that is unstoppable, irrevocable, and forever unattributable to the voter. In particular, our approach enhances security of online, remote, public-sector elections, for which there is a growing need and the threat of improper influence is most acute. We introduce the new approach, give detailed cryptographic protocols, show how it can be applied to several voting settings, and describe our implementation. The protocols compose a full voting system, which we call {\it {\votexx}}, including registration, voting, nullification, and tallying---using an anonymous communication system for registration, vote casting, and other communication in the system. We demonstrate how the technique can be applied to known systems, including where ballots can be mailed to voters and voters use codes on the ballot to cast their votes online. In comparison with previous proposals, our system makes fewer assumptions and protects against a strong adversary who learns all of the voter's keys. In {\votexx}, each voter has two public-private key pairs. Without revealing their private keys, each voter registers their public keys with the election authority. Each voter may share their keys with one or more hedgehogs. During nullification, the voter, or one or more of their hedgehogs, can interact through the anonymous communication system to nullify a vote by proving knowledge of one of the voter's private keys via a zero-knowledge proof without revealing the private key. We describe a fully decentralizable implementation of {\votexx}, including its public bulletin board, which could be implemented on a blockchain.