  The field of device-independent quantum cryptography has seen enormous
success in the past several years, including security proofs for key
distribution and random number generation that account for arbitrary
imperfections in the devices used. Full security proofs in the field so far are
long and technically deep. In this paper we show that the concept of the mirror
adversary can be used to simplify device-independent proofs. We give a short
proof that any bipartite Bell violation can be used to generate private random
numbers. The proof is based on elementary techniques and is self-contained.
