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We present a device-independent randomness expansion protocol, involving onlya constant number of non-signaling quantum devices, that achieves\emph{infinite expansion}: starting with $m$ bits of uniform privaterandomness, the protocol can produce an unbounded amount of certifiedrandomness that is $\exp(-\Omega(m^{1/3}))$-close to uniform and secure againsta quantum adversary. The only parameters which depend on the size of the inputare the soundness of the protocol and the security of the output (both areinverse exponential in $m$). This settles a long-standing open problem in thearea of randomness expansion and device-independence.  The analysis of our protocols involves overcoming fundamental challenges inthe study of \emph{adaptive} device-independent protocols. Our primarytechnical contribution is the design and analysis of device-independentprotocols which are \emph{Input Secure}; that is, their output is guaranteed tobe secure against a quantum eavesdropper, \emph{even if the input randomnesswas generated by that same eavesdropper}!  The notion of Input Security may be of independent interest to other areassuch as device-independent quantum key distribution.

Infinite Randomness Expansion and Amplification with a Constant Number of Devices