  Device-Independent (DI) security is the best form of quantum cryptography,
providing information-theoretic security based on the very laws of nature. In
its highest form, security is guaranteed against adversaries limited only by
the no-superluminal signalling rule of relativity. The task of randomness
amplification, to generate secure fully uniform bits starting from weakly
random seeds, is of both cryptographic and foundational interest, being
important for the generation of cryptographically secure random numbers as well
as bringing deep connections to the existence of free-will. DI no-signalling
proof protocols for this fundamental task have thus far relied on esoteric
proofs of non-locality termed pseudo-telepathy games, complicated multi-party
setups or high-dimensional quantum systems, and have remained out of reach of
experimental implementation. In this paper, we construct the first practically
relevant no-signalling proof DI protocols for randomness amplification based on
the simplest proofs of Bell non-locality and illustrate them with an
experimental implementation in a quantum optical setup using polarised photons.
Technically, we relate the problem to the vast field of Hardy paradoxes,
without which it would be impossible to achieve amplification of arbitrarily
weak sources in the simplest Bell non-locality scenario consisting of two
parties choosing between two binary inputs. Furthermore, we identify a deep
connection between proofs of the celebrated Kochen-Specker theorem and Hardy
paradoxes that enables us to construct Hardy paradoxes with the non-zero
probability taking any value in $(0,1]$. Our methods enable us, under the
fair-sampling assumption of the experiment, to realize up to $25$ bits of
randomness in $20$ hours of experimental data collection from an initial
private source of randomness $0.1$ away from uniform.
