Device-independent randomness extraction from an arbitrarily weak min-entropy source

Expansion and amplification of weak randomness play a crucial role in many security protocols. Using quantum devices, such procedure is possible even without trusting the devices used, by utilizing correlations between outcomes of parts of the devices. We show here how to extract random bits with an arbitrarily low bias from single arbitrarily weak min-entropy block source in a device independent setting. To do this we use Mermin devices that exhibit superclassical correlations. The number of devices used scales polynomially in the length of the random sequence $n$. Our protocol is robust; it can tolerate devices that malfunction with probability decreasing polynomially in $n$ at the cost of minor increase in the number of devices used.

Figure 1

Depicted is the value of Mermin variable $v=f\left(\varepsilon \right)$ needed to certify the bias of the output bit to be at most $\varepsilon$.

Figure 2

Depiction of a single-round protocol. Bit string drawn from the flat random source is hashed into $m$ inputs into Mermin devices so that at least one device receives perfectly random distribution. This guarantees that at least one result is almost perfectly random, which also holds for the product of individual results.

Figure 3

Matrix $M$ for $Rn=4$.