Byzantine faults present a major obstacle to unbiased distri-buted random number generation, because it allows an adversary to abort the protocol if they can predict a random value they do not like. There are computationally/communication heavy mechanisms to recover from such faults. Our approach avoids this by accepting a limited bias in favour of an adversary; in exchange, our protocols can run many times in the time it takes the heavier mechanisms to recover. Our protocols assume the strongest Byzantine adversary, as opposed to assuming an honest majority. Both theoretical analysis and simulation have been used to confirm and further evaluate the protocols’ viability. The application of the randomness used in the paper is creating private pseudonyms.