# Distinguishing uniform probability with non uniform probability

Suppose a verfier gets N independent random numbers from an untrusted guy. These random numbers can take values 0 or 1. An honest guy will send the verifier each random number with a probability p. So the expected total value the verifier receives is Np. However a dishonest guy can send these random numbers with probabilities different from p such that the total expected value is still Np. Can the verifier devise a test on all N or a subset of random numbers to distinguish honest guy from dishonest guy with a good enough probability?

• What are your thoughts on the problem at hand? what have you tried so far?? – The Integrator May 14 '18 at 18:54
• It is a problem to prove the security of a protocol which is dependent on how well I can distinguish the honest guy from the adversary. The honest case probability p = 1 - exp(-2|a|^2/2), where |a|^2 is typically very low(<0.5). The probability for a dishonest guy looks like p' = 2 - (exp(-|x-a|^2/2) + exp(-|x+a|^2/2)). It is easy to see for an honest guy, |a| = |x| for all N random numbers. But an adversary can play around with x. It is easy to distinguish the honest and adversary distributions for large |a| but not apparently not when |a| is very small. – Niraj Kumar May 14 '18 at 19:33