# 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, 2018 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, 2018 at 19:33
• The problem is unclear as presently written. Are you saying that $p$ is the probability of getting the number one, or are you saying that it is the probability of "sending" either number to the verifier? Because if it is the latter, that does not give an expected total value of $Np$.
– Ben
Feb 9, 2020 at 20:28