Editing Zero-knowledge proof (section)

=== Two balls and the colour-blind friend ===

Imagine Victor is red-green [[color blindness|colour-blind]] (while Peggy is not) and Peggy has two balls: one red and one green, but otherwise identical. To Victor, the balls seem completely identical. Victor is skeptical that the balls are actually distinguishable. Peggy wants to ''prove to Victor that the balls are in fact differently coloured'', but nothing else. In particular, Peggy does not want to reveal which ball is the red one and which is the green.

Here is the proof system: Peggy gives the two balls to Victor and he puts them behind his back. Next, he takes one of the balls and brings it out from behind his back and displays it. He then places it behind his back again and then chooses to reveal just one of the two balls, picking one of the two at random with equal probability. He will ask Peggy, "Did I switch the ball?" This whole procedure is then repeated as often as necessary.

By looking at the balls' colours, Peggy can, of course, say with certainty whether or not he switched them. On the other hand, if the balls were the same colour and hence indistinguishable, Peggy's ability to determine whether a switch occurred would be no better than random guessing. Since the probability that Peggy would have randomly succeeded at identifying each switch/non-switch is 50%, the probability of having randomly succeeded at ''all'' switch/non-switches approaches zero.

Over multiple trials, the success rate would [[Law of large numbers|statistically converge]] to 50%, and Peggy could not achieve a performance significantly better than chance. If Peggy and Victor repeat this "proof" multiple times (e.g. 20 times), Victor should become convinced that the balls are indeed differently coloured.

The above proof is ''zero-knowledge'' because Victor never learns which ball is green and which is red; indeed, he gains no knowledge about how to distinguish the balls.<ref>{{Cite news|url=https://www.linkedin.com/pulse/demonstrate-how-zero-knowledge-proofs-work-without-using-chalkias|title=Demonstrate how Zero-Knowledge Proofs work without using maths|last=Chalkias|first=Konstantinos|work=CordaCon 2017|access-date=2017-09-13|language=en}}</ref>