Electronic Colloquium on Computational Complexity, Report No. 52 (2004) Non-Abelian Homomorphism Testing, and Distributions

Abstract. In this paper, we study two questions related to the problem of testing whether a function is close to a homomorphism. For two finite groups (not necessarily Abelian), an arbitrary map, and a parameter, say that is-close to a homomorphism if there is some homomorphism such that and differ on at most elements of, and say that is-far otherwise. For a given and, a homomorphism tester should distinguish whether is a homomorphism, or if is-far from a homomorphism. When is Abelian, it was known that the test which! "#% picks random & pairs!' and tests that gives a homomorphism tester. Our first result shows $/.0()&1+ that such a test works for all groups. Next, we consider functions that are close to their self-convolutions. 23. Let 5 G6HJILK DMGMN A D 5 G. It is known that 2C.O2 = exactly =A.CB-DFE when is the uniform distribution over a subgroup of. We show that there is 2 a sense in which this characterization is robust – that 2 is, if is close in statistical 2> = distance to 2, then must be close to uniform over some subgroup of