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 *G,H* (not necessarily Abelian), an arbitrary map and a parameter , say that *f *is -close to a homomorphism if there is some homomorphism *g* such that *g* and *f* differ on at most |*G*| elements of *G*, and say that* f* is -far otherwise. For a given *f *and , a homomorphism tester should distinguish whether* f* is a homomorphism, or if *f* is -far from a homomorphism. When *G* is Abelian, it was known that the test which picks random pairs* x, y* and tests that *f(x) + f(y) = f(x + y)* gives a homomorphism tester. Our first result shows that such a test works for all groups *G*. Next, we consider functions that are close to their self-convolutions. Let *A* = be a distribution on *G*. The self-convolution of *A, A'* = , is defined by *a'** _{x}* = It is known that

*A = A'*exactly when

*A*is the uniform distribution over a subgroup of

*G*. We show that there is a sense in which this characterization is robust -- that is, if

*A*is close in statistical distance to

*A'*, then

*A*must be close to uniform over some subgroup of

*G*.

^{1}

By:* Michael Ben Or; Don Coppersmith; Mike Luby; Ronnitt Rubinfeld*

Published in: RC23465 in 2004

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