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 **A**belian), **a**n **a**rbitrary map **a**nd **a** parameter , say that *f *is -close to **a** homomorphism if there is some homomorphism *g* such that *g* **a**nd *f* differ on **a**t most |*G*| elements of *G*, **a**nd say that* f* is -far otherwise. For **a** given *f ***a**nd , **a** homomorphism tester should distinguish whether* f* is **a** homomorphism, or if *f* is -far from **a** homomorphism. When *G* is **A**belian, it was known that the test which picks random pairs* x, y* **a**nd tests that *f(x) + f(y) = f(x + y)* gives **a** homomorphism tester. Our first result shows that such **a** test works for **a**ll groups *G*. Next, we consider functions that **a**re close to their self-convolutions. Let * A* = be

**a**distribution on

*G*. The self-convolution of

*= , is defined by*

**A**,**A**'

**a**'*= It is known that*

_{x}*exactly when*

**A**=**A**'*is the uniform distribution over*

**A****a**subgroup of

*G*. We show that there is

**a**sense in which this characterization is robust -- that is, if

*is close in statistical distance to*

**A***, then*

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

**A***G*.

^{1}

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

Published in: RC23465 in 2004

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