Non-Abelian homomorphism testing, and distributions close to their self-convolutions

• Authors: Michael Ben-or;Don Coppersmith;Mike Luby;Ronitt Rubinfeld
• Venue: Random Structures & Algorithms
• Year: 2008

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 f : G,H, and a parameter 0 ε f is ε-close to a homomorphism if there is some homomorphism g such that g and f differ on at most ε | G | elements of G, and say that f is ε-far otherwise. For a given f and ε, a homomorphism tester should distinguish whether f is a homomorphism, or if f is ε-far from a homomorphism. When G is Abelian, it was known that the test which picks O(1/ε) 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 = {ag | g ε G} be a distribution on G. The self-convolution of A, A′ = {a g′ | g ε G}, is defined by $$a^{\prime}_x = \sum_{y,z \in G; yz=x}a_y a_z.$$ 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. Finally, we show a relationship between the question of testing whether a function is close to a homomorphism via the above test and the question of characterizing functions that are close to their self-convolutions. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2008 A preliminary version of this paper appeared in RANDOM 2004 [3].