  Two Boolean functions f, g : F_2^{n} \to {-1, 1} are called linearly
isomorphic if there exists an invertible matrix M \in F_2^{n\times n} such that
f\circ M = g. Testing linear isomorphism is a generalization of, now classical
in the context of property testing, isomorphism testing between Boolean
functions. Linear-invariance of Boolean functions has also been extensively
studied in other areas like coding theory and cryptography, and mathematics in
general. In this paper, we will study the following two variants of this
problem:
  [1] [Communication complexity: ] Assume that Boolean functions f and g on
F_2^{n} are given to Alice and Bob respectively, and the goal is to test linear
isomorphism between f and g by exchanging a minimum amount of communication,
measured in bits, between Alice and Bob. Our main result is an efficient
two-party communication protocol for testing linear isomorphism in terms of the
approximate spectral norm of the functions. We will crucially exploit the
connection between approximate spectral norm and sign-approximating
polynomials.
  [2] [Query complexity: ] If f: F_2^{n} \to { -1, 1 } is a known function and
g : F_2^{n} \to { -1, 1 } be an unknown function with a query access. We study
the query complexity of testing linear isomorphism between f and g in terms of
the approximate spectral norm of f. As in the case of communication complexity,
we will use properties of the approximate spectral norm.
