A bent function is a Boolean function all of whose Fourier coefficients are equal in absolute value. These functions have been extensively studied in cryptography and play an important role in cryptanalysis and design of cryptographic systems.
We study bent functions in the framework of property testing. In particular, we show that testing whether a given Boolean function on $n$ variables is bent, or $\frac{1}{8}$-far from being bent, requires $\Omega(n^2)$ queries.
As an intermediate step in our proof, we show that the query complexity of testing if a given function is a quadratic bent function, or $\frac{1}{4}$-far from being so, is $\Theta(n^2)$. We remark that this problem is equivalent to testing affine-isomorphism to the inner product function.
Our proof exploits the recent connection between property testing and parity decision trees due to Chakraborty and Kulkarni. We believe our techniques might be useful in proving lower bounds for other properties of quadratic polynomials.

Added reference to Lemma 7

A bent function is a Boolean function all of whose Fourier coefficients are equal in absolute value. These functions have been extensively studied in cryptography and play an important role in cryptanalysis and design of cryptographic systems.
We study bent functions in the framework of property testing. In particular, we show that testing whether a given Boolean function on $n$ variables is bent, or $\frac{1}{8}$-far from being bent, requires $\Omega(n^2)$ queries.
As an intermediate step in our proof, we show that the query complexity of testing if a given function is a quadratic bent function, or $\frac{1}{4}$-far from being so, is $\Theta(n^2)$. We remark that this problem is equivalent to testing affine-isomorphism to the inner product function.
Our proof exploits the recent connection between property testing and parity decision trees due to Chakraborty and Kulkarni. We believe our techniques might be useful in proving lower bounds for other properties of quadratic polynomials.