  In EUROCRYPT 2018, Cid $et\;al.$ introduced a new concept on the
cryptographic property of S-boxes to evaluate the subtleties of boomerang-style
attacks. This concept was named as boomerang connectivity table (BCT for short)
. For a power function, the distribution of BCT can be directly determined by
its boomerang spectrum. In this paper, we investigate the boomerang spectra of
two classes power functions over even characteristic finite fields via their
differential spectra. The boomerang spectrum of the power function $
{x^{{2^{m+1}} - 1}} $ over $ {\mathbb{F}_{{2^{2m}}}} $ is determined, where
$2^{m+1}-1$ is a kind of Niho exponent. The boomerang spectrum of the Gold
function $G(x)=x^{2^t+1}$ over $ {\mathbb{F}_{{2^n}}} $ is also determined. It
is shown that the Gold function has two-valued boomerang spectrum.
