  Cosmic dust is an important component of the Universe, and its origin,
especially at high redshifts, is still unknown. I present a simple but powerful
method of assessing whether dust observed in a given galaxy could in principle
have been formed by asymptotic giant branch (AGB) stars or supernovae (SNe).
Using this method I show that for most of the galaxies with detected dust
emission between z=4 and z=7.5 (1.5-0.7 billion years after the Big Bang) AGB
stars are not numerous and efficient enough to be responsible for the measured
dust masses. Supernovae could account for most of the dust, but only if all of
them had efficiencies close to the maximal theoretically allowed value. This
suggests that a different mechanism is responsible for dust production at high
redshifts, and the most likely possibility is the grain growth in the
interstellar medium.
