The comets are thought to be the most primitive objects in the solar system. However, there is still no consensus as for similarity of the cometary nuclei dust material to that of the primordial protosolar nebula and to the one of the interstellar matter. The in situ measurements using the PIA and PUMA dust analyzers have revealed the unexpectedly high number densities of very small dust particles near Comet Halley (Utterback and Kissel, 1990) corresponding to a few percent of the total dust loss from Halley during its entire apparition. Interpreting the soft X-ray EUVE and ROSAT observations of another comet (Hyakutake) also may favor to possible presence there of a significant attogram dust fraction (Krasnopolsky, 1996). Such a high abundance of the attogram dust is inconsistent with its purely collisional origin. On the other hand, the contemporary concepts imply roughly 0.1 mum as the "classical" size of the interstellar grains, i.e. much greater size than that of the attogram one. The authors attempt to explain some possible mechanism for the attogram dust formation in cometary nuclei due to specific irradiation processing of their material at different evolution stages. Primary cosmic ray bombardment of the cometary nuclei during their formation and growth has been considered from the viewpoint of microstructure thermoelastic fragmentation effects of heavy ions in solids (Kalinichenko et al., 1996). For the Comet Halley case, its orbital period of about 75 years has been found insufficient for the near-surface origin and accumulation of the attogram dust involving this mechanism only. However, considering the reasonable growth times of cometary nuclei on the planetesimal stage, the accumulated fraction of the ion-produced ultramicron grains can be sufficient to explain the observed attogram dust flux.