Hello, Ari.
I am not an expert, but my family taught me how to distinguish names based on nationalities. Your last name may be a combination of two words - one Russian and the other Asian in origin perhaps Mongolian or coming from the "-stan" or Turkic countries.
Attila comes to mind. I was told by my mother that the ending in "ilo" has a "mongoloid" [There is no other word I can use for "slanted-eyed" people of Asian origin as there are three main types in the region - Mongoloid, Indian and "White". In Nepal, for example, one can find the Indian, the Mongoloid and the mix between the two.] or Hungarian background. Hungarians are the descendants of the Huns. Now, нашa (nasha - "our" feminine) can be mixed with "tilo", whose meaning I don't know. You may want to look for the meaning of the name "Attila" or anything ending in "ilo" (
http://www.behindthename.com/name/attila#).
As an aside, the word "nasha" in Romanian means "godmother" so every time I heard "nasha Russia", I used to think Russians were talking about Godmother Russia instead of Our Russia.
Hope this helps.