This depends on your definition of civilization. Also, in a given population, surnames could be restricted to a certain part of the population, for example the nobility.

Surnames aren't ubiquitous even now. Most Icelanders don't have surnames, and a lot of ethnic minorities all over the globe don't, either.

It also depends on your definition of surname. Middle ages common folks didn't have dynastic family names, but you had to differentiate which Jim you were talking about. There were two basic way that this was resolved, patronymics/matronymics and descriptive names.

Patronymics are formed by taking name of the father and tweaking it (or sometimes not even that) - Russian Ilya Vladislavovich is Ilya who has Vladislav as a father, and some English surnames come from this custom too, forexample, Jhonson, as in Jhon's son. Incidentally, that means that Ilya Vladislavson is in some sense more accurate translation of the name Ilya Vladislavovich, and people in middle ages did sometimes translate names like this.

Matronymics are tha same as patronymics, but with mother instead of a father. Simple.

Last are descriptive names, and those are subdivided into several categories, mostly profession, origin and miscellaneous. If Ilya is a potter, then Ilya the Potter is what people will refer to him, because that's his most important thing for them.

Once Ilya decides to do his pottery abroad, he may well become known as Ilya from Novgorod, because he became that weird guy who came from way over there, and the fact he's a potter doesn't describe him that much.

Last are miscellaneous, those could be anything: the Legless, the Lame, the Good, the All Right, and so on.

All of these were used in medieval legal documents to distinguish individuals, so in that sense, they can be considered surnames. However, they weren't inherited, and you certainly weren't restricted to just one "surname" of this kind, there were people who had several (Mark d'Hauteville aka Bohemond of Taranto, Andrew III Arpad aka Andrew the Venetian, etc).

So, if you count these as surnames, then the answer to your question is that they always have been used, because these are essentially nicknames. If your definition is something else, then a) it's unavoidably arbitrary and b) we need to know what it is to give a good answer.