Both the cigarettes and the chestnuts are symbolic of friendship in the short story "Cranes."  

The two items are not only symbolic of the friendship that exists between Songsam and Tokchae, but the two items are also symbolic of different times.  Toward the beginning of the story, the author does a brief flashback to when the two men were young boys.  One of the things that the boys used to do together was steal chestnuts from another villager's tree.  Songsam was caught in the act and had to flee; therefore, he had no chestnuts to eat.  Tokchae shared his chestnuts with his friend, because that is what friends do.  

In the present time of the story, Tokchae is Songsam's prisoner, and the two men recognize each other.  As Songsam begins escorting Tokchae, Songsam lights up a cigarette.  While he smokes his cigarette, he thinks that he should offer Tokchae a cigarette too.  He would do this because sharing is part of being friends, and despite being on different sides of a conflict, Songsam still harbors feelings of friendship for Tokchae.  

I would like to point out one thing though.  Songsam decides not to offer the friendship cigarette.  He is still mad at Tokchae at this point in the story.  The Enotes summary states that the cigarette was offered.  It was not.  

Songsam threw away the cigarette he had just lit. Then he made up his mind not to light another while he was escorting Tokchae.

By the end of the story though, Songsam offers a much better gift of friendship than chestnuts or cigarettes.  Songsam gives Tokchae his freedom.