Digital Cellular Automata
CJam, 21 bytes

l~_,\2/f-l:~fm&gt;:.+Af%


Test it here.

Explanation

l~   e# Read and evaluate N.
_,   e# Duplicate and turn into range [0 1 ... N-1]
\2/  e# Swap with other copy and (integer) divide by 2.
f-   e# Subtract this from each element in the range to get
     e# [-(N-1)/2 ... -1 0 1 ... (N-1)/2]
l:~  e# Read string and evaluate each digit separately.
fm&gt;  e# Make one copy of the result for each element i in the range, shifting the array
     e# i cells to the right, cyclically.
:.+  e# Sum the columns of the resulting matrix.
Af%  e# Take each of those sums modulo 10.