Cyclic convolution ($a{\ast }_{k}b$)

For arrays $a,b$ indexed by integers modulo $k$, their cyclic convolution isWhen $a_i$ and $b_j$ count choices with respective residue classes $i$ and $j$, the cyclic convolution counts pairs whose sum has residue class $r$. Repeated cyclic convolution therefore counts sums of longer sequences. The discrete Fourier transform converts cyclic convolution into pointwise multiplication.