The Caffe strategy for convolution is to reduce the problem to matrix-matrix multiplication. This linear algebra computation is highly-tuned in BLAS libraries and efficiently computed on GPU devices.

For more details read Yangqing's Convolution in Caffe: a memo.

As it turns out, this same reduction was independently explored in the context of conv. nets by

K. Chellapilla, S. Puri, P. Simard, et al. High performance convolutional neural networks for document processing. In Tenth International Workshop on Frontiers in Handwriting Recognition, 2006.