Many basic tasks in computational biology involve operations on individual DNA and protein sequences. These sequences, even when anonymized, are vulnerable to re-identification attacks and may reveal highly sensitive information about individuals. We present a relatively efficient, privacy-preserving implementation of fundamental genomic computations such as calculating the edit distance and Smith-Waterman similarity scores between two sequences. Our techniques are cryptographically secure and significantly more practical than previous solutions. We evaluate our prototype implementation on sequences from the Pfam database of protein families, and demonstrate that its performance is adequate for solving real-world sequence-alignment and related problems in a privacy-preserving manner. Furthermore, our techniques have applications beyond computational biology. They can be used to obtain efficient, privacy-preserving implementations for many dynamic programming algorithms over distributed datasets.