We consider the problem of detecting and locating modifications in signed data to ensure partial data integrity. We assume that the data is divided into $n$ blocks (not necessarily of the same size) and that a threshold $d$ is given for the maximum amount of modified blocks that the scheme can support. We propose efficient algorithms for signature and verification steps which provide a reasonably compact signature size, for controlled sizes of $d$ with respect to $n$. For instance, for fixed $d$ the standard signature size gets multiplied by a factor of $O(\log n)$, while allowing the identification of up to $d$ modified blocks. Our scheme is based on nonadaptive combinatorial group testing and cover-free families.