The advent of high throughput next-generation sequencing (NGS) machines made DNA sequencing cheaper, but also put pressure on the genomic life-cycle, which includes aligning millions of short DNA sequences, called reads, to a reference genome. On the performance side, efficient algorithms have been developed, and parallelized on public clouds. On the privacy side, since genomic data are utterly sensitive, several cryptographic mechanisms have been proposed to align reads securely, with a lower performance than the former, which in turn are not secure. This manuscript proposes a novel contribution to improving the privacy × performance product in current genomic studies. Building on recent works that argue that genomics data needs to be treated according to a threat-risk analysis, we introduce a multi-level sensitivity classification of genomic variations. Our classification prevents the amplification of possible privacy attacks, thanks to promoting and partitioning mechanisms among sensitivity levels. Thanks to this classification, reads can be aligned, stored, and later accessed, using different security levels. We then extend a recent filter, which detects the reads that carry sensitive information, to classify reads into sensitivity levels. Finally, based on a review of the existing alignment methods, we show that adapting alignment algorithms to reads sensitivity allows high performance gains, whilst enforcing high privacy levels. Our results indicate that using sensitivity levels is feasible to optimize the performance of privacy preserving alignment, if one combines the advantages of private and public clouds.