Given the rapidly growing utility of critical health information revealed in the human genome, secure genomic computation is essential to moving forward, especially as genome sequencing becomes commonplace. We devise and implement proof-of-principle computational operations for precisely identifying causal variants in Mendelian patients using secure multiparty computation methods based on Yao's protocol. We show multiple real scenarios (small patient cohorts, trio analysis, two hospital collaboration) where the causal variant is discovered jointly, while keeping up to 99.7% of all participants' most sensitive genomic information private. All similar operations performed today to diagnose such cases are done openly, keeping 0% of participants' genomic information private. Our work will help usher in an era where genomes can be both utilized and truly protected.