Genotype imputation is a fundamental step in genomic data analysis, where missing variant genotypes are predicted using the existing genotypes of nearby "tag"variants. Although researchers can outsource genotype imputation, privacy concerns may prohibit genetic data sharing with an untrusted imputation service. Here, we developed secure genotype imputation using efficient homomorphic encryption (HE) techniques. In HE-based methods, the genotype data are secure while it is in transit, at rest, and in analysis. It can only be decrypted by the owner. We compared secure imputation with three state-of-the-art non-secure methods and found that HE-based methods provide genetic data security with comparable accuracy for common variants. HE-based methods have time and memory requirements that are comparable or lower than those for the non-secure methods. Our results provide evidence that HE-based methods can practically perform resource-intensive computations for high-throughput genetic data analysis. The source code is freely available for download at https://github.com/K-miran/secure-imputation.