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Genomic data hold salient information about the characteristics of a living organism. Throughout the past decade, pinnacle developments have given us more accurate and inexpensive methods to retrieve genome sequences of humans. However, with the advancement of genomic research, there is a growing privacy concern regarding the collection, storage and analysis of such sensitive human data. Recent results show that given some background information, it is possible for an adversary to reidentify an individual from a specific genomic data set. This can reveal the current association or future susceptibility of some diseases for that individual (and sometimes the kinship between individuals) resulting in a privacy violation. Regardless of these risks, our genomic data hold much importance in analyzing the well-being of us and the future generation. Thus, in this article, we discuss the different privacy and security-related problems revolving around human genomic data. In addition, we will explore some of the cardinal cryptographic concepts, which can bring efficacy in secure and private genomic data computation. This article will relate the gaps between these two research areas-Cryptography and Genomics.