Suppose you have some confidential data – a business secret, for example, or your health data – and you encrypt this and send this to me. You don’t trust me, though, so you don’t want to give me the decryption key. Is there some way that I can do computations with your secret and send you the (encrypted) answers, without me ever needing to decrypt anything or learn your secret? Yes! This is Homomorphic Encryption, the next frontier in cryptography. It will be essential for truly secure cloud computing.
Making it work, and work efficiently, involves deep results from across pure mathematics, including number theory, group theory, and lattice theory. See e.g. https://www.microsoft.com/en-us/research/project/microsoft-seal/
