Fully Homomorphic Encryption for Mathematicians

We give an introduction to Fully Homomorphic Encryption for mathematicians. Fully Homomorphic Encryption allows untrusted parties to take encrypted data Enc(m1), . . . ,Enc(mt) and any efficiently computable function f , and compute an encryption of f(m1, . . . ,mt), without knowing or learning the decryption key or the raw data m1, . . . ,mt. The problem of how to do this was recently solved by Craig Gentry, using ideas from algebraic number theory and the geometry of numbers. In this paper we discuss some of the history and background, give examples of Fully Homomorphic Encryption schemes, and discuss the hard mathematical problems on which the cryptographic security is based.