BFV

Around 2012 Brakerski proposed a new efficient FHE scheme whose security is based on the LWE problem. Later on, this scheme was ported to the ring-LWE setting by Fan and Vercauteren. The various optimisations achieved by the last two authors made the scheme suitable for implementation. One such implementation is available in Microsoft SEAL is an actively maintained library which makes homomorphic encryption available in an easy-to-use form both to experts and to non-experts.

Overview of the BFV scheme

In BFV the plaintext space consists of polynomials of degree less than $n$ with coefficients modulo $t$, more precisely ${\mathcal {R}}_{t}=\mathbb {Z} _{t}[x]/(x^{n}+1)$. This is a ring, where addition is just the usual addition of polynomials. Multiplication is also quite intuitive, in the sense that multiplication of two elements is just multiplication of the underlying polynomials with $x^{n}$ being converted to $-1$. In this way, the result of ring operations on ${\mathcal {R}}_{t}$ is always a polynomial of degree strictly less than $n$.

The homomorphic operations on ciphertext, that will be described later, will carry through encryption to addition and multiplication operations in the plaintext ${\mathcal {R}}_{t}$.

If we wish to encrypt an integer or a rational number, then we need to encode it first into a plaintext polynomial in ${\mathcal {R}}_{t}$ and the result can be encrypted only after that. We mention that distinction should be made between encoding and encrypting data. The first one is a public operation, i.e. anybody can reverse it by decoding , whereas the latter can be reverse just by someone who knows the (secret) encryption key. In order to be able to compute additions and multiplications on, for example, integers in encrypted form, the encoding must be such that addition and multiplication of encoded polynomials in ${\mathcal {R}}_{t}$ carry over correctly to the integers when the result is decoded. The library Microsoft SEAL provides a few different

encoders for the user’s convenience