BGV & BFV Schemes for Integer Operations

BGV (Brakerski-Gentry-Vaikuntanathan, 2012) is a leveled FHE scheme based on RLWE. It supports arbitrary additions and multiplications on packed integer plaintexts. "Leveled" means it handles circuits up to a fixed depth L without bootstrapping.

BGV and BFV encode plaintexts as polynomials in Z_t[x]/(x^n+1) where t is a small plaintext modulus (e.g., t=65537). Each polynomial encodes n integer values (one per coefficient). Arithmetic on ciphertexts operates on all n values simultaneously — SIMD parallelism.