Function secret sharing (FSS) for a class \(\mathcal {F}\) allows to split a secret function \(f \in \mathcal {F}\) into (succinct) secret shares \(f_0,f_1\) , such that for all \(x\in \{0,1\}^n\) it holds \(f_0(x)-f_1(x)=f(x)\) . FSS has numerous applications, including private database queries, nearest neighbour search, private heavy hitters and secure computation in the preprocessing model, where the supported class \(\mathcal {F}\) translates to richness in the application. Unfortunately, concretely efficient FSS constructions are only known for very limited function classes. In this work we introduce the notion of pseudorandom generators with encoded-output homomorphism (EOH-PRGs), and give direct FSS constructions for branching programs and more based on this primitive. Further, we give constructions of FSS for deterministic finite automatas (DFAs) from a KDM secure variant of EOH-PRGs.