# Verifiable Computing Frameworks from Functional Encryption and Functional Signatures

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## Abstract

In addition to proof or argument based verifiable computing schemes and constructions that rely on homomorphic encryption or homomorphic authenticators, verifiable computing schemes can also be constructed using functional encryption or functional signatures. Thus, in this chapter we present the verifiable computing schemes using one of these primitives. Functional encryption refers to encryption schemes where ciphertexts can be decrypted only if they fulfill certain requirements. There are basically two approaches that use functional encryption to build a verifiable computing scheme. “Verifiable Computation from Attribute Based Encryption” by Parno et al. uses (key-policy) attribute-based encryption, a specific instantiation of functional encryption, while the approach presented in “Delegatable Homomorphic Encryption with Applications to Secure Outsourcing of Computation” by Barbosa and Farshim is constructed directly from functional encryption schemes. Functional signatures come with a secondary parameterized signing key, in addition to the master signing key, that allows to sign messages, but restricts the signing capabilities to messages in a certain range. This property allows to build verifiable computing schemes as shown by Boyle et al. in “Functional Signatures and Pseudorandom Functions”.

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