We construct a delegation scheme for verifying non-deterministic computations, with complexity proportional only to the non-deterministic space of the computation. Specifi cally, letting n denote the input length, we construct a delegation scheme for any language veri fiable in non-deterministic time and space ( T ( n ); S ( n )) with communication complexity pol y ( S ( n )) , verifi er runtime npolylo g ( T ( n )) + p ol y ( S ( n )) , and prover runtime pol y ( T ( n )) .

Our scheme consists of only two messages and has adaptive soundness, assuming the existence of a sub-exponentially secure private information retrieval (PIR) scheme, which can be instantiated under standard (albeit, sub-exponential) cryptographic assumptions, such as the sub-exponential LWE assumption. Specifi cally, the verifi er publishes a (short) public key ahead of time, and this key can be used by any prover to non-interactively prove the correctness of any adaptively chosen non-deterministic computation. Such a scheme is referred to as a noninteractive delegation scheme. Our scheme is privately veri fiable, where the veri fier needs the corresponding secret key in order to verify proofs.

Prior to our work, such results were known only in the Random Oracle Model, or under knowledge assumptions. Our results yield succinct non-interactive arguments based on subexponential LWE, for many natural languages believed to be outside of P.