# Tight Verifiable Delay Functions

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## Abstract

A Verifiable Delay Function (VDF) is a function that takes at least *T* sequential steps to evaluate and produces a unique output that can be verified efficiently, in time essentially independent of *T*. In this work we study *tight* VDFs, where the function can be evaluated in time not much more than the sequentiality bound *T*.

On the negative side, we show the impossibility of a black-box construction from random oracles of a VDF that can be evaluated in time \(T + O(T^\delta )\) for any constant \(\delta < 1\). On the positive side, we show that any VDF with an inefficient prover (running in time *cT* for some constant *c*) that has a natural self-composability property can be generically transformed into a VDF with a tight prover efficiency of \(T+O(1)\). Our compiler introduces only a logarithmic factor overhead in the proof size and in the number of parallel threads needed by the prover. As a corollary, we obtain a simple construction of a tight VDF from any succinct non-interactive argument combined with repeated hashing. This is in contrast with prior generic constructions (Boneh et al., CRYPTO 2018) that required the existence of incremental verifiable computation, which entails stronger assumptions and complex machinery.

## Notes

### Acknowledgments

S. Garg is supported in part from DARPA SIEVE Award, AFOSR Award FA9550-15-1-0274, AFOSR Award FA9550-19-1-0200, AFOSR YIP Award, NSF CNS Award 1936826, DARPA and SPAWAR under contract N66001-15-C-4065, a Hellman Award, a Sloan Research Fellowship and research grants by the Okawa Foundation, Visa Inc., and Center for Long-Term Cybersecurity (CLTC, UC Berkeley). The views expressed are those of the author and do not reflect the official policy or position of the funding agencies.

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