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Proofs of Useful Work from Arbitrary Matrix Multiplication (Invited Talk)

Authors Ilan Komargodski , Omri Weinstein

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LIPIcs.ESA.2026.2.pdf
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Subject Classification

ACM Subject Classification
  • Security and privacy → Cryptography
  • Theory of computation → Cryptographic protocols
  • Theory of computation → Problems, reductions and completeness
Keywords
  • Proof of work
  • blockchain
  • matrix multiplication
  • fine-grained complexity
  • random self-reducibility

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Abstract

We revisit the longstanding open problem of implementing Nakamoto’s proof-of-work (PoW) consensus based on a real-world computational task T(x) (as opposed to artificial random hashing), in a truly permissionless setting where the miner itself chooses the input x. The challenge in designing such a Proof-of-Useful-Work (PoUW) protocol is to use the native computation of T(x) to produce a PoW certificate with prescribed hardness and with negligible computational overhead over the worst-case complexity of T(⋅). This ensures malicious miners cannot "game the system" by fooling the verifier to accept with higher probability than honest miners while using similar resources. Indeed, obtaining a PoUW with O(1)-factor overhead is trivial for any task T, but also useless.
Our main result is a PoUW for the task of matrix multiplication MatMul(A,B) of arbitrary matrices, with 1+o(1) multiplicative overhead compared to naïve MatMul. We conjecture that our protocol has optimal security, in the sense that a malicious prover cannot obtain any significant advantage over an honest prover. This conjecture reduces the hardness of our protocol to the task of solving a batch of correlated low-rank random linear systems, which is of independent interest.
Since matrix multiplications are the bottleneck of AI compute as well as countless industry-scale applications, this primitive suggests a concrete design of a new L1 base-layer protocol which nearly eliminates the energy waste of Bitcoin mining, allowing GPU consumers to reduce their AI training and inference costs by "re-using" them for blockchain consensus, in exchange for block rewards (2-for-1).

Cite As Get BibTex

Ilan Komargodski and Omri Weinstein. Proofs of Useful Work from Arbitrary Matrix Multiplication (Invited Talk). In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 2:1-2:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ESA.2026.2

Author Details

Ilan Komargodski
  • School of Computer Science and Engineering, Hebrew University of Jerusalem, Israel
Omri Weinstein
  • School of Computer Science and Engineering, Hebrew University of Jerusalem, Israel

Acknowledgements

We thank Yonatan Sompolinsky for multiple discussions on the topic of this work and for valuable comments on the manuscript. We thank Ohad Klein for valuable feedback and particularly for suggesting the FMM-based algorithm for computing the intermediates. We thank Eylon Yogev and Mark Zhandry for useful remarks and suggestions on this manuscript.

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