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Abstract The advent of quantum computing threatens blockchain protocols and networks because they utilize non-quantum resistant cryptographic algorithms. When quantum computers become robust enough to run Shor’s algorithm on a large scale, the most used asymmetric algorithms, utilized for digital signatures and message encryption, such as RSA, (EC)DSA, and (EC)DH, will be no longer secure. Quantum computers will be able to break them within a short period of time. Similarly, Grover’s algorithm concedes a quadratic advantage for mining blocks in certain consensus protocols such as proof of work. Today, there are hundreds of billions of dollars denominated in cryptocurrencies and other digital assets that rely on blockchain ledgers as well as thousands of blockchain-based applications storing value in blockchain networks. Cryptocurrencies and blockchain-based applications require solutions that guarantee quantum resistance in order to preserve the integrity of data and assets in these public and immutable ledgers. The quantum threat and some potential solutions are well understood and presented in the literature. However, most proposals are theoretical, require large QKD networks, or propose new quantum-resistant blockchain networks to be built from scratch. Our work, which is presented in this paper, is pioneer in proposing an end-to-end framework for post-quantum blockchain networks that can be applied to existing blockchain to achieve quantum-resistance. We have developed an open-source implementation in an Ethereum-based (i.e., EVM compatible) network that can be extended to other existing blockchains. For the implementation we have (i) used quantum entropy to generate post-quantum key pairs, (ii) established post-quantum TLS connections and X.509 certificates to secure the exchange of information between blockchain nodes over the internet without needing a large QKD network, (iii) introduced a post-quantum second signature in transactions using Falcon-512 post-quantum keys, and (iv) developed the first on-chain verification of post-quantum signatures using three different mechanisms that are compared and analyzed: Solidity smart-contracts run by the validators for each transaction, modified EVM Opcode, and precompiled smart contracts.

Abstract We present a new quantum heuristic algorithm aimed at finding satisfying assignments for hard K-SAT instances using a continuous time quantum walk that explicitly exploits the properties of quantum tunneling. Our algorithm uses a Hamiltonian $$H_A(F)$$ H A ( F ) which is specifically constructed to solve a K-SAT instance F. The heuristic algorithm aims at iteratively reducing the Hamming distance between an evolving state $${|{\psi _j}\rangle }$$ | ψ j ⟩ and a state that represents a satisfying assignment for F. Each iteration consists on the evolution of $${|{\psi _j}\rangle }$$ | ψ j ⟩ (where j is the iteration number) under $$e^{-iH_At}$$ e - i H A t , a measurement that collapses the superposition, a check to see if the post-measurement state satisfies F and in the case it does not, an update to $$H_A$$ H A for the next iteration. Operator $$H_A$$ H A describes a continuous time quantum walk over a hypercube graph with potential barriers that makes an evolving state to scatter and mostly follow the shortest tunneling paths with the smaller potentials that lead to a state $${|{s}\rangle }$$ | s ⟩ that represents a satisfying assignment for F. The potential barriers in the Hamiltonian $$H_A$$ H A are constructed through a process that does not require any previous knowledge on the satisfying assignments for the instance F. Due to the topology of $$H_A$$ H A each iteration is expected to reduce the Hamming distance between each post measurement state and a state $${|{s}\rangle }$$ | s ⟩ . If the state $${|{s}\rangle }$$ | s ⟩ is not measured after n iterations (the number n of logical variables in the instance F being solved), the algorithm is restarted. Periodic measurements and quantum tunneling also give the possibility of getting out of local minima. Our numerical simulations show a success rate of 0.66 on measuring $${|{s}\rangle }$$ | s ⟩ on the first run of the algorithm (i.e., without restarting after n iterations) on thousands of 3-SAT instances of 4, 6, and 10 variables with unique satisfying assignments.