Consensus algorithms that function in permissionless blockchain systems must randomly select new block proposers in a decentralised environment. Our contribution is a new blockchain consensus algorithm called Proof-of-Publicly Verifiable Randomness (PoPVR). It may be used in blockchain design to make permissionless blockchain systems function as pseudo-random number generators and to use the results for decentralised consensus. The method employs verifiable random functions to embed pseudo-random number seeds in the blockchain that are confidential, tamper-resistant, unpredictable, collision-resistant, and publicly verifiable. PoPVR does not require large-scale computation, as is the case with Proof-of-Work and is not vulnerable to the exclusion of less wealthy stakeholders from the consensus process inherent in stake-based alternatives. It aims to promote fairness of participation in the consensus process by all participants and functions transparently using only open-source algorithms. PoPVR may also be useful in blockchain systems where asset values cannot be directly compared, for example, logistical systems, intellectual property records and the direct trading of commodities and services. PoPVR scales well with complexity linear in the number of transactions per block.