Several emerging proof-of-work (PoW) blockchain protocols rely on a "parallel-chain" architecture for scaling, where instead of a single chain, multiple chains are run in parallel and aggregated. A key requirement of practical PoW blockchains is to adapt to mining power variations over time (Bitcoin's total mining power has increased by a 10 14 factor over the decade). In this paper, we consider the design of provably secure parallel-chain protocols which can adapt to such mining power variations.The Bitcoin difficulty adjustment rule adjusts the difficulty target of block mining periodically to get a constant mean inter-block time. While superficially simple, the rule has proved itself to be sophisticated and successfully secure, both in practice and in theory [11,13]. We show that natural adaptations of the Bitcoin adjustment rule to the parallel-chain case open the door to subtle, but catastrophic safety and liveness breaches. We uncover a meta-design principle that allow us to design variable mining difficulty protocols for three popular PoW blockchain proposals (Prism [3], OHIE [26], Fruitchains [21]) inside a common rubric.The principle has three components: (M1) a pivot chain, based on which blocks in all chains choose difficulty, (M2) a monotonicity condition for referencing pivot chain blocks and (M3) translating additional protocol aspects from using levels (depth) to using "difficulty levels". We show that protocols employing a subset of these principles may have catastrophic failures. The security of the designs is also proved using a common rubric -the key technical challenge involves analyzing the interaction between the pivot chain and the other chains, as well as bounding the sudden changes in difficulty target experienced in non-pivot chains. We empirically investigate the responsivity of the new mining difficulty rule via simulations based on historical Bitcoin data, and find that the protocol very effectively controls the forking rate across all the chains.

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