Security Analysis of Majority and Selfish Mining Attacks in a Blockchain with Sharding

Sharding is a promising technology to enhance the scalability of a Proof-of-Work (PoW) blockchain by dividing the nodes or miners into multiple disjoint groups (shards). Each shard processes a subset of transactions so that the throughput can be significantly improved. However, since the number of nodes or the total mining rate in a shard becomes much smaller, previous works claimed that sharding mechanisms decrease security level of a blockchain. In this paper, we mathematically show that the claim is not necessarily true for all types of attacks. We consider both the majority and selfish mining attacks in a sharded blockchain. Three key metrics are generally utilized to evaluate the effectiveness of security attacks, namely, the probability of single shard takeover (<tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$P_{S S T}$</tex>) for majority attack, expected fraction of rewards earned by selfish miners <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(E R)$</tex> and profitable threshold <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(P T)$</tex> for selfish mining attack. These metrics are formulated and obtained via mathematical analyses, numerical methods, or simulations. We show that the security level decreases with the increasing number of shards in terms of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$P_{S S T}$</tex> in majority attacks and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$P T$</tex> in selfish mining attacks. However, in the case of selfish mining attacks, increasing the number of shards does not necessarily reduce the security level of a sharded blockchain, as measured by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$E R$</tex>.