A Circuit Approach to Constructing Blockchains on Blockchains

TL;DR

The paper introduces blockchain circuits, a framework for constructing overlay blockchains on top of multiple underlay ledgers using serial and triangular compositions. It provides a complete characterization of safety and liveness tradeoffs under partial synchrony, showing that any achievable $(k,s,l)$ point must satisfy $ig floor k/2ig floor+1 \\le l \\le k$ and $s \\ge 2(k-l)+1$, with a matching impossibility result, and extends these ideas to synchronous networks. The authors realize these tradeoffs by instantiating the primitives with certificate-based, cross-chain architectures (e.g., OFT-inspired validators) and by recasting the problem in a circuit-composition paradigm, enabling scalable, modular overlay protocols. They also develop a general framework for permutation-invariant overlays, provide both constructive achievability results and converses, and discuss efficiency improvements via light-client implementations and timestamp-based approaches. The work advances cross-chain consensus by decoupling safety and liveness from underlay properties, enabling a broad spectrum of secure, portable interchain constructions with potential practical impact in multi-chain ecosystems.

Abstract

Since the creation of Bitcoin 15 years ago, there has been an explosion in the number of permissionless blockchains. Each of these blockchains provides an open ledger that anyone can read from and write to. In this multi-chain world, an important question emerges: how can we build a more secure overlay blockchain by reading from and writing to a given set of blockchains? Drawing an analogy with switching circuits, we approach the problem by defining two basic compositional operations between blockchains, serial and triangular compositions, and use these operations as building blocks to construct general overlay blockchains. Under the partially synchronous setting, we have the following results: 1) the serial composition, between two blockchains, yields an overlay blockchain that is safe if at least one of the two underlay blockchains is safe and that is live if both of them are live; 2) the triangular composition between three blockchains, akin to parallel composition of switching circuits, yields an overlay blockchain that is safe if all underlay blockchains are safe and that is live if over half of them are live; 3) repeated composition of these two basic operations can yield all possible tradeoffs of safety and liveness for an overlay blockchain built on arbitrary number of underlay chains. The results are also extended to the synchronous setting.

Figures (3)

• Figure 1: Region of safety-liveness guarantee. The integer grids in the blue area consists of all points which are achievable, while the integer points in the red area are not achievable under partial synchrony. We highlight the two extreme achievable tuples $(k,1,k)$ and $(k,k,1+\lfloor \frac{k}{2} \rfloor)$.
• Figure 2: Serial composition. The $\Pi_A$ blocks (brown) are denoted by $b_1, b_2, \ldots$ and the $\Pi_B$ blocks (blue) are denoted by $B_1, B_2, \ldots$. Certificates of the $\Pi_A$ blocks are denoted by the medals. In (a), both $\Pi_A$ and $\Pi_B$ are safe. Thus, every client observes the same $\Pi_B$ ledger with certified snapshots $\mathsf{snp}_1 = (\mathrm{tx}_1, \mathrm{tx}_2)$ and $\mathsf{snp}_2 = (\mathrm{tx}_1, \mathrm{tx}_2, \mathrm{tx}_3, \mathrm{tx}_4)$. Upon sanitizing the snapshots, clients obtain $\textsc{Clean}(\mathsf{snp}_1, \mathsf{snp}_2) = (\mathrm{tx}_1, \mathrm{tx}_2, \mathrm{tx}_3, \mathrm{tx}_4)$ as the $\Pi_s$ ledger. In (b), the $\Pi_B$ ledger is not safe, and two clients $x$ and $y$ observe conflicting $\Pi_B$ ledgers $L^{x}_{B,t_1}$ and $L^{y}_{B,t_2}$ with blocks $B_1, B_2$ and $B_1, B'_2$ respectively. The blocks $B_1$, $B_2$ and $B'_2$ contain the certified snapshots $\mathsf{snp}_1 = (\mathrm{tx}_1, \mathrm{tx}_2)$, $\mathsf{snp}_2 = (\mathrm{tx}_1, \mathrm{tx}_2, \mathrm{tx}_3, \mathrm{tx}_4)$ and $\mathsf{snp}'_2 = (\mathrm{tx}_1, \mathrm{tx}_2)$ respectively. Note that $(\mathrm{tx}'_3, \mathrm{tx}'_4)$ is not part of the certified snapshot $\mathsf{snp}'_2$ as they are not included in a certified $\Pi_A$ block. Upon sanitizing the snapshots, clients again obtain consistent $\Pi_s$ ledgers $L^{x}_{B,t_1} = \textsc{Clean}(\mathsf{snp}_1, \mathsf{snp}_2) = (\mathrm{tx}_1, \mathrm{tx}_2, \mathrm{tx}_3, \mathrm{tx}_4)$ and $L^{y}_{B,t_2} = \textsc{Clean}(\mathsf{snp}_1, \mathsf{snp}'_2) = (\mathrm{tx}_1, \mathrm{tx}_2)$. In (c), the $\Pi_A$ ledger is not safe, and two clients $x$ and $y$ observe conflicting $\Pi_A$ ledgers $L^{x}_{A,t_1}$ and $L^{y}_{A,t_2}$ with blocks $b_1, b_2$ and $b_1, b_3$ respectively. However, both clients observe the same $\Pi_B$ ledger with blocks $B_1, B_2$, $B_3$ and their certified snapshots $\mathsf{snp}_1, \mathsf{snp}_2$, $\mathsf{snp}_3$. Hence, upon sanitizing the snapshots, clients obtain the same (consistent) $\Pi_s$ ledgers $L^{x}_{s,t_1} = L^{y}_{s, t_2} = \textsc{Clean}(\mathsf{snp}_1, \mathsf{snp}_2, \mathsf{snp}_3) = (\mathrm{tx}_1, \mathrm{tx}_2, \mathrm{tx}_3, \mathrm{tx}_4, \mathrm{tx}'_3, \mathrm{tx}'_4)$.
• Figure 3: Triangular composition. An overlay OFT protocol run on top of $3$ underlay blockchains. A smart contract on each of the underlays emulates a validator of the OFT protocol and outputs a finalized OFT ledger. The client reads the underlay chains' ledgers and outputs the OFT ledger finalized by a majority of the emulated validators.

Theorems & Definitions (35)

• Theorem 1
• Theorem 2: Informal, Theorem \ref{['lem:converse-symmetric-psync']}
• Theorem 3: Informal, Theorems \ref{['lem:circuit-symmetric-sync']} and \ref{['lem:converse-symmetric-sync']}
• Definition 4: Definition 3.2 of lewispyeroughgardenccs
• Theorem 5
• Theorem 6
• Lemma 7
• Lemma 8
• Lemma 9
• Theorem 10