Fractal principles for constructing second-order cryptographic accumulators

Abstract

Cryptographic accumulators are a core component of blockchain verification systems, including zero-knowledge bridges, where proof size directly impacts on-chain cost and prover performance. Existing accumulator designs, such as Merkle Mountain Ranges and their variants, provide strong asymptotic guarantees but do not adequately exploit the temporally biased access patterns typical of real-world verification workloads. The aim of  work is to present and analyze a new cryptographic accumulator data structure that minimizes the expected proof size under realistic, time-varying verification workloads.In this work, a new class of accumulator data structures, the Fractal Merkle Tree  and Fractal Loom Tree, is introduced and analyzed with the goal of minimizing expected proof size under such workloads. Proof cost is formalized in terms of cryptographic hash operations and input word counts, reflecting both on-chain verification expenses and in-circuit prover workload. The Fractal Loom Tree construction employs multi-scale pointer selection, yielding verification costs that grow logarithmically with the temporal distance from the most recent state. The parameter space of Fractal Loom Tree is characterized, showing how radix, tree depth, and pointer allocation shape the time-dependent distribution of proof sizes. An anchor caching optimization is presented, enabling reuse of persistent intermediate states and further reducing worst-case verification cost. Directions for future research are proposed to extend the construction to alternative commitment primitives, adaptive parameter selection, and more general graph-based structures. The main result of this work is the development of a cryptographic accumulator that achieves substantially smaller proofs than existing designs under realistic, temporally biased blockchain verification workloads.