Abstract

Stateless clients are an Ethereum scaling direction in which full nodes validate blocks without storing the entire state trie locally. Instead, block proposers attach a “witness” - a cryptographic proof that the specific state elements accessed during block execution are members of the canonical state. The accumulator underpinning this scheme must support compact membership proofs, efficient updates when elements change, and aggregation of multiple proofs into a single verification. This article examines the accumulator constructions under active consideration for stateless Ethereum, focusing on Verkle trees (covered separately in the literature) and the RSA and class-group accumulator alternatives that offer different security and performance profiles.

What an Accumulator Must Do

A cryptographic accumulator allows a prover to produce a short proof that an element x is included in a set S, given a compact digest of S called the accumulator value. For stateless clients the set S is the entire Ethereum state - approximately 30 GB of account and storage data as of mid-2025. The witness for a block must prove membership for every state element touched during execution (reads and writes), and must be verifiable against the current state root without requiring the verifier to hold the full state. Critically, the accumulator must support non-membership proofs (for creating new accounts) and update proofs (showing how the accumulator value changes after writes), and these proofs should be aggregatable so that all state accesses in a block produce a single compact witness.

RSA Accumulators

RSA accumulators use a group of unknown order ZN where N = pq, representing accumulated elements as products of prime representatives. A membership proof is a quotient: given accumulator A = g^(e1 * e2 * … * en), a proof for element ei is g^(e1 * … * en / ei). Verification checks that proof^ei = A. The “unknown order” requirement means no one knows phi(N), so updating the accumulator after a state change does not require trusted setup beyond the initial RSA modulus generation. Batched membership proofs using the “PoKE” (Proof of Knowledge of Exponent) technique from Boneh, Bunz, and Fisch (2019) allow multiple membership proofs to be aggregated into a single short proof, which is attractive for block witnesses.

The main drawback is that RSA accumulators require a trusted setup or a multi-party ceremony to generate N without anyone knowing p and q. Ethereum’s preference for setup-free constructions has led researchers to explore class groups of imaginary quadratic fields, where groups of unknown order can be constructed without trusted setup. The Lipa Long-Li construction and subsequent work by Dobson and Galbraith have advanced the efficiency of class-group accumulators, though prover time remains 10-100x slower than RSA accumulators at equivalent security levels.

Witness Size and Network Impact

The witness size for a typical Ethereum block depends on the number of state accesses. Empirical measurements on historical blocks show that complex DeFi transactions touch 500 to 2000 distinct state elements. An RSA accumulator witness for 1000 elements, using batched PoKE, is approximately 3 to 5 KB - comparable to a Verkle tree witness for the same workload. The main difference is that RSA accumulator provers require modular exponentiation with 2048-bit or 3072-bit moduli for adequate security, while Verkle witnesses involve 256-bit elliptic curve operations, making Verkle proving faster on typical validator hardware.

Status in the Ethereum Roadmap

As of mid-2025, the Ethereum core dev consensus favors Verkle trees as the primary path to stateless clients, with the Verkle migration expected in the “Glamsterdam” or subsequent hard fork. RSA and class-group accumulators remain active research topics within the Cryptography and Privacy Research Group at the Ethereum Foundation, and may appear as components of more specialized constructions (such as accumulators for EIP-4444 historical data access proofs) even if they are not the primary state witness scheme.