arxiv: 2602.23464 · v2 · submitted 2026-02-26 · 💻 cs.CR · cs.DC· cs.DS

Recognition: no theorem link

2G2T: Constant-Size, Statistically Sound MSM Outsourcing

Majid Khabbazian

Authors on Pith no claims yet

Pith reviewed 2026-05-15 18:42 UTC · model grok-4.3

classification 💻 cs.CR cs.DCcs.DS

keywords multi-scalar multiplicationverifiable outsourcingstatistical soundnessconstant-size proofRistretto255cryptographic delegation

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The pith

2G2T outsources multi-scalar multiplication to an untrusted server with constant-size responses and statistical soundness.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces 2G2T as a protocol that lets a client delegate the heavy MSM computation to a server. The server returns the claimed result A together with one auxiliary group element B. The client checks correctness using one field inner product plus three group operations. The protocol is statistically sound: any adversarial server succeeds in fooling the client with probability at most 1/q per query in a prime-order group of order q. In addition, nearly all client work can be performed before the server replies, so final verification reduces to one scalar multiplication and one group addition.

Core claim

2G2T achieves verifiable outsourcing of MSM(P, x) = sum x_i P_i by having the server compute and return both A = MSM(P, x) and an auxiliary element B. Client verification then consists of a single length-n field inner product followed by three group operations. The scheme is statistically sound: the probability that any (even unbounded) server makes the client accept an incorrect result is at most 1/q per query and at most e/q after e adaptive queries.

What carries the argument

The 2G2T verification equation, which uses the auxiliary group element B to bind the server's claimed MSM result A through a random linear combination check.

If this is right

Verification cost becomes independent of vector length n once precomputation is done.
The protocol works in any prime-order group without further cryptographic assumptions.
Soundness holds against unbounded adversaries and is not based on computational hardness.
Most verification work can be overlapped with server computation to hide latency.
Communication size remains two group elements regardless of input vector length.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same binding technique may apply to outsourcing other linear operations common in cryptographic proofs.
Resource-constrained devices could use 2G2T to participate in protocols whose dominant cost is MSM.
Implementations could combine the precomputable part with hardware accelerators for additional speed gains.

Load-bearing premise

The underlying group must have prime order q so that the statistical soundness bound of 1/q holds information-theoretically.

What would settle it

An experiment in which an adversarial server returns incorrect A and B yet passes client verification with probability measurably higher than 1/q.

Figures

Figures reproduced from arXiv: 2602.23464 by Majid Khabbazian.

**Figure 1.** Figure 1: 2G2T speedup on Ristretto255 (up to ∼ 300× vs. optimized MSM and ∼ 3000× vs. naïve MSM, for n ≤ 2 18). of accepting an incorrect A is at most 1/q per execution, and at most e/q over e adaptive executions, requiring only that G has prime order q. Empirically, a Rust implementation over Ristretto255 confirms that verification is extremely fast in the large-n regime: for n up to 2 18, verification is up to ∼ … view at source ↗

read the original abstract

Multi-scalar multiplication (MSM), MSM(vec{P},vec{x}) = sum_{i=1}^n x_i P_i, is a dominant computational kernel in discrete-logarithm-based cryptography and often becomes a bottleneck for verifiers and other resource-constrained clients. We present 2G2T, a simple protocol for verifiably outsourcing MSM to an untrusted server. 2G2T is efficient for both parties: the server performs only two MSM computations and returns only two group elements to the client, namely the claimed result A = MSM(vec{P},vec{x}) and an auxiliary group element B. Client-side verification consists of a single length-n field inner product and only three group operations (two scalar multiplications and one group addition). In our Ristretto255 implementation, verification is up to about 300x faster than computing the MSM locally using a highly optimized MSM routine (for n up to 2^18). Moreover, 2G2T enables latency-hiding verification: nearly all verifier work can be performed while waiting for the server's response, so once (A,B) arrives the verifier completes the check with only one scalar multiplication and one group addition (both independent of n). Finally, despite its simplicity and efficiency, we prove that 2G2T achieves statistical soundness: for any (even unbounded) adversarial server, the probability of accepting an incorrect result is at most 1/q per query, and at most e/q over e adaptive executions, in a prime-order group of size q.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

2G2T gives a straightforward constant-size outsourcing protocol for MSM with statistical soundness 1/q and practical verification speedups.

read the letter

This paper's main contribution is a simple outsourcing protocol for multi-scalar multiplication that returns only two group elements and achieves statistical soundness of 1/q against unbounded adversaries in prime-order groups. The verification is lightweight and supports latency hiding, which is a practical plus. The protocol itself is new in its specific combination: server sends A (the MSM result) and B (auxiliary), client checks with one inner product and three group operations. This keeps communication constant regardless of n, and the numbers from the Ristretto255 implementation suggest verification can be hundreds of times faster than local computation for large n. The soundness bound over multiple adaptive queries is also stated explicitly as e/q, which follows from standard random challenge arguments once the server commits first. It does well on simplicity and concrete efficiency. The latency-hiding verification means the client can precompute most of the work while waiting, leaving just one scalar mult and addition at the end. That's the kind of detail that makes a protocol deployable. The soft spots are minor but worth noting. The full proof isn't in the abstract, so the exact steps in the soundness argument need checking for any overlooked cases, like how the inner product interacts with the group elements. The implementation is mentioned but without full benchmark details or code, reproducibility of the 300x claim would need verification. No circularity or hidden assumptions beyond the prime-order group property, which is fine. This paper is for people building systems with discrete log crypto who want to offload MSM without heavy machinery. A reader working on verifiable outsourcing or optimized crypto libraries would get value from the concrete protocol and bounds. It deserves a serious referee because the claims are grounded in standard techniques and the efficiency looks real. I would recommend sending it out for peer review.

Referee Report

1 major / 2 minor

Summary. The manuscript presents 2G2T, a protocol for outsourcing multi-scalar multiplication MSM(vec{P}, vec{x}) = sum x_i P_i to an untrusted server in a prime-order group. The server returns two group elements A (the claimed result) and B after performing two MSM operations. Client verification requires one length-n field inner product plus three group operations (two scalar multiplications and one addition). The paper claims statistical soundness: for any unbounded adversarial server the acceptance probability for an incorrect result is at most 1/q per query and at most e/q over e adaptive queries. An implementation in Ristretto255 reports verification up to ~300x faster than local MSM computation for n up to 2^18, together with a latency-hiding mode in which nearly all verifier work occurs before the server response arrives.

Significance. If the soundness argument holds, the result is significant for cryptographic applications in which MSM is a bottleneck for resource-constrained verifiers. The protocol achieves constant-size communication, information-theoretic soundness that depends only on the prime-order group property, and concrete efficiency gains that are supported by the reported Ristretto255 timings. The latency-hiding feature is a practical advantage not commonly emphasized in prior outsourcing schemes.

major comments (1)

[Soundness analysis] The soundness claim (probability at most 1/q) is stated to follow from the client sampling a uniform random challenge after the server commits to A and B. The manuscript should explicitly describe the protocol flow (including the exact timing of the challenge relative to the commitment) and provide the full probability calculation showing that any incorrect A is accepted with probability exactly 1/q.

minor comments (2)

[Abstract and §1] The notation 'e' for the number of adaptive executions in the soundness bound should be introduced earlier and used consistently throughout the text.
[Implementation] The implementation section would benefit from a brief description of the machine and compiler flags used for the timing experiments to improve reproducibility.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the thorough review and the recommendation for minor revision. The comment on the soundness analysis is well-taken, and we will update the manuscript accordingly to provide greater clarity on the protocol flow and the probability calculation.

read point-by-point responses

Referee: [Soundness analysis] The soundness claim (probability at most 1/q) is stated to follow from the client sampling a uniform random challenge after the server commits to A and B. The manuscript should explicitly describe the protocol flow (including the exact timing of the challenge relative to the commitment) and provide the full probability calculation showing that any incorrect A is accepted with probability exactly 1/q.

Authors: We agree that the current presentation of the soundness argument could be more explicit. In the revised version, we will insert a detailed protocol flow description immediately following the high-level overview. Specifically, we will state that the server commits to A and B in its response to the client's outsourcing request, prior to the client sampling the challenge. The client samples the uniform random challenge r ∈ F_q only after receiving (A, B). We will then provide the full probability calculation as follows: Suppose the server returns an incorrect A ≠ MSM(P, x). The verification check performed by the client can be rewritten as an equation of the form f(r) = 0, where f is a non-zero linear polynomial in the random challenge r (the non-zero coefficient arises from the difference A - A* multiplied by a non-zero group element in the prime-order group). Since F_q is a field, such an equation has at most one root, so the probability that a random r satisfies the check is exactly 1/q. This argument holds information-theoretically against any (even unbounded) server, as it relies only on the prime order of the group. We believe this addition will fully address the referee's concern. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper introduces the 2G2T protocol and derives its statistical soundness bound of 1/q directly from the algebraic properties of prime-order groups of size q combined with a random challenge sampled after the server commits to its response. This is a standard information-theoretic argument (equivalent to a random linear check) that does not rely on any fitted parameters, self-referential definitions, or load-bearing self-citations. The protocol description, efficiency claims, and proof are self-contained; the claimed soundness follows from the group order without reducing the result to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests only on the standard domain assumption that the group is prime-order of size q; no free parameters are fitted, no new entities are postulated, and no ad-hoc axioms are introduced beyond ordinary group properties.

axioms (1)

domain assumption The underlying structure is a prime-order group of size q.
Required for the statistical soundness probability bound of 1/q per query.

pith-pipeline@v0.9.0 · 5580 in / 1303 out tokens · 42526 ms · 2026-05-15T18:42:08.893946+00:00 · methodology

discussion (0)

Reference graph

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