2G2T: Constant-Size, Statistically Sound MSM Outsourcing

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.