upTPM: Unbounded Preprocessing for Schnorr Multi-Signatures on TPM

Schnorr-based multi-signature schemes support offline preprocessing of nonce commitments to reduce online signing to a single round. However, preprocessing is inherently bounded: each preprocessed nonce pair consumes signer-side storage, and once exhausted, an interactive commitment round is required to refill. This limitation is particularly severe for TPM~2.0 devices, where usable NVRAM is typically 6--16\,KB and connectivity is intermittent. This paper presents upTPM, a framework that achieves unbounded preprocessing with constant signer storage. Each TPM stores a single 32-byte secret seed from which an unlimited sequence of nonce commitments is deterministically derived. Commitments are published to an untrusted coordinator before use; nonce scalars never leave the TPM. We formalize three properties not provided by existing schemes: (1)~unbounded deterministic preprocessing with constant storage; (2)~asynchronous commitment refill, allowing any signer to unilaterally extend its commitment pool; and (3)~TPM-attested commitments, a hardware-backed authenticity and state-binding mechanism that strengthens resistance to host-software compromise. We prove EU-CMA security in the random oracle model under the discrete logarithm assumption and Pseudo Random Function (PRF) security, with a one-time-use invariant enforced by TPM hardware state. We extend the construction to $(t,n)$-threshold signatures and provide a detailed analysis of coordinator trust, crash recovery, and performance evaluations.