String commitment from unstructured noise

TL;DR

This work addresses two-party string commitment under adversarial channels that may exhibit memory effects. It introduces the unstructured noisy channel (UsNC) model, formalizes a secure commitment protocol using linear codes and 2-universal hashing, and proves completeness, hiding, and binding with explicit finite-n bounds. An asymptotic rate is derived, showing how the rate depends on entropic bounds and channel parameters, and the model is instantiated by the noisy-quantum-storage (NQS) setting, linking physical limitations to the UsNC framework. Collectively, the results bridge two major approaches to commitment—noisy-channel based and physically constrained schemes—and demonstrate feasible commitment against stronger, non-i.i.d. adversaries, with practical relevance to quantum storage scenarios.

Abstract

Noisy channels are a foundational resource for constructing cryptographic primitives such as string commitment and oblivious transfer. The noisy channel model has been extended to unfair noisy channels, where adversaries can influence the parameters of a memoryless channel. In this work, we introduce the unstructured noisy channel model as a generalization of the unfair noisy channel model to allow the adversary to manipulate the channel arbitrarily subject to certain entropic constraints. We present a string commitment protocol with established security and derive its achievable commitment rate, demonstrating the feasibility of commitment against this stronger class of adversaries. Furthermore, we show that the entropic constraints in the unstructured noisy channel model can be derived from physical assumptions such as noisy quantum storage. Our work thus connects two distinct approaches to commitment, i.e., the noisy channel and physical limitations.

Figures (3)

• Figure 1: (a) The honest protocol is modelled by random interactive systems. $\alpha_C,\alpha_R$ are Alice's protocol in the commit phase and reveal phase. $\beta_C,\beta_R$ are Bob's protocol in the commit phase and reveal phase. The arrow $\nleftrightarrow$ denotes multi-rounds of interactions through the noiseless channel. The order of the use of noiseless channel and the use of noisy channel $W$ can be arbitrary. Variables $R_\alpha, R_\beta$ denote all registers of system $\alpha_C,\beta_C$, respectively. $\hat{M}$ is the message inferred by $\beta_R$. $F \in \{\textrm{"acc"},\textrm{"rej"} \}$ is a flag representing the acceptance or rejection of the revealed message. (b) An illustration of dishonest Bob in the commit phase. $T_C$ is the Bob's transcript after interacting with Alice through $\mathcal{B}_C$. (c) An illustration of dishonest Alice. $\mathcal{A}_{R0}, \mathcal{A}_{R1}$ and $\mathcal{A}_C$ are interactive systems representing Alice's strategy. $\beta_{R0}$ and $\beta_{R1}$ are two copies of $\beta_R$.
• Figure 2: An illustration of Protocol \ref{['pt:com']}. The dotted boxes $\alpha_C, \beta_C$ are protocol in the commit phase. The dotted boxes $\alpha_R, \beta_R$ are protocol in the reveal phase.
• Figure 3: The rate $R$ as a function of $\xi_A$ and $\xi_B$ when $p=0.1$ ($h(p) \approx 0.469$). The X and Y axes are $\xi_A$ and $\xi_B$ respectively. The rate hits zero at the red line and below, and hits maximum value $h(p)$ at the point $\xi_A=\xi_B=h(p)$.

Theorems & Definitions (22)

• Lemma 1: winter2003commitment, adapted
• Definition 2: The unstructured noisy channel model
• Remark 3
• Definition 4: Achievable commitment in the UsNC model
• Theorem 5: Main result
• Lemma 6
• Theorem 7
• proof
• Lemma 8
• Corollary 9