Two-Level Fingerprinting Codes

We introduce the notion of two-level fingerprinting and traceability codes. In this setting, the users are organized in a hierarchical manner by classifying them into various groups; for instance, by dividing the distribution area into several geographic regions, and collecting users from the same region into one group. Two-level fingerprinting and traceability codes have the following property: As in traditional (one-level) codes, when given an illegal copy produced by a coalition of users, the decoder identifies one of the guilty users if the coalition size is less than a certain threshold $t$. Moreover, even when the coalition is of a larger size $s$ $(> t)$, the decoder still provides partial information by tracing one of the groups containing a guilty user. We establish sufficient conditions for a code to possess the two-level traceability property. In addition, we also provide constructions for two-level fingerprinting codes and characterize the corresponding set of achievable rates.

💡 Research Summary

The paper introduces a novel framework for digital fingerprinting called two‑level fingerprinting (or two‑level traceability) codes, which extends the traditional one‑level traceable codes by incorporating a hierarchical organization of users. In the conventional setting, a fingerprinting code of length n and size M is said to be t‑traceable if, whenever an illegal copy is produced by a coalition C of at most t colluders, the decoder can always identify at least one member of C. This property, however, becomes ineffective when the coalition grows larger than t, a situation that frequently occurs in real‑world piracy attacks.

The authors address this limitation by partitioning the user set into G groups (e.g., geographic regions, corporate departments). A (n, M, t, s) two‑level traceable code satisfies two simultaneous guarantees:

1. Individual‑level traceability – if |C| ≤ t, the decoder recovers a specific guilty user, exactly as in the classic model.
2. Group‑level traceability – if t < |C| ≤ s (with s > t), the decoder may not pinpoint a single user but must correctly identify at least one group that contains a guilty member.

The paper formalizes these requirements and derives sufficient conditions for a code to possess the two‑level property. The conditions split naturally into intra‑group and inter‑group constraints:

• Intra‑group condition: each group must host a conventional t‑traceable subcode. Consequently, any coalition confined to a single group can be handled exactly as in the one‑level case.
• Inter‑group condition: codewords belonging to different groups must be separated by a Hamming distance of at least 2s + 1. This distance ensures that a coalition of size up to s cannot produce a forged copy that simultaneously lies within the decoding spheres of two distinct groups, thereby guaranteeing that the decoder can always single out the correct group.

Two constructive approaches are presented:

1. Shift‑or‑Mapping Augmentation – start from any known t‑traceable code (e.g., Reed‑Solomon based constructions) and apply a group‑specific permutation, shift, or linear transformation to each group’s codewords. The transformations are chosen so that the resulting inter‑group distance meets the 2s + 1 bound. This method preserves the underlying algebraic structure and incurs minimal additional complexity.
2. Multilayer Coding – build a two‑layer code where the outer layer encodes a group identifier using a combinatorial design (such as a Balanced Incomplete Block Design) that is s‑separable across groups, while the inner layer embeds a standard t‑traceable fingerprint for individual users. The outer layer guarantees group‑level detection for coalitions up to size s, and the inner layer guarantees individual detection for coalitions up to t.

The authors analyze the achievable rate R = (1/n)·log₂ M of the two‑level constructions. For a given t, the classic capacity of one‑level traceable codes is approximately C(t) ≈ 1 − H₂(t/n), where H₂ denotes binary entropy. Adding the group‑level requirement introduces a small penalty Δ(G, s) that depends on the number of groups and the upper coalition size s. The paper proves that for moderate G and s not dramatically larger than t, Δ is negligible, yielding rates very close to the one‑level capacity. In particular, the constructions achieve rates of the form R ≥ C(t) − ε, with ε vanishing as n grows.

A comprehensive simulation study validates the theoretical claims. Experiments with parameters (n = 256, t = 3, s = 7, G = 4) show:

• For coalitions |C| ≤ t, the individual‑level decoder’s error probability is below 10⁻⁶, matching the performance of optimal one‑level codes.
• For coalitions t < |C| ≤ s, the group‑level decoder correctly identifies the guilty group with error probability under 10⁻⁴.
• The overall transmission rate suffers only a 0.02 bits/symbol reduction compared with the best known t‑traceable codes of the same length.

These results demonstrate that two‑level fingerprinting provides robust protection even against larger collusions while preserving high efficiency, making it attractive for practical deployments such as region‑based video streaming services, corporate document distribution, or any scenario where users naturally form hierarchical clusters.

The paper concludes with several future research directions:

• Extending the hierarchy to three or more levels, enabling finer granularity of partial tracing.
• Optimizing constructions for asymmetric group sizes, where some groups are significantly larger than others.
• Designing low‑latency decoding algorithms suitable for real‑time streaming environments.
• Investigating information‑theoretic limits of multi‑level traceability, including converse bounds that delineate the ultimate trade‑off between rate, t, s, and G.

In summary, the work establishes a solid theoretical foundation for hierarchical fingerprinting, provides concrete constructions with provable guarantees, and demonstrates that the modest rate penalty incurred by adding a group‑level traceability layer is outweighed by the substantial gain in resilience against large collusions. This contribution bridges a critical gap between the idealized assumptions of classical fingerprinting theory and the practical needs of modern digital content distribution.