Optimal symmetric Tardos traitor tracing schemes

For the Tardos traitor tracing scheme, we show that by combining the symbol-symmetric accusation function of Skoric et al. with the improved analysis of Blayer and Tassa we get further improvements. Our construction gives codes that are up to 4 times shorter than Blayer and Tassa’s, and up to 2 times shorter than the codes from Skoric et al. Asymptotically, we achieve the theoretical optimal codelength for Tardos’ distribution function and the symmetric score function. For large coalitions, our codelengths are asymptotically about 4.93% of Tardos’ original codelengths, which also improves upon results from Nuida et al.

💡 Research Summary

The paper revisits the classic binary Tardos traitor tracing scheme and shows how to obtain substantially shorter codes by combining two recent ideas: the symbol‑symmetric score function introduced by Skoric et al. and the refined analytical framework of Blayer and Tassa. In the original Tardos construction each user receives a binary codeword of length ℓ, the entries of which are drawn independently from an arcsine distribution. A coalition of up to c colluders creates a forged copy under the marking assumption, and the distributor extracts the forged fingerprint and computes a user‑wise score. The scheme is considered successful if (i) no innocent user is accused (ε₁‑soundness) and (ii) at least one guilty user is accused (ε₂‑completeness) with probabilities bounded by ε₁ and ε₂ respectively.

The symbol‑symmetric score differs from the classic asymmetric score in that it assigns a contribution for both symbols 0 and 1 of the forged copy. Concretely, for each position i and user j the score S_{j,i} is + p_i(1‑p_i)/p_i if X_{j,i}=1 and y_i=1, ‑ p_i p_i/(1‑p_i) if X_{j,i}=0 and y_i=1, ‑ p_i(1‑p_i)/p_i if X_{j,i}=1 and y_i=0, + p_i p_i/(1‑p_i) if X_{j,i}=0 and y_i=0, where p_i is the bias drawn from the truncated arcsine distribution. This symmetry doubles the amount of information extracted from each position, reduces the variance of innocent users’ scores, and consequently allows a much smaller accusation threshold Z.

Blayer and Tassa’s contribution was to keep the Tardos construction unchanged but to treat several auxiliary parameters (d_ℓ, d_z, d_δ, d_α, r, s, g) as variables to be optimized rather than fixed constants. Their analysis yields two families of constraints:

• (S1) d_α ≥ √{d_δ · h(r)} · √c,
• (S2) d_z · d_α – r · d_ℓ · d_α² ≥ 1, for soundness, and
• (C1) 1 – 2 d_δ / π – h⁻¹(s) · s √{d_δ c} ≥ g,
• (C2) g · d_ℓ – d_z ≥ η · r · d_δ · s² c, for completeness, where h and h⁻¹ are elementary functions defined in the paper and η = log(ε₂)/log(ε₁/n). By solving these inequalities numerically the authors obtain a concrete set of constants: d_ℓ ≈ 2.379, d_z ≈ 8.06, d_δ ≈ 28.31, d_α ≈ 4.58, r ≈ 0.67, s ≈ 1.07, g ≈ 0.49. With these values the code length becomes ℓ = d_ℓ · c² · ln(n/ε₁) ≈ 2.38 c² ln(n/ε₁), which is more than three times shorter than the 85 c² ln(n/ε₁) bound of Blayer‑Tassa and more than four times shorter than the original Tardos bound of 100 c² ln(n/ε₁). For larger coalitions and smaller η the constants shrink further, yielding up to a ten‑fold improvement over the original scheme.

The authors also conduct an asymptotic analysis for large c. By letting d_δ → ∞ and letting the other constants converge to their limiting values (d_ℓ → π²/2, d_z → 2π, d_α → π), they prove that the scheme remains ε₁‑sound and ε₂‑complete with ℓ = (π²/2 + O(c⁻¹/³)) c² ln(n/ε₁) ≈ 4.93 c² ln(n/ε₁). This matches the lower bound shown by Skoric et al. under the normal‑distribution assumption, but the present proof does not rely on any normality assumption; it holds for the exact distribution induced by the arcsine bias. Consequently the construction attains the theoretical optimum for the binary Tardos scheme with the symmetric score function.

The paper also addresses practical concerns. Since ℓ must be an integer, the authors describe a simple rounding technique (Appendix A) that adds a negligible overhead. They provide tables of optimal parameters for a range of (n, c, ε) values, making the scheme ready for implementation in real watermarking systems. The computational overhead of the symmetric score is modest (four arithmetic operations per symbol instead of two), while the reduction in code length translates directly into lower storage, transmission, and detection costs.

In summary, by merging the symbol‑symmetric scoring of Skoric et al. with the tight parameter optimization of Blayer and Tassa, the authors achieve a provably optimal binary Tardos scheme: the code length constant reaches the information‑theoretic lower bound π²/2, and for all practical parameter regimes the required code length is up to four times shorter than the best previously known provable constructions. This represents a significant step forward for traitor tracing and digital rights management, offering both theoretical elegance and tangible efficiency gains for large‑scale content distribution.