Optimization of Memory Usage in Tardos's Fingerprinting Codes
Koji Nuida, Manabu Hagiwara, Hajime Watanabe, Hideki Imai
TL;DR
This work addresses the gap between the theoretical optimality of Tardos's fingerprinting codes and practical deployment, where continuous bias distributions incur substantial memory overhead. It establishes that $c$-indistinguishability characterizes all feasible finite bias distributions and proves a one-to-one correspondence with symmetric quadrature systems of degree $c-1$, identifying Gauss-Legendre distributions as the optimal choice with only $\lceil c/2\rceil$ outputs and memory $\lceil \log_2 \lceil c/2\rceil \rceil$ bits. The authors derive an improved code-length formula, account for approximation errors via an approximated tracing algorithm, and show that the code length relative to Tardos asymptotically drops to about $0.206$ as $c \to \infty$, outperforming prior finite-distribution schemes. Numerical results corroborate substantial memory savings and shorter code lengths, while the GL approach maintains robustness under practical finite-precision implementations. Overall, the paper delivers both theoretical and practical advances that enable memory-efficient, explicitly implementable Tardos-type codes for collusion-secure watermarking.
Abstract
It is known that Tardos's collusion-secure probabilistic fingerprinting code (Tardos code; STOC'03) has length of theoretically minimal order with respect to the number of colluding users. However, Tardos code uses certain continuous probability distribution in codeword generation, which creates some problems for practical use, in particular, it requires large extra memory. A solution proposed so far is to use some finite probability distributions instead. In this paper, we determine the optimal finite distribution in order to decrease extra memory amount. By our result, the extra memory is reduced to 1/32 of the original, or even becomes needless, in some practical setting. Moreover, the code length is also reduced, e.g. to about 20.6% of Tardos code asymptotically. Finally, we address some other practical issues such as approximation errors which are inevitable in any real implementation.