Optimal probabilistic fingerprinting codes using optimal finite random variables related to numerical quadrature

We investigate candidates of finite random variables for c-secure random fingerprinting codes, in viewpoints of both code lengths and required memories. We determine, under a natural assumption, the random variables with the minimal number of outputs (i.e. optimal in a viewpoint of memory) among all candidates, by revealing their deep relation with theory of Gauss-Legendre quadrature (a famous method for approximating definite integrals by finite sums). Our proposal also reduces the code lengths significantly; e.g. it is asymptotically about 20% of Tardos codes. Moreover, any innocent user is unlikely to be falsely accused by our code, even if the collusion is larger than the setting. These properties make our codes more desirable for a practical use.