Necessary and Sufficient Numbers of Cards for Securely Computing Two-Bit Output Functions

In 2015, Koch et al. proposed a five-card finite-runtime committed protocol to compute securely the AND function, showing that their protocol was optimal: there is no protocol computing the AND function with four cards in finite-runtime fashion and committed format. Thus, necessary and sufficient numbers of cards for computing single-bit output functions are known. However, as for two-bit output functions, such an exact characterization is unknown. This paper gives a six-card (or less) protocol for each of all two-bit output functions and proves that our finite-runtime committed protocols are optimal by providing a lower bound. In other words, we give the necessary and sufficient number of cards for any two-bit output function to be computed by a finite-runtime committed protocol. Our lower bound can also be applied to any function which outputs more than two bits.