Playing President with Virtual Players: How to Play Multiple Cards of a Kind

President is a popular card game in which players may play one to four cards of the same rank. Since it is less enjoyable with too few human players, to address this, we study the player-simulation problem for President: realizing the moves of a virtual player while keeping its hand hidden. We propose a selection protocol, which selects multiple cards of the same rank uniformly at random from a hidden virtual player’s hand, whose rank exceeds the latest played cards. Our construction reduces the task to secure sorting, so the overall efficiency is dominated by the underlying sorting protocol. To address this bottleneck, we design an efficient sorting protocol, which reduces the number of steps from O(m log m) to O(m), compared to the existing sorting protocols.