Designing a protocol to exchange a secret key is one of the most fundamental subjects in cryptography. Using a random deal of cards, pairs of card players (agents) can share information-theoretically secure keys that are secret from an eavesdropper. In this paper we first introduce the notion of an Eulerian secret key exchange, in which the pairs of players sharing secret keys form an Eulerian circuit passing through all players. Along the Eulerian circuit any designated player can send a message to the rest of players and the message can be finally returned to the sender. Checking whether the returned message is the same as the original one, the sender can know whether the message circulation has been completed without any false alteration. We then give three efficient protocols to realize such an Eulerian secret key exchange. Each of the three protocols is optimal in a sense. The first protocol requires the minimum number of cards under a natural assumption that the same number of cards are dealt to each player. The second requires the minimum number of cards dealt to all players when one does not make the assumption. The third forms the shortest Eulerian circuit, and hence the time required to send the message to all players and acknowledge the secure receipt is minimum in this case.