This paper shows card-based cryptographic protocols to calculate several Boolean functions using private operations under the semi-honest model. Private operations, introduced by Nakai et al. are the most powerful operations for card-based protocols. We showed that copy, logical AND, and logical XOR can be calculated with the minimum number of cards using three private operations, private random bisection cuts, private reverse cuts, and private reveals. This paper shows that by using these private operations, all of the following Boolean functions can be calculated without additional cards other than the input cards: (1) Any three input Boolean functions, (2) Half adder and full adder, and (3)Any n-input symmetric Boolean functions. The numbers of cards used in these protocols are smaller than the ones without private operations.

https://link.springer.com/chapter/10.1007/978-3-030-79987-8_33