Efficient Card-Based Protocols for Basic Arithmetic Operations

Card-based cryptography studies the problem of representing the functionality and security of cryptographic primitives visually by using physical cards, to demonstrate their security properties for those who are unfamiliar with cryptography. In this study, we propose efficient card-based protocols for secure computation of the four basic arithmetic operations (addition, subtraction, multiplication, and division). Existing protocols for securely performing these operations on ℓ-bit integers either require an exponential number of card manipulations in ℓ or need Ω (ℓ) additional cards. The proposed protocols simultaneously resolves these drawbacks for the first time by achieving a polynomial number of card manipulations in ℓ while requiring only a constant number of additional cards. The construction of our protocols is based on the long-hand algorithms for each of the four arithmetic operations. The technical novelty lies in demonstrating that the number of additional cards can be kept constant by freeing the cards representing intermediate values that are no longer needed during the process.