Card-Minimal Protocols for Symmetric Boolean Functions of More than Seven Inputs

Securely computing the n-variable equality function with 2n cards

“…In 2021, Toyoda et al [15] proposed a six-card protocol to compute the majority function (S 3 {2,3} ). Very recently, in 2023, Shikata et al [12] introduced a technique to compute any threevariable function S n X such that 0 ∈ X if and only if 3 ∈ X with six cards. This solved the ) in the case n = 3, finally making the problem settled for this case.…”

“…Functions S 3 {1} and S 3 {1,2} can be computed with eight cards using the newly developed technique of Shikata et al [11]. However, their protocols are Las Vegas and does not guarantee a finite number of shuffles.…”

“…In 2022, Shikata et al [12] proved that there exists a 2n-card protocol to compute any n-variable symmetric Boolean function for n ≥ 8. This limits the open problems to only n = 5, 6, and 7.…”

“…Given two commitments of bits a, b ∈ {0, 1}, this subprotocol produces either E ♣ 3 (a + b) or E ♥ 3 (a + b) without using any helping card. It was developed by Shikata et al [12].…”

“…We use an idea similar to the one in [11], but extends their idea further to any number of variables.…”