Csirmaz's Duality Conjecture and Threshold Secret Sharing

We conjecture that the smallest possible share size for binary secrets for the t-out-of-n and (n− t+1)out-of-n access structures is the same for all 1 ≤ t ≤ n. This is a strenghtening of a recent conjecture by Csirmaz (J. Math. Cryptol., 2020). We prove the conjecture for t = 2 and all n. Our proof gives a new (n − 1)-out-of-n secret sharing scheme for binary secrets with share alphabet size n. 2012 ACM Subject Classification Theory of computation → Randomness, geometry and discrete structures; Theory of computation → Cryptographic primitives; Mathematics of computing → Information theory; Security and privacy → Mathematical foundations of cryptography