As we know, a $(t,r,n)$-ramp scheme is described by means of two thresholds $t$ and $r$. Every set with at most $t$ participants is forbidden, while every set with at least $r$ participants is authorized. In other words, we require that any subset of at least $r$ participants can recover the secret, while any subset of size at most $t$ cannot learn anything about the secret. In the ramp schemes proposed in [1], the length (or entropy) of every share is $1/(r − t)$ times the length (or entropy) of the secret, which is also optimal.

My question. As we mention above, the optimal $(t,r,n)$-ramp scheme always exists. However, what is known when we consider a certain secret? To be specific, what is the lower bound (given by a proof) and the upper bound (given by a construction) of information ratio for sharing a $1$-bit binary secret? Could the ratio reach $1/(r-t)$, i.e., each participant holds $1/(r-t)$-bit information? What is the best result known to us?

[1]: G. R. Blakley, C. Meadows. Security of Ramp Schemes. Advances in Cryptology, Crypto’84. Lecture Notes in Comput. Sci. 196 (1985) 242–268.


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