Introduction to Cryptography with Coding Theory, 3rd Edition

17.3 Exercises

1. Suppose you have a secret, namely 5. You want to set up a system where four persons A, B, C, D are given shares of the secret in such a way that any two of them can determine the secret, but no one alone can determine the secret. Describe how this can be done. In particular, list the actual pieces of information (i.e., numbers) that you could give to each person to accomplish this.

2. Persons $A$, $B$, $C$ participate in a Shamir $(3\text{,}2)$ secret sharing scheme. They work mod 11. $A$ receives the share $(1\text{,}5)$, $B$ receives $(2\text{,}9)$, and $C$ receives $(3\text{,}3)$.

   1. Show that at least one of the three shares is incorrect.

   2. Suppose $A$ and $C$ have correct shares. Find the secret.

3. You set up a (2, 30) Shamir threshold scheme, working mod the prime 101. Two ...