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follows that Pr[P(0) = 3] = 11 even given Alice’s partial knowledge of Bob’s share. (b) (10 points) Suppose Alice also ﬁnds out that Carol’s share, which is P(2), is equal to 4. Given
this additional new information, what is the probability that the secret is equal to 3. Justify your
answer (2 points for the correct answer without proper justiﬁcation).
Note: A brute force answer will take you too long. There is an easy way to solve the problem.
Solution: We’ll ﬁrst assume the secret is 3 and determine what Bob’s share (P(4)) is in this
case. If Bob’s share is 10 in this case, we know that the probability that the secret is equal to 3
is 0 (because Bob’s share cannot be 10). Otherwise, the probability that the secret is equal to 3
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is 10 , since Bob’s share can be any value in GF (11) other than 10.
To determine Bob’s share given that the secret is 3, we use Property 1 and Lagrange interpolation. Property 1 tells us that P is uniquely determined by 3 points, which are P(0) = 3, P(5) = 2,
and P(2) = 4 in this case. Lagrange interpolation (or a system of linear equations) allows us to
ﬁnd the polynomial P(x) = 6x2 + 5x + 3. Then Bob’s share, P(4), is equal to 9. As we argued
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above, this means the probability...
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 Winter '09

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