Recover a shared secret
Cryptographic protocols, Autumn 2009, 5th home exercise
A dealer used Shamir's secret sharing scheme to share a secret v
among ten parties, such that any four of them could recover it. The
secret value, as well as the shares of parties belong to the field
Z_{13}. At the recovery time, the parties reported the
following shares:
P_{1}  12 
P_{2}  8 
P_{3}  2 
P_{4}  11 
P_{5}  0 
P_{6}  7 
P_{7}  9 
P_{8}  4 
P_{9}  5 
P_{10}  10

(the value reported by P_{i} is the value of the sharing
polynomial at the point i).
It is known that up to three parties may have lied when reporting the value of
their share. Find v.