Simply said, the mirror theory of Patarin stated the following. Consider bit strings and additions of those bit strings (e.g. 1011 + 1100 = 0111). Consider the following system of equations: P_1 + Q_1 = h_1, P_2 + Q_2 = h_2, P_3 + Q_3 = h_3, ... P_l + Q_l = h_l, wWhere h_1,...,h_l are given. The mirror theory states a LOWER bound on the number of solutions {P_1,...,P_l,Q_1,...,Q_l} that satisfy these equations and where the P_i's are distinct and the Q_i's are distinct. See also Section 1.1. Theorem 2 in that paper states a lower bound for a general setting, but the proof is disputed. A proof proof has later been given. The goal of the thesis would be to simulate this, e.g., make a smart computer program for small scale bit strings and number of equations that derives the actual lower bound.

If you are interested in this topic, please send an email to Bart Mennink via bart.mennink@ru.nl .