The authors of this paper “propose a slight modification of the Berlekamp-Massey algorithm for obtaining the minimal polynomial of a given linearly recurrent sequence.”

After reading the paper several times, I was only able to discover two differences between algorithm one, which is the usual Berlekamp-Massey algorithm, and algorithm two, which is the new one. The first difference is the occurrence of the local variable m, which gets an initial value and does not vary. The second is that the extended Euclidean algorithm is performed, in the first case, for the polynomials x²ⁿ and P(x), and in the second case for x²ⁿ and x° ^PP(1/x), where the coefficients of P(x) are members of a linear recursive sequence.

The authors state that both algorithms find the minimal polynomial of the sequence, but I am not convinced that their proof is complete.