I am trying to optimize GCM multiplication, I am reading the pdf that explains such a thing, it is this:
The algorithms are in section 4.1
In algorithm 3 I have to multiply table M  with the element P that represents the polynomial α. Formulas (3) and (4) appear on page 9, from what I can understand this would be:
if V127 =0 then V ← rightshift(V ) else V ← rightshift(V ) ⊕ R
The polynomial f would be 0xe1, no?
But I don't understand in algorithm 3 this, M[i] ← M[2i] · P, What would be the value of the element P, if it corresponds to the polynomial α?
Someone knows some code of the implementation of the tables that I can read, I have not found any code, and this could help me understand all this.