Kneser theorem states that given two proper subsets $A$ and $B$ of a group $G$ of prime order p, the size of the sumset is lower-bounded:
$|A + B| ≥ |A| + |B| − 1$
given that $|A|, |B| ≤ p−2$ and $|A+B| ≤ p.$ Vosper theorem classifies the subsets that attain the bound. They are in the form of arithemetic progressions with the same gap.
In this talk, we will discuss multiplicative analogs of these theorems where
we replace sets by vector subspaces, orders of sets with dimensions, and sumset of two subsets with the Schur product of spaces.
Lastly, we will see the relation of this result with cryptanalysis of McEliece
cryptosystems.