# What is the extra defense that McEliece 6960119 and 6688128 have that 8192128 does not?

The 8192128 parameter set is bigger, but the 6960119 and 6688128 parameter sets include an extra defense explained in the submission. People paranoid enough to imagine 2^306 vs. 2^270 making a difference should also be paranoid enough to imagine this defense making a difference.

I looked in the Classic McEliece round 1, round 2, and round 3 submissions but I was not able to find any defense applying to 6960119 and 6688128 but not 8192128. The only sections which seem kinda relevant to me are 4.3 and 8.3 in the round 3 submission https://classic.mceliece.org/nist/mceliece-20201010.pdf but I am not able to understand their effect (and they do not mention both 6960119 and 6688128).

Sendrier’s “support splitting” algorithm quickly finds $$\alpha_1, \cdots, \alpha_n$$ given $$g$$ provided that $$n = q$$. More generally, whether or not $$n = q$$, support splitting finds $$\alpha_1, \cdots, \alpha_n$$ given $$g$$ and given the set $$\{\alpha_1, \cdots, \alpha_n\}$$. (This can be viewed as a reason to keep $$n$$ somewhat smaller than $$q$$, since then there are many possibilities for the set, along with many possibilities for $$g$$; most of our proposed parameter sets provide this extra defense.)