This talk discusses a few small results related to cube-style algebraic attacks. It is shown that in a white-box scenario with \[ f = ml + r\] and (\(deg l = 1\))), the complexity of extracting \(l\) is much less than \(2^{\deg m}\) oracle queries -- this means the attacker does not need to sum over the entire cube, just over a much smaller subset. An algorithm is provided to construct such smaller subsets.

It is further shown that generalizing cube-style attacks to decompositions of the form \[f = pl+r\] (where \(p\) is a polynomial) is unlikely to yield many benefits over the monomial scenario.


The second talk presents some results that show that the ring of boolean functions, even though it is not euclidian, allows for an algorithm that performs an equivalent task: Given a set of \(n\) generators for an ideal, the algorithm can calculate the single generator for the entire ideal (the ring of boolean functions is a principal ideal domain) in \(n\) polynomial multiplications and \(n+1\) polynomial additions. 

A number of interesting results follow from this: 

- Calculating the product of a set of boolean functions which have one solution in common is equivalent to solving them
- Estimating the number of monomials in the product gives information about the hamming weight of the solution
- Restrictions of boolean functions to particular subsets can be calculated easily