  Linear (or differential) cryptanalysis may seem dull topics for a
mathematician: they are about super simple invariants characterized by say a
word on n=64 bits with very few bits at 1, the space of possible attacks is
small, and basic principles are trivial. In contract mathematics offers an
infinitely rich world of possibilities. If so, why is that cryptographers have
ever found so few attacks on block ciphers? In this paper we argue that
black-box methods used so far to find attacks in symmetric cryptography are
inadequate and we work with a more recent white-box algebraic methodology.
Invariant attacks can be constructed explicitly through the study of roots of
the so-called Fundamental Equation (FE). We also argue that certain properties
of the ring of Boolean polynomials such as lack of unique factorization allow
for a certain type of product construction attacks to flourish. As a proof of
concept we show how to construct a complex and non-trivial attack where a
polynomial of degree 7 is an invariant for any number of rounds for a complex
block cipher.
