  Block ciphers are in widespread use since the 1970s. Their iterated structure
is prone to numerous round invariant attacks for example in Linear
Cryptanalysis (LC). The next step is to look at non-linear polynomial
invariants cf. Eurocrypt'95. Until recently, researchers have found extremely
few such attacks, with some impossibility results. Eventually recent papers
show how to construct polynomial invariant attacks for block ciphers, however
many such results were of degree 2. In this paper we propose a new incremental
methodology for constructing high degree polynomial invariant attacks on block
ciphers. A trivial attack on one cipher setup will be transposed to show the
existence of a more advanced attack on a stronger cipher in several steps. The
key tool is the manipulation of the roots of the so called Fundamental
Equation. Examples are constructed with an old historical block cipher T-310.
