Differential equations of addition

In cryptography, differential equations of addition (DEA) are one of the most basic equations related to differential cryptanalysis that mix additions over two different groups (e.g. addition modulo 232 and addition over GF(2)) and where input and output differences are expressed as XORs.
Examples.
Differential equations of addition (DEA) are of the following form:
formula_1
where formula_2 and formula_3 are formula_4-bit unknown variables and formula_5, formula_6 and formula_7 are known variables. The symbols formula_8 and formula_9 denote "addition modulo" formula_10 and "bitwise exclusive-or" respectively. The above equation is denoted by formula_11.
Let a set 
formula_12 
for integer formula_13 denote a system of formula_14 DEA where formula_14 is a polynomial in formula_4. It has been proved that the satisfiability of an arbitrary set of DEA is in the complexity class P when a brute force search requires an exponential time. 
In 2013, some properties of a special form of DEA were reported by Chengqing Li et al., where formula_17 and formula_3 is assumed known. Essentially, the special DEA can be represented as formula_19. Based on the found properties, an algorithm for deriving formula_2 was proposed and analyzed.
Applications.
Solution to an arbitrary set of DEA (either in batch and or in adaptive query model) was due to Souradyuti Paul and Bart Preneel. The solution techniques have been used to attack the stream cipher Helix.