Sufficient variables for transition function of a modified additive generator

We consider the class of bijective shift registers of length $n$ over the set $V_r$ of binary vectors of length $r$. In this paper, we research registers constructed on the additive generators modulo $2^r$ modified by a transformation of $V_r$. The feedback function of such a register is a composition of additive generator feedback function and the transformation of $V_r$. It is known that determination of sufficient variables for the composition of nonlinear functions is a complicated problem. By using combinative properties of the bijection $\mathbb Z_{2^r}\leftrightarrow V_r$, we describe the set of all sufficient variables for feedback function of the registers researched.