Algebraic Properties of Modular Addition Modulo a Power of Two

Modular addition modulo a power of two, is one of the most applicable operators in symmetric cryptography; therefore, investigating cryptographic properties of this operator has a significant role in design and analysis of symmetric ciphers. Algebraic properties of modular addition modulo a power of two have been studied for two operands by Braeken in fse’05. Also, the authors of this paper, have studied this operator, in some special cases, before. In this paper, taking advantage of previous researches in this area, we generalize algebraic properties of this operator for more than two summands. More precisely, we determine the algebraic degree of the component Boolean functions of modular addition of arbitrary number of summands modulo a power of two, as a vectorial Boolean function, along with the number of terms and variables in these component functions. As a result, algebraic degrees of the component Boolean functions of Generalized Pseudo-Hadamard Transforms are computed. Keywords; Modular addition modulo a power of two; Boolean function; Algebraic Normal Form; Algebraic degree; Pseudo-Hadamard Transform.