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Compact formulas are derived to represent the Algebraic Normal Form (ANF) of f(x + a mod 2n) and f(x ◊ a mod 2n) from the ANF of f, where f is a Boolean function on Fn 2 and a is a constant of Fn 2. We compare the algebraic degree of the composed functions with the algebraic degree of the original function f. As an application, the formula for addition modulo 2n is applied in an algebraic attack on the summation generator and the E0 encryption scheme in the Bluetooth keystream generator.

The ANF of the Composition of Addition and Multiplication mod 2 n with a Boolean Function