Analysis of Addition Modulo 2n on Boolean Function

In order to analyze the impact on the security of cryptographic algorithm produced by the linear approximations of addition modulo 2ⁿ with XOR, this paper firstly translates the operation of addition modulo 2ⁿ into vector Boolean function with dimension of n. The component functions can be obtained through a recurrence formula which does not require a recall of the carry function. Then we explore the cryptographic properties of n-dimensional Boolean functions of addition modulo 2ⁿ, and the results indicate that the component functions are all kth-order quasi-Bent functions. The n-dimensional vector has the first order-correlation immunity, rather than the second-order correlation immunity.