Addition modulo $2^n$ in block ciphers

Cryptographic properties of the addition modulo $2^n$ and bitwise addition modulo $2$ are analysed in this article. For the first operation, the author proposes some linear and non-linear approximations and their usage in cryptanalysis. Also, a modification of the linear cryptanalysis method is proposed. In some cases, this modification allows a more efficient way for attack. For example, an attack on eight rounds GOST 28147-89 can be carried out with this modification and cannot be done without it. The author gives examples how the approximations are used for known plaintext attack on ciphers using the addition modulo $2^n$ for key mixing. The author shows that the usage of the addition modulo $2^n$ instead of XOR increases the resistance of block ciphers to linear cryptanalysis and its non-linear modification.