Polynomials over ℤ2n and their applications in symmetric cryptography

Components which are constructed via the application of basic instructions of modern processors are common in symmetric ciphers targeting software applications; among them are polynomials over $\mathbb{Z}_{2^{n}}$, which fit n-bit processors. For instance, the AES finalist RC6 uses a quadratic polynomial over $\mathbb{Z}_{2^{32}}$. In this paper, after some mathematical examination, we give the explicit formula for the inverse of RC6-like polynomials over $\mathbb{Z}_{2^{n}}$ and propose some degree-one polynomials as well as some self-invertible (involutive) quadratic polynomials with better cryptographic properties, instead of them, for the use in modern software-oriented symmetric ciphers. Then, we provide a new nonlinear generator with provable period, which could be used in stream ciphers and pseudo-random number generators.