Polynomials over ℤ2n and their applications in symmetric cryptography

Abstract

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.