Research on Nonlinear Invariants of a Power Function over a Binary Field

Abstract

The nonlinear invariant attack is a new and powerful cryptanalysis for lightweight block ciphers. The core step of such cryptanalysis is to find the nonlinear invariant(s) of its cascade round. Generally, for an n-bit width function, we need time complexity O(23n) to find the nonlinear invariants. In this paper, we take consider of the power function xm over the finite field GF(2n), which is one of the most important cryptographic functions of last decades. Firstly, we study the nonlinear invariants of xm, we provide mathematical toolboxes named the～m periodical point and the ～m equivalence class. Secondly, we present an algorithm to get all the nonlinear invariants of xm over GF(2n) at the price of time complexity O(2n). Moreover, if the growth of n exceeds our tolerance above, we also provide a method to get parts of nonlinear invariants of xm. Finally, we take consider of the nonlinear invariants of x3 over GF(2129) as application, which is used in the block cipher MiMC and seems impractical by existing methods. Our results allow us to find several (but not all) nontrivial nonlinear invariants of such function for the first time.