Degree Evaluation of NFSR-Based Cryptosystems

In this paper, we study the security of NFSR-based cryptosystems from the algebraic degree point of view. We first present a general framework of iterative estimation of algebraic degree for NFSR-based cryptosystems, by exploiting a new technique, called numeric mapping. Then based on this general framework we propose a concrete and efficient algorithm to find an upper bound on the algebraic degree for Trivium-like ciphers. Our algorithm has linear time complexity and needs a negligible amount of memory. As illustrations, we apply it to Trivium, Kreyvium and TriviA-SC, and reveal various upper bounds on the algebraic degree of these ciphers by setting different input variables. By this algorithm, we can make use of a cube with any size in cube testers, which is generally believed to be infeasible for an NFSR-based cryptosystem before. Due to the high efficiency of our algorithm, we can exhaust a large set of the cubes with large size. As such, we obtain the best known distinguishing attacks on reduced Trivium and TriviA-SC as well as the first cryptanalysis of Kreyvium. Our experiments on Trivium show that our algorithm is not only efficient in computation but also accurate in estimation of attacked rounds. The best cubes we have found for Kreyvium and TriviA-SC are both of size larger than 60. To the best of our knowledge, our tool is the first formalized and systematic one for finding an upper bound on the algebraic degree of an NFSR-based cryptosystem, and this is the first time that a cube of size beyond practical computations can be used in cryptanalysis of an NFSR-based cryptosystem. It is also potentially useful in the future applications to key recovery attacks and more cryptographic primitives.