Cryptanalysis of Stream Cipher Grain Family

Grain v1 is one of the 7 final candidates of ECRYPT eStream project, which involves in the 80-bit secret key. Grain-128 is a variant version with 128-bit secret key, and Grain v0 is the original version in the first evaluation phase. Firstly, we describe a distinguishing attack against the Grain family with weak Key-IVs. Utilizing the second Walsh spectra of the nonlinear functions, we show that there are 2/2/2 weak Key-IVs among total 2/2/2 Key-IVs, and to distinguish a weak Key-IV needs about 2/2/2 keystream bits and 2/2/ 2 operations for Grain v0, Grain v1 and Grain-128 respectively. Secondly, we apply algebraic attacks to the Grain family with a weak Key-IV, and can recover the secret key in about 2 seconds and 150 keystream bits for Grain v0 and Grain v1, and reveal the key of Grain-128 with about 100 keystream bits and 2 operations. Furthermore, we discuss the period of the keystream with a weak Key-IV for any Grain-like structure which can lead in self-sliding attack.