A class of stream ciphers similar to the well-known SNOW 2.0 cipher is investigated. The formal description of the ciphers from this class is given and the relation between keysream-generation process of a SNOW 2.0-like cipher and the message encryption with Even-Mansour scheme is determined. The security of SNOW-2.0-like stream ciphers against attacks based on the existence of shifting equivalent keys is analyzed. These attacks are related key attacks and can be applied to many stream ciphers, particularly, to SNOW 2.0. The main result of this paper is a sufficient condition for SNOW 2.0-like ciphers to be secure againt mentioned attacks. This condition is convenient for prac-tical appliances and allows constructing the affine mappings (that proceed keys and initialization vectors inserting into the keystream generator) guarantee the security of corresponding ciphers against mentioned attacks. The two examples of such mappings that can be used for constructing of new SNOW 2.0-like ciphers are proposed.