The paper presents a new approach to generating enciphering sequences useful in information protection, with an illustration on images. The procedure is both theoretically and experimentally supported by combining elements derived from the running-key cipher, information theory and statistics. The enciphering key generator is based on the logistic map, and its theoretical properties are demonstrated by statistical tests. The new enciphering sequences comply with the fair coin model, and the randomly chosen initial conditions of the logistic map (defining the enciphering sequence) can be part of the secret key. Although dealing with a particular case of chaotic system, the paper contains the necessary elements so that the overall procedure can be applied to other chaotic maps (e.g., tent map).