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Every individual has a distinctive way of walking. For this reason gait can be a key element of biometric techniques aimed at authenticating and/or identifying the user of a wearable device. This paper presents a lightweight method that uses the acceleration collected at the user's wrist for authentication purposes. The user's typical gait pattern is learned during the first period of use, then detection of anomalies in a set of acceleration-based features is used to understand if a new user, a possible impostor or a thief, is wearing the device. The method has been successfully evaluated with 15 volunteers, showing an Equal Error Rate of 2.9%. These results suggest that gait-based authentication with a wrist-worn device can be carried out with high accuracy levels. A comparison with a similar method executed on a pocket-worn device is also included.