In this paper, we present a construction of (n,k,d,m) secure regenerating codes for distributed storage systems against eavesdroppers that can observe either data stored in at most m storage nodes or downloaded data for repairing at most m failed nodes in a network where m<k≤d≤n-1. The (n,k,d,m) secure regenerating code is based on an (n,k,d) minimum bandwidth regenerating (MBR) code, which was proposed by Rashmi, Shah and Kumar as optimal exact-regenerating codes, for all values of the parameters (n,k,d). The (n,k,d,m) secure regenerating codes have the security as a secret sharing scheme such that even if an eavesdropper knows either data stored in at most m storage nodes or downloaded data for repairing at most m failed nodes, no information about data leaks to the eavesdropper.