The paper presents group-oriented (t,n) threshold digital signature schemes based on the difficulty of solving the discrete logarithm problem. By employing these schemes, any t out of n users in a group can represent this group to sign the group signature. The size of the group signature and the verification time of the group signature are equivalent to that of an individual digital signature. In other words, the (t, n) threshold signature scheme has the following five properties: (i) any group signature is mutually generated by at least t group members; (ii) the size of the group signature is equivalent to the size of an individual signature; (iii) the signature verification process is simplified because there is only one group public key required; (iv) the group signature can be verified by any outsider; and (v) the group holds the responsibility to the signed message. In addition to the above properties, two of the schemes proposed do not require the assistance of a mutually trusted party. Each member selects its own secret key and the group public key is determined by all group members. Each group member signs a message separately and sends the individual signature to a designated clerk. The clerk validates each individual signature and then combines all individual signatures into a group signature. The (n, n) threshold signature scheme can be easily extended to become a digital multisignature scheme.