Several broadcast encryption (BE) constructions have been proposed since Fiat and Naor introduced the concept, some achieving short parameters size while others achieve better security. Since 1994, a lot of alternatives to BE have moreover been additionally proposed, such as the broadcast and trace (BT) primitive which is a combination of broadcast encryption and traitor tracing. Among the other variants of BE, the notion of augmented BE (AugBE), introduced by Boneh and Waters in 2006, corresponds to a BE scheme with the particularity that the encryption algorithm takes an index as an additional parameter. If an AugBE scheme is both message and index hiding, it has been proved that it can generically be used to construct a secure BT scheme. Hence, any new result related to the former gives an improvement to the latter. In this paper, we rst show that both BE and AugBE can be obtained by using an identity-based encryption scheme with wildcard (WIBE). We also introduce the new notion of anonymous AugBE, where the used users set is hidden, and prove that it implies index hiding. We then provide two dierent WIBE constructions. The rst one has constant size ciphertext and used to construct a new constant size ciphertext BE scheme with adaptive CPA security, in the standard model (under the SXDH assumption). The second WIBE provides pattern-hiding, a new denition we introduced, and serves as a basis for the rst anonymous AugBE scheme (and subsequently a BT scheme since our scheme is also index hiding by nature) in the literature, with adaptive security in the standard model (under the XDLin assumption).