This paper considers the Broadcast Channel with Confidential Message (BCCM) where the sender attempts to send altogether a common message to two receivers and a confidential message to one of them. The achievable rate regions are derived for the power-constrained Gaussian BCCM with finite input alphabet using various transmission strategies. Namely, time sharing, superposition modulation and superposition coding are used as broadcast strategies. For superposition modulation and superposition coding, the maximal achievable rate regions are obtained by maximizing over both constellation symbol positions and the joint probability distribution. The maximization of the secrecy rate for wiretap channels is also studied as a particular case of the BCCM problem. We compare the considered transmission strategies in terms of percentage gains in achievable rates. We concentrate on the impact of the finite alphabet constraint on achievable rates, and show that this constraint may change well known results obtained in the Gaussian case. We show also that the secrecy constraint can change the shape of the achievable rate region in superposition modulation used in some standards when symbols are equiprobable. On a more practical side, it is shown that a performance close to the optimum can be obtained by strategies with reduced complexity.