Summary: The first practical identity based encryption (IBE) scheme was proposed by {\it D. Boneh} and {\it M. Franklin} [see SIAM J. Comput. 32, 586‒615 (2003; Zbl 1046.94008)]. In this work we point out that there is a flawed step in the security reduction exhibited by the authors. Fortunately, it is possible to fix it without changing the scheme or the underlying assumption. In the second place, we introduce a variant of the seminal IBE scheme which allows a more efficient security reduction. This variant is simpler, and has more compact ciphertexts than Boneh-Franklin’s proposal, while keeping the computational cost. Finally, we observe that the flawed step pointed out here is present in several works, and that our techniques can be applied to obtain tighter reductions for previous relevant schemes.