Undeniable signatures were introduced in 1989 by Chaum and van Antwerpen to limit the self-authenticating property of digital signatures. An extended concept – the convertible undeniable signatures – proposed by Boyar, Chaum, Damgård and Pedersen in 1991, allows the signer to convert undeniable signatures to ordinary digital signatures. We present a new efficient convertible undeniable signature scheme based on bilinear maps. Its unforgeability is tightly related, in the random oracle model, to the computational Diffie-Hellman problem and its anonymity to a non-standard decisional assumption. The advantages of our scheme are the short length of the signatures, the low computational cost of the signature and the receipt generation. Moreover, a variant of our scheme permits the signer to universally convert signatures pertaining only to a specific time period. We formalize this notion as the time-selective conversion.