This paper studies the concrete security of BBS signatures (Boneh, Boyen, Shacham, CRYPTO ’04; Camenisch and Lysyanskaya, CRYPTO ’04), a popular algebraic construction of digital signatures which underlies practical privacy-preserving authentication systems and is undergoing standardization by the W3C and IRTF. Schäge (Journal of Cryptology ’15) gave a tight standard-model security proof under the q-SDH assumption for a less efficient variant of the scheme, called BBS+—here, q is the number of issued signatures. In contrast, the security proof for BBS (Tessaro and Zhu, EUROCRYPT ’23), also under the q-SDH assumption, is not tight. Nonetheless, this recent proof shifted both standardization and industry adoption towards the more efficient BBS, instead of BBS+, and for this reason, it is important to understand whether this tightness gap is inherent. Recent cryptanalysis by Chairattana-Apirom and Tessaro (ASIACRYPT ’25) also shows that a tight reduction to q-SDH is the best we can hope for. This paper closes this gap in two different ways. On the positive end, we show a novel tight reduction for BBS in the case where each message is signed at most once–this case covers in particular the common practical use case which derandomizes signing. On the negative end, we use a meta-reduction argument to prove that if we allow generating multiple signatures for the same message, then no algebraic reduction to q-SDH (and its variants) can be tight.