# Is BLS signature scheme strongly unforgeable?

I would like to know if BLS signature scheme is strongly unforgeable under adaptive chosen message attack. If it is not, is it possible to modify the BLS scheme in order to reach this property ?

BLS signatures are computed by hashing the message $m$ from the message space $\mathcal{M}$ to a source group of a bilinear pairing $e: \mathbb{G} \times \mathbb{G} \to \mathbb{G}_T$ using a hash function $H: \mathcal{M} \to \mathbb{G}$ and exponentiating $h$ with the secret key $x$, i.e., a signature $\sigma$ is of the form $$\sigma := H(m)^x.$$
The hash function maps each message to a uniquely determined value (there could be multiple messages which are mapped to the same value), and $x$ is fixed by the public key $g^x$. This means that BLS signatures are even unique. That is there is only one signature for each message. This is an even stronger property than unforgeability, i.e., every existentially unforgeable and unique signature scheme is also strongly existentially unforgeable.