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Studying the bls signatures, I am wondering how in practice should we handle the parameter G.

In many signatures scheme, the generator used is a well known constant, for example eddsa with ed25519 has a well known fixed generator g. However, in the bls example from the C source code of the pbc library of Dan Boneh they generate a random G. In the literature they suppose the G is known by all parties. In jPBC sources , they save that system parameter G to a file.

Is there not a well defined constant G somewhere so client / signers don't have to exchange that value?

If I were to deploy a bls signature system, how would I chose this G? There does not seem to be any constraint over this system parameters since it's taken at random.

Do I include it in the signature and generate a new one for each signature? (providing more randomness at the expense of a bigger size?) or should I generate one element at random and call it G and make it public? Or save to a file and maybe have multiple files for different recipients ?

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