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Sorry if that's a naive question but why do you need several signed messages? If you have a quantum computer and a quantum period finding function don't you get immediately the discrete log? Assuming you have one public key (not hashed) doesn't that give you the private key immediately?

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Broadly speaking, more signed messages can get you more points on the curve you're trying to guess.

https://www.cs.umd.edu/~amchilds/teaching/w08/l03.pdf

May help if you're actually interested.

Edit: More signed transactions help with the classical and not the quantum part of schor.

Edit2: Schor has not yet even been able to factor the integer 35 with current quantum hardware, too much interference.

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