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I found that homomorphic signatures allows an agency to carry out arbitrary computation $f$ on the signed data $m$ and accordingly gain a signature for the computation result $f(m)$ with respect to $f$. Another related notion is functional signatures, in which DO hands out a secondary key $sk$ to allow a specified agency to sign messages $m'$ which satisfy that $m' = f(m)$.

Could you please let me know: What are the differences between these two schemes? Because to me they appear to be the same.

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The difference is that in homomorphic signatures anyone can compute on the signatures, whereas in functional signatures only the party holding the functional secret key $sk_f$ can compute $f$ on signatures (signed by master secret key corresponding to $sk_f$)

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