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Why is proof-by-reduction needed (for Elgamal proof of security, for example)?
The textbook proof for Elgamal encryption basically reduces to the Decisional Diffie-Hellman assumption (DDH).
Elgamal:
$Gen(.): x \xleftarrow{R} \mathbb{Z}_p$; $Enc(m,g^x): r \xleftarrow{R} ...
