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An asymmetric cipher is an encryption scheme using a pair of keys, one to encrypt and a second to decrypt a message. This way the encrypting key need not be kept secret to ensure a private communication. Similarly in public key authentication, the verification key can be public and the signing key private.
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Understanding Practical Differences Between ElGamal and Diffie-Hellman
in ElGamal you use an additional variable K which is different for each message. you don't run the algorithm from scratch again for every new message.