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This is much younger than other approaches and was first published in 1975. As we have discussed this has distinctive feature of only needing one key per individual/organization requiring encrypted authenticated messaging
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It has nontrivial infrastructure to distribute the N public keys for N organizations but this is better than N2 keys for secret key cryptography
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Roughly the public key is a very large number that is the product of two primes. The private key is (related to) one of these primes.
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It is used differently in two cases which following foils do in detail
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Transmission over insecure network where one encodes with public key of receiver (and receiver decodes with his or her private key)
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Authentication where you encode your digital signature with your private key and receiver checks the signature with your public key -- only you can encode signature so it is correctly decoded with public key
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