What is Public and Private Key Encryption?

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How does public and private key encryption work? How is asynchronous encryption different than symmetric encryption?

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Contents
0:00 Types of cryptography - Caesar DES AES asymmetric and hashing
0:45 Symmetric vs asymmetric encryption
1:15 public vs private keys
1:30 Whitfield Diffie New Directions in Cryptography
1:55 PGP Phil Zimmerman 1991 US Customs
2:50 Symmetric Encryption Standard Password Problems
4:10 Asymmetric encryption advantages
4:50 GPG gen-key
5:10 Share public key in GPG
5:35 import public key into gpg encrypt for user