A problem of the symmetric cryptosystems that we have described so far is key distribution and key management. When Alice and Bob use a symmetric cryptosystem, they must exchange a secret key before they can secretly communicate. For the key exchange, they need, for example, a secure channel or a courier. The key exchange problem becomes even more difficult if many people want to exchange encrypted messages, for example, on the Internet. If a communication network has n users and any two of them exchange a key, then n(n − 1)/2 secret key exchanges are necessary and all those keys have to be stored securely. According to  there were approximately 6 · 108 Internet users in 2002. If any two Internet users exchanged a secret key then 1.8 · 1017 keys would be necessary. This would be impossible to organize.
KeywordsDiscrete Logarithm Chinese Remainder Theorem Choose Ciphertext Attack Extended Euclidean Algorithm ElGamal Encryption
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