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I am new to cryptography and I cant get my head around this issue:
If symmetric cryptography is faster then asymmetric and given that Diffie-Hellman key exchange algorithm allows you to exchange keys safely why bother using Asymmetric cryptography if I can exchange symmetric keys safely ?
As mentioned in the comment by user93353, asymmetric cryptography is mostly used for symmetric key exchanging and signing of messages. I will give a simple explanation of the need for asymmetric cryptography.
Image a complete graph $G$ with $n$ nodes in which every node needs to communicate with each other. If we use symmetric encryption we need $O(n^2)$, a key for every edge. If we use asymmetric encryption we need $O(n)$ keys. As you can see symmetric encryption doesn't scale well. I have seen it in some cryptography textbooks referenced as The squaring problem. The simple solution to this one is whenever a node needs to communicate with another node to generate a shared symmetric secret key and use it for onward communication. This is what TLS does basically.
Asymmetric Cryptography was the start of a new era in cryptography.
Diffie-Hellman key exchange is asymmetric cryptography.
We also use digital signature algorithms. These allow someone with a private key to digitally "sign" a message, such that anyone with the corresponding public key can verify that the signature was made on that message by that private key. This allows proving the authorship of messages and provides the non-repudiation. The non-repudiation is only possible with asymmetric cryptography and Rabin Signature Scheme was the first genuine digital signature.
Key exchange and digital signature schemes are the most common use of asymmetric cryptography.
