# Problems equivalent to the existence of secure cryptosystems? [closed]

What are some necessary and sufficient conditions for the existence of following?

• Symmetric encryption (defined here as satisfying $\text{IND-CPA}$ and $\text{INT_CTXT}$)
• Asymmetric encryption
• Digital signatures
• Key exchange

Ignore quantum computing for the moment.

• Have you checked a textbook on foundations of cryptography? A good one should cover all of these. – D.W. May 29 '17 at 4:53