I am a student of Pure Mathematics.I want to get some information on the following:
$1$.What do modern day cryptographers work on?
$2$. How does pure mathematics influence modern day cryptography?
$3$. What are some open problems in modern Cryptography and how does mathematics affect them?
Regarding 1 or 3:
E.g. Homomorphic encryption has been mentioned as "Holy grail of Cryptography".
It would allow to provide encrypted inputs to some service provider, who would be able to perform useful calculations on the inputs, without him knowing the decrypted form.
Kind of putting some ingredients into a locked box, giving it to the provider who shakes and bakes it, and when you open the returned box, you get some nice pizza.