$1.$ What are three broad categories of applications of public$-$key cryptosystems?

$2-$Users Alice and Bob use the Diffie$-$Hellman key exchange technique with a

common prime q $=23$ and a primitive root a $=5.$

a$.$ If user Alice has private key XA $=6,$ what is Alice$$s public key YA$?$

b$.$ If user Bob has private key XB $=15,$ what is Bob$$s public key YB$?$

c$.$ What is the shared secret key?

Answer the following questions about Cryptographic Hash Functions?

a$.$ What are the security requirements for Cryptographic Hash Functions?

b$.$ Given the following input $(5241,9235,3237,6948,8259,6072)$ and the hash

function x mod $5,$ which of the above inputs hash to the same value?

$4.$ In the context of public key distribution, why is key revocation important,

and what mechanisms are used to revoke compromised or outdated public

keys?

$5.$ Explain the difference between a digital signature and a digital certificate.

How are these two concepts related, and how do they complement each other

in digital security protocols?