RSA encryption exponent In RSA public-key material (e, N), why must the encryption exponent e be relatively prime to $\phi(N)$? Why must any public key encryption algorithm resist CPA?
why must the encryption exponent e be relatively prime to $\phi(N)$


If it's not, then the decryption will not be unique.  That is, there will be multiple messages $M_1 \ne M_2$ such that $M_1^e \bmod N = M_2^e \bmod N$; hence if the decryptor receives that common value, he will not be able to determine if the original message was $M_1$ or $M_2$


  Why must any public key encryption algorithm resist CPA


When we're analyzing the security of a public key algorithm, we assume that an attacker has the public key (after all, we assume that we can give it out freely, so it is not unreasonable that the attacker can get a copy).  With that public key, the attacker can pick list of message he wants, and encrypt those messages.  Hence, a public key encryption algorithm must resist CPA because, practically speaking, an attacker can actually perform that attack.