can session key be be used as digital signature? if two communicating parties share a (symmetric)  session key, k.

When one of the party wants to send a message 'm' to other, is it okay use {m}k as a digital signature?
No. ​ Let || be concatenation, let [E',D'] be any symmetric encryption scheme,

and then let [E,D] be given by ​ ​ ​ E(k,x) ​ = ​ 0 || E'(k,x) ​ ​ ​ and

D(k,empty_string) = empty_string ​ ​ ​ and ​ ​ ​ D(k,0||c) = D'(k,c) ​ ​ ​ and ​ D'(k,1||c) = c .

The efficiency and correctness conditions for [E,D] are trivial. ​ Since E can be simulated

from E' by just prepending a zero to each ciphertext, [E,D] will be IND-CPA if [E',D'] is.

However, with [E,D], one can trivially break {m}k as an authentication

scheme by just outputting something that starts with 1.