1. Let II = (Gen, Enc, Dec) be a private-key encryption scheme that has indistinguishable en-
cryptions in the presence of an eavesdropper. Which of the following encryption schemes are
also necessarily secure against an eavesdropper? If you think a scheme is secure, sketch a
proof, if not, provide a counterexample.
Here, for a bit string s, parity(s) is 1 if the number of 1's in s is odd, and 0 otherwise.
The || symbol stands for concatenation. So, for strings if x = 00 and y = 11, x||y = 0011.
(a) $Enc_k^1(m) = 0||Enc_k(m)$
(b) $Enc_k^2(m) = Enc_k(m)||parity(m)$
(c) $Enc_k^3(m) = Enc_k(m)||Enc_k(m)$
(d) $Enc_k^4(m) = Enc_k(m)||Enc_k(m + 1)$. Here think of m as an integer.
