Problem 1. A public-key encryption scheme (PKE) is the same as SKE except with the fol- lowing changes to the syntax of (Gen, Enc, Dec): Gen outputs a key pair (pk, sk) where pk is the public key and sk is the secret key; Enc takes the pair (msg, pk) as input; Dec takes the pair (ct, sk) as input. Suppose that we have two encryption schemes: a PKE scheme (PKE.Gen, PKE.Enc, PKE.Dec) and a SKE scheme (SKE.Gen,SKE.Enc,SKE.Dec). Prove correctness and security for the following combined PKE scheme (Gen, Enc, Dec): Gen(): Draw (PKE.pk, PKE.sk) PKE.Gen() and output (pk, sk) = (PKE.pk, PKE.sk). Enc(msg, pk): Draw SKE.sk SKE.Gen() and SKE.ct SKE.Enc(msg, SKE.sk). Additionally, parse pk = PKE.pk and draw PKE.ct PKE.Enc(SKE.sk, PKE.pk). Output ct = (PKE.ct, SKE.ct). So, intuitively, we are using SKE to encrypt the message with a temporary key, and we are using PKE to encrypt the temporary key. Dec(ct, sk): Parse ct = (PKE.ct, SKE.ct) and sk = PKE.sk. Compute SKE.sk = PKE.Dec(PKE.ct, PKE.sk) and output msg = SKE.Dec(SKE.ct, SKE.sk).