Problem $1.$ Use exhaustive key search to decrypt the following ciphertext, which was encrypted using a Shift Cipher:BEEAKFYDJXUQYHYJIQRYHTYJIQFBQDUYJIIKFUHCQD Problem $2.$ Suppose we are told that the plaintextbreathtakingyields the ciphertextwhere the Hill Cipher is used. Determine the encryption matrix.Problem $3.$ Let M $=\{$a$,$b$\}$ with Pr$[$a$]=0\mathrm{.}25,$ Pr$[$b$]=0\mathrm{.}75.$ Let K $=\{$k$1,$k$2,$k$3\}$ with Pr$[$k$1]=0\mathrm{.}5,$ Pr$[$k$2]=$ Pr$[$k$3]=0\mathrm{.}25.$ Let C $=\{1,2,3,4\},$ and suppose the encryption functions are defined to be:$1.$ Enc$($k$1,$ a$)=1,$ Enc$($k$1,$ b$)=22.$ Enc$($k$2,$ a$)=2,$ Enc$($k$2,$ b$)=33.$ Enc$($k$3,$ a$)=3,$ Enc$($k$3,$ b$)=4$Compute the probability distribution on C$.$Problem $4.$ Let M $=\{$a$,$ b$\}$ and let K $=\{$k$1,$ k$2,$ k$3,$ k$4,$ k$5\}.$ Let C $=\{1,2,3,4,5\},$ and suppose the encryptionfunctions are defined to be:$1.$ Enc$($k$1,$ a$)=1,$ Enc$($k$1,$ b$)=22.$ Enc$($k$2,$ a$)=2,$ Enc$($k$2,$ b$)=33.$ Enc$($k$3,$ a$)=3,$ Enc$($k$3,$ b$)=14.$ Enc$($k$4,$ a$)=4,$ Enc$($k$4,$ b$)=55.$ Enc$($k$5,$ a$)=5,$ Enc$($k$5,$ b$)=4$Now choose two positive real numbers s and t such that s $+$ t $=1,$ and define Pr$[$k$1]=$ Pr$[$k$2]=$ Pr$[$k$3]=3$s and Pr$[$k$4]=$ Pr$[$k$5]=2$t $.$ Prove that this cryptosystem achieves perfect secrecy$.$