Block cipher modes of operation $[8$ Marks$].$

Let $F$:$\{0,1{\}}^n\{0,1{\}}^n\{0,1{\}}^n$ be a length$-$preserving pseudorandom permutation, i$.$e$.,$ a block cipher with

block length $n.$ Suppose all messages being encrypted have length a multiple of $n,$ i$.$e$.,m=m_1{m}_{2}dots{m}_l$ with

each $m_{i}in\{0,1{\}}^n$ being a block of the message. The Fischer Spiffy Mixer $($FSM$)$ mode of operation works as

follows.

Gen: On input $1^n,$ output a uniform key $k$ of length $n,$ i$.$e$.,$ klarr$\{0,1{\}}^n.$

Enc: To encrypt a message $m=m_1{m}_{2}dots{m}_l$ with each $m_{i}in\{0,1{\}}^n,$ do the following. Choose two random

initialization vectors $I{V}_1$ and $I{V}_2$ of length $n$ each and set $m_0$:$=I{V}_1$larr$\{0,1{\}}^n$ and $c_0$:$=I{V}_2$larr$\{0,1{\}}^n.$ For

$i=1$ to $l,$ set $c_i$:$=m_{i-1}o+F_k(m_{i}o+c_{i-1}).$ Output the ciphertext as $c=($:$c_{-1},c_0,c_1,c_2,$dots,$c_l$:$)$ where $c_{-1}=m_0.$

$($a$)(2$ marks$)$ Describe how you would decrypt the ciphertext in the FSM mode.

$($b$)(6$ marks$)$ Suppose a single bit error occurs in transit in the ciphertext block $c_i$ for $i=-1,0,1,$dots,$l.$

Describe the effect of this single bit error on the reconstructed plaintext. In particular, specify which

message blocks are affected and which are unchanged when a receiver decrypts the modified ciphertext.