Design a symmetric$-$key encryption method. Describe the idea of your design. $(15$ points$)$

For a CPU with only bit operations $($bit$-$addition, bit$-$multiplication, bit$-$or$,$ etc.$),$ how many bit operations are needed to compute $x+y,x.y,$ and xmody, where $x$ is $m-$bit long and $y$ is $n-$bit long. $(15$ points$)$

In CFB mode of DES with $s=16,$ a message $m=m_1{m}_{2}dots{m}_{16}$ is encrypted as $c_1{c}_{2}dots{c}_{16},$ where each $m_i$ is $16-$bit long. Assume that upon receiving the ciphertext, $c_2$ is lost and $c_9$ is wrong. What parts of $m$ can be decrypted correctly? $(20$ points$)$

Use the Chinese Remainder Theorem to compute $x$ for xmod$3=2,$xmod$7=5,$ and xmod$13=3.$ Provide all intermediate steps. $(25$ points$)$

In the SubBytes of AES, $f(x)=x^{-1}mod{x}^8+x^4+x^3+x+1.$ Compute $f(10101100).$ Provide all

intermediate steps. $(25$ points$)$ Please answer all questions