(Solved): Complete the two tables. Discuss "patterns" you observe. Note that 11 is a prime and 15 is not. ...

Complete the two tables. Discuss "patterns" you observe. Note that 11 is a prime and 15 is not. $m=11,Z_m=\{0,1,2,3,4,5,6,7,8,9,10\}$ 2. Compute the Euler totients of the following three numbers: $1061,3239,1147$ 3. Explain how the Diffie-Hellman key exchange works, and the assumptions under which it is secure. 4. In an RSA-system the public encryption function is $\mathbf{C}={\mathbf{M}}^{\mathrm{e}}\mathrm{mod}\mathbf{n}$ and the secret decryption function is $\mathbf{M}=\mathbf{C}\mathbf{mod}\mathbf{n}$, where $\mathbf{M}$ is the plaintext and $\mathbf{C}$ is the ciphertext. Let the public parameters of the RSA-system be denoted (n, e), where $\mathbf{n}=\mathbf{pq}$. a) Find a valid value for the pair (n, e) such that each prime is larger than 1000 . b) Give the corresponding secret parameters $(p,q,d,\phi (n))$. c) Decrypt the ciphertext $\mathbf{C}=2$ in your system. d) Assume that we append a digital signature $S$ to a message $M$, using $RSA$, by $S=M^d\mathrm{mod}\mathbf{n}$ Assume that Alice signed and sent a message $(M,S)$. Show how this signed message can be used to construct other signed messages (not $0,1,-1$ ) that Alice did not sign. How do we modify this scheme to overcome the problem?