Diffie Hellman key exchange protocol is used

Alice's public key is $\text{'}$A$\text{'}$ and a private key $\text{'}$a$\text{'}$

Bob's public key is $\text{'}$B$\text{'}$ and a private key $\text{'}$a$\text{'}$

Quin has a public key $\text{'}$Q$\text{'}$ and a private key $\text{'}$q$\text{'}$

They all know the generator alpha

Consider a secret$-$sharing scheme, which is a way for any three people $($call them Alice,

Bob and Quinn$)$ to determine a secret $S$ that all three of them will know, without the need

to exchange any information privately in advance. Using Algorithm $3$d$\mathrm{.}4,$ suppose that Alice

and Bob set up and publish Diffie$-$Hellman keys, using a modulus $p$ that is big enough that the

Discrete Logarithm Problem remains a Hard Problem. Quinn chooses a secret number $q$ in $Z_p^{**}$

with $Q={}^{q}QqS,aS,bS,qSS={}^{\text{a}\text{b}\text{q}}S={}^{a+b+q}S={}^{ba+qa}0,$ computes $Q={}^q,$ and publishes $Q$ but keeps $q$ secret.

Suppose that the details $of$ the secret$-$sharing scheme will $be$ publicly available, and $tobe$

effective, $it$ must follow four rules:

$R1a.$ Alice has enough information $to$ compute $S,$ accessing only $,$a and the public directory.

$R1b.$ Bob has enough information $to$ compute $S,$ accessing only $,b$ and the public directory.

$R1q.$ Quinn has enough information $to$ compute $S,$ accessing only $,q$ and the public directory.

$R2.No$ eavesdroppers can compute $S$ based $on$ information from the public directory, assuming

that they cant solve the Discrete Logarithm Problem.

For each $of$ the following schemes, verify $R1a,R1b,R1q,R2in$ that order, justifying whether

the rule $is$ met $or$ broken. Once you have justified why a rule $is$ broken, you are done with that

scheme.

$(a)S={}^{\text{a}\text{b}\text{q}}$

$(b)S={}^{a+b+q}$

$(c)S={}^{ba+qa}$