Alice wants to set up a private key to use with Bob. She selects the prime p 684

Question

Alice wants to set up a private key to use with Bob. She selects the prime p 6841 and primitive root r- 22, and publishes these numbers so they are public. Alice chooses private key ki-3859, and Bob chooses private key k2 5773. Using the Diffie-Hellman key exchange protocol, (1) what information will Alice send to Bob (over a possibly insecure channel), (2) what is the common shared private key that Bob will then compute, (3) what information will Bob then send back to Alice (over a possibly insecure channel), and (4) how will Alice then retrieve the private shared key? Given that the prime p is sufficiently big, why is this method secure, even though somebody could know the public information p, r and "listen in" and recover the information transmitted back and forth between Alice and Bob? 125 pts]

Explanation / Answer

The “mod” operation is a way to select the remainder of to divide two numbers, For example:

In this exercise, the key is to know the “mod” operation with another challenge inside:

The solution would be:

223859 mod 6841= x1

225773mod 6841= x2

In this example we are using numbers so big, so the solution will be indicated

225773/6841=x1= remainder of this division

We also can get the results of x2 that it would be:

223859/6841=x2= remainder of this division

When we join the answers of the both operation is possible to get the same result.

x23859 mod 6841= share secret

x15773mod 6841= share secret

The share secret is the number that we can get with the previous equations, they are the same number. In other words:

223859(5773)=225773(3859)=share secret

c)Bob will send: 225773mod 6841= x2

d) Alice will have to make the next operation:

x23859 mod 6841=...=share secret

Where X2 is the last number that Bob public, after that she can make the comparison with this operation

x15773mod 6841=...=share secret

And finally get the Share secret.

I advise to use number smaller and show off this method, and let indicated this exercise, its answers are number > of # *10^5180

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