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In the picture below, for the text underlined in red color: 7X(MOD 11) = 72(MOD 11) =49(MOD 11)

My questions are:

(1) obviously there is no equal relationship between 72(MOD 11) and 49 (MOD 11), and where does 49 come from?

(2) X and Y are picked up randomly, are 7 and 7 in 7X and 11 in MOD 11 also picked up randomly, or is it some algorithm? there is no any explanation in the textbook I have

(3) in the text marked in blue color, where does 2401 come from?

enter image description here

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(1) obviously there is no equal relationship between 72(MOD 11) and 49 (MOD 11), and where does 49 come from?

It's not $72$, it's $7^2 = 7 \cdot 7 = 49$.

(2) X and Y are picked up randomly, are 7 and 7 in 7X and 11 in MOD 11 also picked up randomly, or is it some algorithm? there is no any explanation in the textbook I have.

No, they are not random. This is simply the Diffie-Hellman algorithm. 7 is the base (or generator) and 11 is the modulus. These are pre-established configuration parameters.

Of course, to be secure, they need to be a whole lot larger - about 2048 bits or higher for the modulus.

(3) in the text marked in blue color, where does 2401 come from?

Similar, this is bad printing, it's just $7^4$.

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  • $\begingroup$ Sorry, I'm in answering mode - should have left this for a newcomer. $\endgroup$
    – Maarten Bodewes
    Aug 6 '21 at 12:08
  • $\begingroup$ it is very clear and thanks $\endgroup$
    – common2k
    Aug 6 '21 at 12:34
  • $\begingroup$ You're welcome. By the way, if you use WolframAlpha and enter 2401 you get this page with the nice remark: "$2401 = 7^4$ is a perfect $4^{th}$ power." $\endgroup$
    – Maarten Bodewes
    Aug 6 '21 at 15:10

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