Science:Math Exam Resources/Courses/MATH312/December 2008/Question 05 (c)

MATH312 December 2008

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Question 05 (c)
Alice wants Bob to send her a secret integer P between 0 and 1250 using RSA encryption with key $\displaystyle (e,n)=(1189,1271)$ (so the exponent is 1189 and the modulus is 1271, it is way too small to be really secure, but basic RSA encryption and decryption methods still work of course). (c) Explain what Alice has to do next to obtain P back.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Recall up to this point that Bob computed $\displaystyle C\equiv P^{e}\mod {n}$ and you've computed a d such that $\displaystyle ed\equiv 1\mod {\phi (n)}$. How can you use this information together?

Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Recall up to this point that Bob computed $\displaystyle C\equiv P^{e}\mod {n}$ and you've computed a d such that $\displaystyle ed\equiv 1\mod {\phi (n)}$ so in fact, we have $\displaystyle ed=1+\phi (n)s$ for some integer s. Using Euler's Theorem, we can compute P back if we compute $\displaystyle C^{d}\equiv (P^{e})^{d}\equiv P^{ed}\equiv P^{1+\phi (n)s}\equiv P(P^{\phi (n)})^{s}\equiv P\mod {n}$. This is computable since Alice has both C and d available to her. This completes the proof.

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