Science:Math Exam Resources/Courses/MATH312/December 2010/Question 07

MATH312 December 2010

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Question 07

Suppose the public key for an RSA cryptosystem is $\displaystyle (n,e)=(85,7)$ and the secret key is $\displaystyle (85,d)$. Show that this cryptosystem is not secure by finding d. Remarks: Recall that a message x is encoded by $\displaystyle x\to x^{e}\mod {n}$ and a received message y is decoded by $\displaystyle y\to y^{d}\mod {n}$. To make this cryptosystem secure, one needs to use a much larger n.

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

In order to decrypt this message, it suffices to find a d such that $ed\equiv 1\mod {\phi (n)}$. This is valid since if you have an encrypted message $x^{e}$ and you take this to the dth power with a d satisfying $ed=1+\phi (n)m$ for some integer m, one gets $(x^{e})^{d}\equiv x^{ed}=x^{1+\phi (n)m}\equiv x(x^{m})^{\phi (n)}\equiv x\mod {n}$ holding from Euler's theorem. So your job is to compute such a d satisfying $ed\equiv 1\mod {\phi (n)}$.

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. As stated in the hint, we seek to find a value of d such that $\displaystyle ed\equiv 1\mod {\phi (n)}$ Notice that $\displaystyle \phi (n)=\phi (85)=\phi (5\cdot 17)=\phi (5)\phi (17)=4(16)=64$. So we need to solve $\displaystyle 7d\equiv 1\mod {64}$. To do this, we use the Euclidean Algorithm. ${\begin{aligned}64&=7(9)+1\end{aligned}}$ Here we are immediately done and so multiplying both sides above by -9 yields $\displaystyle d\equiv -9\equiv 55\mod {64}$.

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