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

Suppose the public key for an RSA cryptosystem is and the secret key is . Show that this cryptosystem is not secure by finding d. Remarks: Recall that a message x is encoded by and a received message y is decoded by . 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 . This is valid since if you have an encrypted message and you take this to the dth power with a d satisfying for some integer m, one gets
holding from Euler's theorem. So your job is to compute such a d satisfying . 
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.

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
Notice that . So we need to solve . To do this, we use the Euclidean Algorithm.
Here we are immediately done and so multiplying both sides above by 9 yields . 