MATH312 December 2010
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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?
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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 .
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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 .
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