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