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Suppose we have public key: $$n= 1015, e= 3$$ and private key: $$d= 635, p= 35, q= 29, \phi(n)= 952$$ For $m = 100$, we have $$c = m^e ~mod~n = 100^3 mod~1015 = 225.$$ To decipher this, let us take $$c^d~mod~n$$ which is $$225^{635}~mod~1015$$ which equals $$680$$ But $680 \neq 100$ so this means that RSA incorrectly decrypted it right? Why does this happen?

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Your public key is not a legal RSA public key. In RSA, $n$ must be a product of two primes, but 35 is not a prime. Therefore, things don't work right: for instance, you got the wrong value of $\phi(n)$.

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    $\begingroup$ RSA works even when the "message $m$ is not relatively prime to the modulus $n$". $\:$ The real problem is that, since $p$ is not prime, he got a wrong value of $\phi(n)$. $\;\;\;\;$ $\endgroup$
    – user12859
    Commented Dec 9, 2015 at 6:36
  • $\begingroup$ @RickyDemer, oh, right, good point! Thanks for the correction. $\endgroup$
    – D.W.
    Commented Dec 9, 2015 at 7:39
  • $\begingroup$ @D.W. My implementation of isPrime(n) was slightly wrong...it went up to √n , when I adjusted it to go up to (√n) + 1 it works correctly. $\endgroup$
    – Ragnar
    Commented Dec 9, 2015 at 15:27

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