1
$\begingroup$

Break the RSA code whose key is $(n, e) = (8369428283, 1234567)$.

Find the deciphering key and then decipher the message under the assumption that the plaintext consists of 7-letter blocks in the alphabet, converted to an integer between 0 and 26^(7) −1 in the usual way, and the ciphertext consists of 8-letter blocks in the same alphabet.

The given answer was CLAIMYOURPRIZE, but this doesn't match the one I got.

I converted the first 7-Letter block in CLAIMYOURPRIZE and I also got the first 8-letter block of the encrypted message. Then in order for them to match the value of the private key (d) must be 2 and that is not possible

$\endgroup$
0
$\begingroup$
  • $n = 8369428283 = 81799×102317$ (2 distinct prime factors by Wolfram Alpha)

The $d$ can be found in two ways;

  1. if Euler's totient function $\varphi(n)= (p-1)(q-1) = 8369244168$ is used then
    • $d = 5788687615 = e^{-1} \bmod 8369244168$
  2. If Carmichael Function $\lambda(n) = \text{LCM}(p-1,q-1) = 4184622084$ is used then
    • $d=1604065531 = e^{-1} \bmod 4184622084$

Carmichael Function $\lambda$ provides us the smallest $d$ (lcm versus phi in RSA) and this can be helpful for reducing the timing of decrypting and signing. Here, $d$ calculated with $\lambda(n)$ is two-bit less than $d$ calculated with $\varphi(n)$.

I've encrypted end decrypted both parts by $\varphi(n)$ and $\lambda$ successfully.

[ C =  2 ][ L = 11 ][ A =  0 ][ I =  8 ][ M = 12 ][ Y = 24 ][ O = 14 ]
plaintext       =     CLAIMYO
plaintext int   =   748676046

ciphertext int  =  1773907495
ciphertext char =    AFTHVVNJ
plaintext char  =     CLAIMYO

[ U = 20 ][ R = 17 ][ P = 15 ][ R = 17 ][ I =  8 ][ Z = 25 ][ E =  4 ]
plaintext       =     URPRIZE
plaintext int   =  6387458406

ciphertext int  =  5283907511
ciphertext char =    ARCSTZFZ
plaintext char  =     URPRIZE

note: the A in the ciphertext means 0 as an integer.

| improve this answer | |
$\endgroup$
  • 1
    $\begingroup$ Hey Kelalaka thank you for the help today, I finally was able to get the correct answer, there was a bug in my code at the final step. I now also get CLAIMYOURPRIZE as the answer. $\endgroup$ – justanothertechdude Dec 25 '19 at 2:59

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.