###### tags: `MATA02-2022`
# Ungraded problems 12

You are to work on these questions only for the benefit of learning the material. These problems are meant for practice, not an assignment to be handed in.

## 1.
<!-- p=11, q=7, phi=60, 52=17^7 mod 77, "SEVENISAVERYLUCKYNUMBER" -->
You have intercepted a message encrypted using a hybrid cryptosystem that uses a combination of RSA and a Caesar cipher.

The RSA modulus is $n=77$, and the exponent is $k=7$. The RSA-encrypted ciphertext, which corresponds to the encrypted Caesar cipher key, is 52. The Caesar-cipher encrypted message is "JVMVEZJRMVIPCLTBPELDSVI"

1. Break the RSA encryption to decrypt "52" to get the Caesar shift key.
2. What was the decrypted Caesar-cipher encrypted message?

## 2.
<!-- p=13, q=11, phi=120, 10^17 mod 143 = 43, "ENCRYPTIONISFOREVERYONE" -->
You have intercepted a message encrypted using a hybrid cryptosystem that uses a combination of RSA and a Caesar cipher.

The RSA modulus is $n=143$, and the exponent is $k=17$. The RSA-encrypted ciphertext, which corresponds to the encrypted Caesar cipher key, is 43. The Caesar-cipher encrypted message is "OXMBIZDSYXSCPYBOFOBIYXO"

1. Break the RSA encryption to decrypt "43" to get the Caesar shift key.
2. What was the decrypted Caesar-cipher encrypted message?

## 3.
<!-- p=7, q=17, phi=96, 9^11 mod 119 = 32, "READYFORTHEFINAL" -->
You have intercepted a message encrypted using a hybrid cryptosystem that uses a combination of RSA and a Caesar cipher.

The RSA modulus is $n=119$, and the exponent is $k=11$. The RSA-encrypted ciphertext, which corresponds to the encrypted Caesar cipher key, is 32. The Caesar-cipher encrypted message is "ANJMHOXACQNORWJU"

1. Break the RSA encryption to decrypt "32" to get the Caesar shift key.
2. What was the decrypted Caesar-cipher encrypted message?

# Selected (partial) solutions
<details>
<summary>Solutions (click to expand)</summary>

1. $\sqrt[7]{52} \text{ mod } 77 \equiv 17$, so that's the Caesar shift. Reversing the Caesar shift gives you "SEVENISAVERYLUCKYNUMBER"
2. $\sqrt[17]{43} \text{ mod } 143 \equiv 10$, so that's the Caesar shift. Reversing the Caesar shift gives you "ENCRYPTIONISFOREVERYONE"
3. $\sqrt[11]{32} \text{ mod } 119 \equiv 9$, so that's the Caesar shift. Reversing the Caesar shift gives you "READYFORTHEFINAL"

</details>