Recover the Secret

Decrypt Ciphertext

The recovered private exponent decrypts the toy ciphertext.

highlighted = computed this step

Use recovered d to decrypt

The ciphertext c=13 is raised to recovered d=27 modulo n.

m=cd mod nm=c^d\bmod n
Decrypt with recovered dThe decryption ladder is recomputed from c, d, and n.Decrypt with recovered d - 13^27 mod 55stepbitpriorsquaremultiplyresult0111131311134525220529skip93192688418977

The plaintext returns

The decryption ladder result is plaintext m=7.

13d mod 55=713^d\bmod{}55=7
Decrypt with recovered dThe decryption ladder is recomputed from c, d, and n.Decrypt with recovered d - 13^27 mod 55stepbitpriorsquaremultiplyresult0111131311134525220529skip93192688418977

It matches the earlier encryption

The public operation maps plaintext 7 back to ciphertext 13.

7e mod 55=137^e\bmod{}55=13
Public encryption link-backThe public encryption ladder links the recovered plaintext back to the ciphertext.Public encryption link-back - 7^3 mod 55stepbitpriorsquaremultiplyresult011177117491313

Honesty boundary

NOTE: toy tiny RSA only; real RSA uses enormous moduli and padding; not a production attack guide; not side-channel, oracle, implementation, quantum, or real key-size guidance.

toy only=1\text{toy only}=1
Decrypt with recovered dThe decryption ladder is recomputed from c, d, and n.Decrypt with recovered d - 13^27 mod 55stepbitpriorsquaremultiplyresult0111131311134525220529skip93192688418977