Cracking private key of asymmetric cryptosystem
A method of cracking a private key of an asymmetric cryptosystem includes extracting a modulus and a public key exponent from a public key, calculating the digital root of the modulus, deriving a set of candidate base pairs corresponding to the digital root and the last digit of the modulus, each of the candidate base pairs including a first candidate base and a second candidate base, iteratively testing values of a multiplier until a sum of one of the candidate bases with ninety times the multiplier is a factor of the modulus, and determining the private key using the public key exponent and the factor of the modulus. The method may further include decrypting an encrypted message using the private key.
1. A method of cracking a private key of an asymmetric cryptosystem, the method comprising:
extracting a modulus and a public key exponent from a public key;
performing modulo division on the modulus;
deriving a set of candidate base pairs corresponding to a result of the modulo division, each of the candidate base pairs including a first candidate base and a second candidate base;
iteratively testing values of a multiplier until a sum of one of the candidate bases with a multiple of the multiplier is a factor of the modulus; and
determining the private key using the public key exponent and the factor of the modulus.
2. The method of claim 1 , wherein said extracting includes parsing an encoded data structure containing the public key.
3. The method of claim 2 , wherein said parsing is performed using an ASN.1 parser.
4. The method of claim 1 , wherein said deriving includes referencing one or more lookup tables associating possible results of the modulo division with possible candidate base pairs.
5. The method of claim 1 , wherein said iteratively testing includes incrementing an integer t and testing, for one or more of the candidate base pairs, the sum of the second candidate base with the multiple of the multiplier p, the multiplier p being an expression that relates the modulus n, the first candidate base B 1 , and the second candidate base B 2 .
6. The method of claim 5 , wherein the expression relates the modulus n, the first candidate base B 1 , the second candidate base B 2 , and a starting value offset t 0 .
7. The method of claim 6 , wherein the starting value offset to is a function of the modulus.
8. The method of claim 1 , wherein said iteratively testing includes checking whether the multiplier is an integer.
9. The method of claim 1 , wherein said iteratively testing includes checking whether an expression derived from the multiplier is a perfect square.
10. The method of claim 1 , wherein said iteratively testing includes, for each of the candidate base pairs, comparing the modulus n to the product of the candidate base pair B 1 *B 2 .
11. The method of claim 1 , wherein said determining includes determining a private key exponent of the private key such that the product of the private key exponent and the public key exponent is congruent to one modulo p, where p is equal to (F1−1)*(F2−1), F2 is the factor of the modulus, and F1 is a value equal to the modulus divided by F2.
12. The method of claim 1 , further comprising decrypting an encrypted message using the private key.
13. The method of claim 12 , wherein said decrypting includes decrypting an encrypted symmetric key using the private key and decrypting the message using the symmetric key.
14. The method of claim 12 , wherein said decrypting includes parsing an encoded data structure including the encrypted message.
15. The method of claim 14 , wherein said parsing is performed using an ASN.1 parser.
16. The method of claim 12 , wherein said decrypting includes determining a plaintext m such that a ciphertext c of the encrypted message raised to the power of the private key exponent is congruent to m modulo n where n is the modulus.
17. A non-transitory program storage medium on which are stored instructions executable by a processor or programmable circuit to perform operations for cracking a private key of an asymmetric cryptosystem, the operations comprising:
extracting a modulus and a public key exponent from a public key;
performing modulo division on the modulus;
deriving a set of candidate base pairs corresponding to a result of the modulo division, each of the candidate base pairs including a first candidate base and a second candidate base;
iteratively testing values of a multiplier until a sum of one of the candidate bases with a multiple of the multiplier is a factor of the modulus; and
determining the private key using the public key exponent and the factor of the modulus.
18. The non-transitory program storage medium of claim 17 , wherein the operations further comprise decrypting an encrypted message using the private key.
19. A system for cracking a private key of an asymmetric cryptosystem, the system comprising:
an n/e extractor for extracting a modulus and a public key exponent from a public key;
a calculator for performing modulo division on the modulus;
a candidate base pair deriving engine for deriving a set of candidate base pairs corresponding to a result of the modulo division, each of the candidate base pairs including a first candidate base and a second candidate base;
an iterator for iteratively testing values of a multiplier until a sum of one of the candidate bases with a multiple of the multiplier is a factor of the modulus; and
a private key determiner for determining the private key using the public key exponent and the factor of the modulus.
20. The system of claim 19 , further comprising a decryptor for decrypting an encrypted message using the private key.