A CCA2 Secure PKE Based on McEliece Assumptions in the Standard Model

Transcription

1 A CCA2 Secure PKE Based on McEliece Assumptions in the Standard Model Jörn Müller-Quade European Institute for System Security KIT, Karlsruhe, Germany 04/23/09 Session ID: CRYP301 Session Classification: Alternative Encryption

2 Agenda The McEliece Crypto System An Attack and IND-CCA2 A First Step: Achieving IND-CCA1 Achieving IND-CCA2 2

3 The McEliece Crypto System

4 The McEliece Public Key Scheme Secret Key P G S Public Key P. G. S = G Encryption m G + e = c Decryption m = Goppa S -1 decoding P -1 algorithm. c McEliece Assumption: 1. A public key G is indistinguishable from random 2. Syndrome decoding is hard

5 The McEliece Public Key Scheme Secret Key P G S Public Key Chosen randomly P. G. S = G Encryption m G + e = c Decryption m = Goppa S -1 decoding P -1 algorithm. c McEliece Assumption: 1. A public key G is indistinguishable from random 2. Syndrome decoding is hard

6 Semantic Security (IND-CPA) Public key G A challenge 1, challenge 2 Enc G (challenge i ) Chosen at random A Public key G Decision

7 Semantic Security (IND-CPA) Public key G A challenge 1, challenge 2 A cannot distinguish better than guessing Enc G (challenge i ) Chosen at random A Public key G Decision

8 Semantic Security (IND-CPA) Padding the message with a random string yields a semantically secure public key scheme (IND-CPA). [Nojima, Imai, Kobara, Morozov, DCC 2008] r m G + e Random padding

9 An Attack and IND-CCA2

10 A Reaction Attack The password identifies the customer and the nonce avoids replay attacks. Pay by sending your password and a nonce We use text book McEliece to securely send the password and the nonce. Password not valid Nonce not new

11 A Reaction Attack Replay an old Enc VM (pwd nonce) Add a new error Pay by sending your password and a nonce Password not valid Nonce not new

12 A Reaction Attack Replay an old Enc VM (pwd nonce) Add a new error Pay by sending your password and a nonce Depending on the message the new error corrected an error from the encryption. Password not valid Nonce not new

13 A Reaction Attack Replay an old Enc VM (pwd nonce) Add a new error Pay by sending your password and a nonce Locate all the errors from the encryption. Decode the codeword. Extract pwd. Depending on the message the new error corrected an error from the encryption. Password not valid Nonce not new

14 IND-CCA2 Decryption Oracle Public key e A challenge 1, challenge 2 Enc e (challenge i ) Chosen at random Decryption Oracle! A Public key e Decision

15 IND-CCA2 Decryption Oracle Public key e A challenge 1, challenge 2 A cannot distinguish better than guessing Enc e (challenge i ) Chosen at random Decryption Oracle! A Public key e Decision

16 IND-CCA2 A decryption oracle yields more information than the reaction of the vending machine. IND- CCA2 security implies security against reaction attacks. IND-CCA2 is the strongest established security notion for PKE and it implies universal composability.

17 A First Step: achieving IND-CCA1

18 A First Step: IND-CCA1 Decryption Oracle Public key e A challenge 1, challenge 2 Enc e (challenge i ) Chosen at random Decryption Oracle! A Public key e Decision

19 A First Step: IND-CCA1 Decryption Oracle Public key e A challenge 1, challenge 2 A cannot distinguish better than guessing Enc e (challenge i ) Chosen at random Decryption Oracle! A Public key e Decision

20 k-repetition For a PKE (KeyGen, Enc, Dec) we define the k-repetition (KeyGen k, Enc k, Dec k ) as follows: KeyGen k calls KeyGen k times. Enc k encrypts with Enc k times using the corresponding public keys and correlated randomness. Dec k decrypts all cipher texts with the corresponding secret keys and outputs m if all individual cipher texts did decrypt to m. Soundness: Dec k (Enc k (m)) = m with overwhelming probability. This construction is derived from [Rosen, Segev, TCC 2009]

21 k-repetition McEliece KeyGen k calls KeyGen k times Enc k encrypts with Enc k times using independent random error vectors, but the same padding r m in all encryptions. Dec k decrypts all cipher texts with the corresponding secret keys and outputs m if all individual cipher texts did decrypt to m. Soundness: Dec k (Enc k (m)) = m with overwhelming probability. This randomized k-repetition McEliece remains IND-CPA secure.

22 Verifiability of k-repetition A k-repetition PKE is called verifiable if given a ciphertext c, the public key pk = (pk 1,...,pk k ) and any secret key sk i it is possible to verify if c is a valid ciphertext (i.e., all individual ciphertexts decrypt to the same message). c = (c 1,...,c i,...,c k ) c = (c 1,...,c i,...,c k ) Dec??? Dec k m m

23 Verifiability of k-repetition k-repetition McEliece is verifiable: c = (c 1,...,c i,...,c k ) Dec m Encode with public key matrix and check the hamming distance c = (c 1,...,c i,...,c k ) m Dec k

24 From k-repetition to IND-CCA1 KeyGen CCA1 gives (pk 10,pk 11,...,pk k0,pk k1 ) and the secret key (sk 1 1r,sk 2r,...,sk 2 r k ) k Enc CCA1 chooses a random string s = s 1,...,s k, encrypts (k-repetition) with pk s 1 1,pk s 2 1,...,pk sk k obtains c = c 1,...,c k and outputs (s,c). Dec CCA1 decrypts whenever a s i = r i with sk r i obtains m and outputs m if the ciphertext was valid (verifiability). This construction follows closely [Rosen, Segev, TCC 2009]

25 From k-repetition to IND-CCA1 KeyGen CCA1 gives (pk 10,pk 11,...,pk k0,pk k1 ) and the secret key (sk 1 1r,sk 2r,...,sk 2 r k k ) Enc CCA1 chooses a random string s = s 1,...,s k, encrypts (k-repetition) with pk s 1 1,pk s 2 1,...,pk sk k obtains c = c 1,...,c k and outputs (s,c). Dec CCA1 decrypts whenever a s i = r i with sk r i obtains m and outputs m if the ciphertext was valid (verifiability). If s is the complement of the string r it is impossible to decode.

26 From k-repetition to IND-CCA1 If s is the complement of the string r it is impossible to decode. Given a successful IND-CCA1 adversary A for PKE CCA1 we construct an adversary A breaking the IND-CPA security of k-repetition PKE k : A obtains (pk 1,...,pk k ) A chooses (pk 10,pk 11,...,pk k0,pk k1 ) and (sk 1r,sk 2r,...,sk k r ) then every cipher (s,c) can be deciphered by A, except for s = complement(r). A can mimic the oracle for A. A gives (pk 10,pk 11,...,pk k0,pk k1 ) to A. 1 2 k

27 From k-repetition to IND-CCA1 If s is the complement of the string r it is impossible to decode. Given a successful IND-CCA1 adversary A for PKE CCA1 we construct an adversary A breaking the IND-CPA security of k-repetition PKE k : given (pk 10,pk 11,...,pk k0,pk k1 ) A outputs m 1, m 2 A outputs m 1, m 2 and obtains a challenge c. A gives (complement(r),c) as a challenge to A. A outputs a decision d. A outputs this decision d.

28 Achieving IND-CCA2

29 From k-repetition to IND-CCA2 This scheme is not yet IND-CCA2 secure: After obtaining the challenge (s,c) just change a bit in s and give this to the decryption oracle. KeyGen CCA2 gives (pk 10,pk 11,...,pk k0,pk k1 ) chooses a signing key and r = r 1,...,r k as the complement of a the corresponding signature verification key. The secret key is chosen as (sk 1r,sk 2r,...,sk k r ).

30 From k-repetition to IND-CCA2 This scheme is not yet IND-CCA2 secure: After obtaining the challenge (s,c) just change a bit in s and give this to the decryption oracle. Enc CCA2 chooses a signing key sk and a verification key vk = vk 1,...,vk k, encrypts with pk vk 1 1,pk vk 2 1,...,pk vk k k obtains c = c 1,...,c k then signs (sk,c) and outputs c, vk and the signature.

31 From k-repetition to IND-CCA2 This scheme is not yet IND-CCA2 secure: After obtaining the challenge (s,c) just change a bit in s and give this to the decryption oracle. Dec CCA1 decrypts whenever a s i = r i with sk r obtains m and outputs m if the ciphertext was valid (verifiability) this includes checking of the signatures. Successfully changing the ciphertext now amounts to forging a signature.

32 Conclusion Based on the work of Rosen and Segev at TCC 2009 we presented an IND-CCA2 secure PKE based on the McEliece assumptions in the standard model. The construction is O(k 2 ) and finding a more efficient construction is an open problem.