Cryptographie basée sur les codes correcteurs d erreurs et arithmétique
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1 with Cryptographie basée sur les correcteurs d erreurs et arithmétique with with Laboratoire Hubert Curien, UMR CNRS 5516, Bâtiment F 18 rue du professeur Benoît Lauras Saint-Etienne France pierre.louis.cayrel@univ-st-etienne.fr 8 juin 2015 Pierre-Louis CAYREL Cryptographie basée sur les correcteurs d erreurs et arithmétique 1/23
2 with with with Pierre-Louis CAYREL Cryptographie basée sur les correcteurs d erreurs et arithmétique 2/23
3 Syndrome decoding problem with with with 1 Input. H : matrix of size r n S : vector of F r 2 t : integer 2 Problem. Does there exist a vector e of F n 2 of weight t such that : Problem NP-complete E.R. BERLEKAMP, R.J. MCELIECE and H.C. VAN TILBORG 1978 Pierre-Louis CAYREL Cryptographie basée sur les correcteurs d erreurs et arithmétique 3/23
4 with with with Pierre-Louis CAYREL Cryptographie basée sur les correcteurs d erreurs et arithmétique 4/23
5 with with with Linear : representation most used in error correction error correcting for which redundancy depends linearly on the information can be defined by a generator matrix : c is a word of the code C if and only if : Figure : G : generator matrix in systematic form The generator matrix G : is a r n matrix; rows of G form a basis for the code C. Pierre-Louis CAYREL Cryptographie basée sur les correcteurs d erreurs et arithmétique 5/23
6 with with with The parity check matrix H is orthogonal to G : it s a r n matrix; it s the generator matrix of the dual; the code C is the kernel of H. c C if and only if Hc = 0. s = H c = H c + H e is the syndrome of the error. Pierre-Louis CAYREL Cryptographie basée sur les correcteurs d erreurs et arithmétique 6/23
7 with with with Code based systems + advantages : faster than RSA ; not based on number theory problem (PQ secure) ; does not need processors ; based on hard problem (syndrome decoding problem...) disadvantages : size of public keys (few hundred bits...) Pierre-Louis CAYREL Cryptographie basée sur les correcteurs d erreurs et arithmétique 7/23
8 with with with Pierre-Louis CAYREL Cryptographie basée sur les correcteurs d erreurs et arithmétique 8/23
9 Niederreiter encryption-scheme with with with Private key : C a [n, r, d] code which corrects t errors, H a parity check matrix of C, a r r invertible matrix Q, a n n permutation matrix P. Public key : H = QH P. : φ n,t : m e, with e of weight t. e y = He Decryption : decode (use the secret key to find e) Q 1 y = (Q 1 Q)H Pe in Pe, then P 1 Pe gives e. Pierre-Louis CAYREL Cryptographie basée sur les correcteurs d erreurs et arithmétique 9/23
10 with with with PKC signature. RSA yes McEliece and Niederreiter no directly Pierre-Louis CAYREL Cryptographie basée sur les correcteurs d erreurs et arithmétique 10/23
11 with with with Problem: McEliece and Niederreiter not invertible. if we take y F n 2 random and a code C[n, k, d] for which we are able to decode d/2 errors, it is almost impossible to decode y in a word of C. Solution: the hash value has to be decodable! Pierre-Louis CAYREL Cryptographie basée sur les correcteurs d erreurs et arithmétique 11/23
12 with with with Pierre-Louis CAYREL Cryptographie basée sur les correcteurs d erreurs et arithmétique 12/23
13 with with with Pierre-Louis CAYREL Cryptographie basée sur les correcteurs d erreurs et arithmétique 13/23
14 with with with we need a dense family of : Goppa binary Goppa t small the probability for a random element to be decodable (in a ball of radius t centered on the codewords) is 1 t! we take n = 2 m, m = 16, t = 9. we have 1 chance over 9! = to have a decodable word. Pierre-Louis CAYREL Cryptographie basée sur les correcteurs d erreurs et arithmétique 14/23
15 with with with cons : decode several words (t!) before to find a good one 70 times slower than RSA t small leads to very big parameters public key of 1 MB new PK size : several MB, time to sign : several weeks... change of representation : structured (smaller parameters) and a GPU to have a signature in less than 2 minutes... Pierre-Louis CAYREL Cryptographie basée sur les correcteurs d erreurs et arithmétique 15/23
16 Springer-Verlag, with with with zero-knowledge, the security is based on the syndrome decoding problem. Pierre-Louis CAYREL Cryptographie basée sur les correcteurs d erreurs et arithmétique 16/23
17 with with with generate a random matrix H of size r n we choose an integer t which is the weight this is the public key (H, t) each user receive e of n bits and weight t. this is the private key each user compute : S = He. just once for H fixed S is public Pierre-Louis CAYREL Cryptographie basée sur les correcteurs d erreurs et arithmétique 17/23
18 A wants to prove to B that she knows the secret but she doesn t want to divulgate it. with with with The protocol is on λ rounds and each of them is defined as follows. Pierre-Louis CAYREL Cryptographie basée sur les correcteurs d erreurs et arithmétique 18/23
19 with with with Pierre-Louis CAYREL Cryptographie basée sur les correcteurs d erreurs et arithmétique 19/23
20 with with with Pierre-Louis CAYREL Cryptographie basée sur les correcteurs d erreurs et arithmétique 20/23
21 with with with Pierre-Louis CAYREL Cryptographie basée sur les correcteurs d erreurs et arithmétique 21/23
22 with with with Change of representation : Replace the random matrix H by :. Let l be an integer. a random double circulant matrix l 2l H is defined as : H = (I A), where A is a cyclic matrix, of the form : a 1 a 2 a 3 a l a l a 1 a 2 a l 1 A =..... a 2 a 3 a 4 a 1, where (a 1, a 2, a 3,, a l ) is a random vector of F l 2. Store H needs only l bits. Pierre-Louis CAYREL Cryptographie basée sur les correcteurs d erreurs et arithmétique 22/23
23 with with with Pierre-Louis CAYREL Cryptographie basée sur les correcteurs d erreurs et arithmétique 23/23
24 with Information Set Decoding with with Pierre-Louis CAYREL Cryptographie basée sur les correcteurs d erreurs et arithmétique 24/23
25 with Information Set Decoding with with Pierre-Louis CAYREL Cryptographie basée sur les correcteurs d erreurs et arithmétique 25/23
26 with with with φ : m x with x of weight t This application is called a constant weight encoder. Enumerative coding: [ ( )[ φ 1 n : W n,t 0, t ( ) i 0 (i 0, i 1,..., i t 1 ) + 1 ( ) ( i i t 1 t ) Pierre-Louis CAYREL Cryptographie basée sur les correcteurs d erreurs et arithmétique 26/23
27 How to choose the weight for an optimal complexity? with with with Pierre-Louis CAYREL Cryptographie basée sur les correcteurs d erreurs et arithmétique 27/23
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