Cryptanalysis of McEliece Cryptosystem Based on Algebraic Geometry Codes and their subcodes · arXivDesk
1401.6025Jan 23, 2014A part of the material of this article has been published at the conferences ISIT 2014 with title "A polynomial time attack against AG code based PKC" and 4ICMCTA with title "Crypt. of PKC that use subcodes of AG codes". This long version includes detailed proofs and new results: the proceedings articles only considered the reconstruction of ECP while we discuss here the reconstruction of ECA
Cryptanalysis of McEliece Cryptosystem Based on Algebraic Geometry Codes and their subcodes
We give polynomial time attacks on the McEliece public key cryptosystem based either on algebraic geometry (AG) codes or on small codimensional subcodes of AG codes. These attacks consist in the blind reconstruction either of an Error Correcting Pair (ECP), or an Error Correcting Array (ECA) from the single data of an arbitrary generator matrix of a code. An ECP provides a decoding algorithm that corrects up to 2d∗−1−g errors, where d∗
Nearby in the stack
denotes the designed distance and
g
denotes the genus of the corresponding curve, while with an ECA the decoding algorithm corrects up to
2d∗−1
errors. Roughly speaking, for a public code of length
n
over
Fq
, these attacks run in
O(n4log(n))
operations in
Fq
for the reconstruction of an ECP and
O(n5)
operations for the reconstruction of an ECA. A probabilistic shortcut allows to reduce the complexities respectively to
O(n3+εlog(n))
and
O(n4+ε)
. Compared to the previous known attack due to Faure and Minder, our attack is efficient on codes from curves of arbitrary genus. Furthermore, we investigate how far these methods apply to subcodes of AG codes.