5 research outputs found
Cryptanalysis of public-key cryptosystems that use subcodes of algebraic geometry codes
We give a polynomial time attack on the McEliece public key cryptosystem
based on subcodes of algebraic geometry (AG) codes. The proposed attack reposes
on the distinguishability of such codes from random codes using the Schur
product. Wieschebrink treated the genus zero case a few years ago but his
approach cannot be extent straightforwardly to other genera. We address this
problem by introducing and using a new notion, which we call the t-closure of a
code
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 errors, where denotes
the designed distance and denotes the genus of the corresponding curve,
while with an ECA the decoding algorithm corrects up to
errors. Roughly speaking, for a public code of length over ,
these attacks run in operations in for the
reconstruction of an ECP and operations for the reconstruction of an
ECA. A probabilistic shortcut allows to reduce the complexities respectively to
and . 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.Comment: A 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 EC
On the unique representation of very strong algebraic geometry codes
This paper addresses the question of retrieving the triple (X,P,E) from the algebraic geometry code C=CL(X,P,E) , where X is an algebraic curve over the finite field Fq,P is an n-tuple of Fq -rational points on X and E is a divisor on X . If deg(E)=2g+1 where g is the genus of X , then there is an embedding of X onto Y in the projective space of the linear series of the divisor E. Moreover, if deg(E)=2g+2 , then I(Y) , the vanishing ideal of Y , is generated by I2(Y) , the homogeneous elements of degree two in I(Y) . If n>2deg(E) , then I2(Y)=I2(Q) , where Q is the image of P under the map from X to Y . These three results imply that, if 2g+2=