224 research outputs found
New Identities Relating Wild Goppa Codes
For a given support and a polynomial with no roots in , we prove equality
between the -ary Goppa codes where
denotes the norm of , that is In
particular, for , that is, for a quadratic extension, we get
. If has roots in
, then we do not necessarily have equality and we prove that
the difference of the dimensions of the two codes is bounded above by the
number of distinct roots of in . These identities provide
numerous code equivalences and improved designed parameters for some families
of classical Goppa codes.Comment: 14 page
Variations of the McEliece Cryptosystem
Two variations of the McEliece cryptosystem are presented. The first one is
based on a relaxation of the column permutation in the classical McEliece
scrambling process. This is done in such a way that the Hamming weight of the
error, added in the encryption process, can be controlled so that efficient
decryption remains possible. The second variation is based on the use of
spatially coupled moderate-density parity-check codes as secret codes. These
codes are known for their excellent error-correction performance and allow for
a relatively low key size in the cryptosystem. For both variants the security
with respect to known attacks is discussed
Coding Theory-Based Cryptopraphy: McEliece Cryptosystems in Sage
Unlike RSA encryption, McEliece cryptosystems are considered secure in the presence of quantum computers. McEliece cryptosystems leverage error-correcting codes as a mechanism for encryption. The open-source math software Sage provides a suitable environment for implementing and exploring McEliece cryptosystems for undergraduate research. Using our Sage implementation, we explored Goppa codes, McEliece cryptosystems, and Stern’s attack against a McEliece cryptosystem
On Rational Interpolation-Based List-Decoding and List-Decoding Binary Goppa Codes
We derive the Wu list-decoding algorithm for Generalised Reed-Solomon (GRS)
codes by using Gr\"obner bases over modules and the Euclidean algorithm (EA) as
the initial algorithm instead of the Berlekamp-Massey algorithm (BMA). We
present a novel method for constructing the interpolation polynomial fast. We
give a new application of the Wu list decoder by decoding irreducible binary
Goppa codes up to the binary Johnson radius. Finally, we point out a connection
between the governing equations of the Wu algorithm and the Guruswami-Sudan
algorithm (GSA), immediately leading to equality in the decoding range and a
duality in the choice of parameters needed for decoding, both in the case of
GRS codes and in the case of Goppa codes.Comment: To appear in IEEE Transactions of Information Theor
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