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### Open Access Journals

AMC

In this paper we study the hardness of some discrete logarithm
like problems defined in linear recurring sequences over finite
fields from a point of view as general as possible. The
intractability of these problems plays a key role in the security
of the class of public key cryptographic constructions based on
linear recurring sequences. We define new discrete logarithm,
Diffie-Hellman and decisional Diffie-Hellman problems for any
nontrivial linear recurring sequence in any finite field whose
minimal polynomial is irreducible. Then, we prove that these
problems are polynomially equivalent to the discrete logarithm,
Diffie-Hellman and decisional Diffie-Hellman problems in the
subgroup generated by any root of the minimal polynomial of the
sequence.

AMC

In previous papers [4,5,6] we gave the
first example of a non-abelian group code using the group ring
$F_5S_4$. It is natural to ask if it is really relevant that the
group ring is semisimple. What happens if the field has
characteristic $2$ or $3$? We have addressed this question, with
computer help, proving that there are also examples of non-abelian
group codes in the non-semisimple case. The results show some
interesting differences between the cases of characteristic $2$
and $3$. Furthermore, using the group $SL(2,F_3)$, we construct a
non-abelian group code over $F_2$ of length $24$, dimension $6$
and minimal weight $10$. This code is optimal in the following
sense: every linear code over $F_2$ with length $24$ and dimension
$6$ has minimum distance less than or equal to $10$. In the case
of abelian group codes over $F_2$ the above value for the
minimum distance cannot be achieved, since the minimum distance
of a binary abelian group code with the given length and dimension
6 is at most 8.

keywords:
semisimplicity
,
permutation equivalence.
,
Abelian group code
,
weight distribution
,
Group code

## Year of publication

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