3,175 research outputs found
A new class of codes for Boolean masking of cryptographic computations
We introduce a new class of rate one-half binary codes: {\bf complementary
information set codes.} A binary linear code of length and dimension
is called a complementary information set code (CIS code for short) if it has
two disjoint information sets. This class of codes contains self-dual codes as
a subclass. It is connected to graph correlation immune Boolean functions of
use in the security of hardware implementations of cryptographic primitives.
Such codes permit to improve the cost of masking cryptographic algorithms
against side channel attacks. In this paper we investigate this new class of
codes: we give optimal or best known CIS codes of length We derive
general constructions based on cyclic codes and on double circulant codes. We
derive a Varshamov-Gilbert bound for long CIS codes, and show that they can all
be classified in small lengths by the building up construction. Some
nonlinear permutations are constructed by using -codes, based on the
notion of dual distance of an unrestricted code.Comment: 19 pages. IEEE Trans. on Information Theory, to appea
Self-Dual Codes
Self-dual codes are important because many of the best codes known are of
this type and they have a rich mathematical theory. Topics covered in this
survey include codes over F_2, F_3, F_4, F_q, Z_4, Z_m, shadow codes, weight
enumerators, Gleason-Pierce theorem, invariant theory, Gleason theorems,
bounds, mass formulae, enumeration, extremal codes, open problems. There is a
comprehensive bibliography.Comment: 136 page
On the Residue Codes of Extremal Type II Z4-Codes of Lengths 32 and 40
In this paper, we determine the dimensions of the residue codes of extremal
Type II Z4-codes for lengths 32 and 40. We demonstrate that every binary doubly
even self-dual code of length 32 can be realized as the residue code of some
extremal Type II Z4-code. It is also shown that there is a unique extremal Type
II Z4-code of length 32 whose residue code has the smallest dimension 6 up to
equivalence. As a consequence, many new extremal Type II Z4-codes of lengths 32
and 40 are constructed.Comment: 19 page
Higher-order CIS codes
We introduce {\bf complementary information set codes} of higher-order. A
binary linear code of length and dimension is called a complementary
information set code of order (-CIS code for short) if it has
pairwise disjoint information sets. The duals of such codes permit to reduce
the cost of masking cryptographic algorithms against side-channel attacks. As
in the case of codes for error correction, given the length and the dimension
of a -CIS code, we look for the highest possible minimum distance. In this
paper, this new class of codes is investigated. The existence of good long CIS
codes of order is derived by a counting argument. General constructions
based on cyclic and quasi-cyclic codes and on the building up construction are
given. A formula similar to a mass formula is given. A classification of 3-CIS
codes of length is given. Nonlinear codes better than linear codes are
derived by taking binary images of -codes. A general algorithm based on
Edmonds' basis packing algorithm from matroid theory is developed with the
following property: given a binary linear code of rate it either provides
disjoint information sets or proves that the code is not -CIS. Using
this algorithm, all optimal or best known codes where and are shown to be -CIS for all
such and , except for with and with .Comment: 13 pages; 1 figur
The codes and the lattices of Hadamard matrices
It has been observed by Assmus and Key as a result of the complete
classification of Hadamard matrices of order 24, that the extremality of the
binary code of a Hadamard matrix H of order 24 is equivalent to the extremality
of the ternary code of H^T. In this note, we present two proofs of this fact,
neither of which depends on the classification. One is a consequence of a more
general result on the minimum weight of the dual of the code of a Hadamard
matrix. The other relates the lattices obtained from the binary code and from
the ternary code. Both proofs are presented in greater generality to include
higher orders. In particular, the latter method is also used to show the
equivalence of (i) the extremality of the ternary code, (ii) the extremality of
the Z_4-code, and (iii) the extremality of a lattice obtained from a Hadamard
matrix of order 48.Comment: 16 pages. minor revisio
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