22 research outputs found
Regular subgroups with large intersection
In this paper we study the relationships between the elementary abelian
regular subgroups and the Sylow -subgroups of their normalisers in the
symmetric group , in view of the interest that
they have recently raised for their applications in symmetric cryptography
On the primitivity of Lai-Massey schemes
In symmetric cryptography, the round functions used as building blocks for
iterated block ciphers are often obtained as the composition of different
layers providing confusion and diffusion. The study of the conditions on such
layers which make the group generated by the round functions of a block cipher
a primitive group has been addressed in the past years, both in the case of
Substitution Permutation Networks and Feistel Networks, giving to block cipher
designers the receipt to avoid the imprimitivity attack. In this paper a
similar study is proposed on the subject of the Lai-Massey scheme, a framework
which combines both Substitution Permutation Network and Feistel Network
features. Its resistance to the imprimitivity attack is obtained as a
consequence of a more general result in which the problem of proving the
primitivity of the Lai-Massey scheme is reduced to the simpler one of proving
the primitivity of the group generated by the round functions of a strictly
related Substitution Permutation Network
Wave-Shaped Round Functions and Primitive Groups
Round functions used as building blocks for iterated block ciphers, both in
the case of Substitution-Permutation Networks and Feistel Networks, are often
obtained as the composition of different layers which provide confusion and
diffusion, and key additions. The bijectivity of any encryption function,
crucial in order to make the decryption possible, is guaranteed by the use of
invertible layers or by the Feistel structure. In this work a new family of
ciphers, called wave ciphers, is introduced. In wave ciphers, round functions
feature wave functions, which are vectorial Boolean functions obtained as the
composition of non-invertible layers, where the confusion layer enlarges the
message which returns to its original size after the diffusion layer is
applied. This is motivated by the fact that relaxing the requirement that all
the layers are invertible allows to consider more functions which are optimal
with regard to non-linearity. In particular it allows to consider injective APN
S-boxes. In order to guarantee efficient decryption we propose to use wave
functions in Feistel Networks. With regard to security, the immunity from some
group-theoretical attacks is investigated. In particular, it is shown how to
avoid that the group generated by the round functions acts imprimitively, which
represent a serious flaw for the cipher
A modular idealizer chain and unrefinability of partitions with repeated parts
Recently Aragona et al. have introduced a chain of normalizers in a Sylow
2-subgroup of Sym(2^n), starting from an elementary abelian regular subgroup.
They have shown that the indices of consecutive groups in the chain depend on
the number of partitions into distinct parts and have given a description, by
means of rigid commutators, of the first n-2 terms in the chain. Moreover, they
proved that the (n-1)-th term of the chain is described by means of rigid
commutators corresponding to unrefinable partitions into distinct parts.
Although the mentioned chain can be defined in a Sylow p-subgroup of Sym(p^n),
for p > 2 computing the chain of normalizers becomes a challenging task, in the
absence of a suitable notion of rigid commutators. This problem is addressed
here from an alternative point of view. We propose a more general framework for
the normalizer chain, defining a chain of idealizers in a Lie ring over Z_m
whose elements are represented by integer partitions. We show how the
corresponding idealizers are generated by subsets of partitions into at most
m-1 parts and we conjecture that the idealizer chain grows as the normalizer
chain in the symmetric group. As an evidence of this, we establish a
correspondence between the two constructions in the case m=2
Some group-theoretical results on Feistel Networks in a long-key scenario
The study of the trapdoors that can be hidden in a block cipher is and has
always been a high-interest topic in symmetric cryptography. In this paper we
focus on Feistel-network-like ciphers in a classical long-key scenario and we
investigate some conditions which make such a construction immune to the
partition-based attack introduced recently by Bannier et al.Comment: Accepted for publication in Advances in Mathematics of Communication
Some group-theoretical results on Feistel Networks in a long-key scenario
Under embargo until: 2021-07-01The study of the trapdoors that can be hidden in a block cipher is and has always been a high-interest topic in symmetric cryptography. In this paper we focus on Feistel-network-like ciphers in a classical long-key scenario and we investigate some conditions which make such a construction immune to the partition-based attack introduced recently by Bannier et al.acceptedVersio
On properties of translation groups in the affine general linear group with applications to cryptography
The affine general linear group acting on a vector space over a prime field is a well-understood mathematical object. Its elementary abelian regular subgroups have recently drawn attention in applied mathematics thanks to their use in cryptography as a way to hide or detect weaknesses inside block ciphers. This paper is focused on building a convenient representation of their elements which suits better the purposes of the cryptanalyst. Several combinatorial counting formulas and a classification of their conjugacy classes are given as well.publishedVersio
On the maximal part in unrefinable partitions of triangular numbers
A partition into distinct parts is refinable if one of its parts can be
replaced by two different integers which do not belong to the partition and
whose sum is , and it is unrefinable otherwise. Clearly, the condition of
being unrefinable imposes on the partition a non-trivial limitation on the size
of the largest part and on the possible distributions of the parts. We prove a
-upper bound for the largest part in an unrefinable partition of
, and we call maximal those which reach the bound. We show a complete
classification of maximal unrefinable partitions for triangular numbers,
proving that if is even there exists only one maximal unrefinable partition
of , and that if is odd the number of such partitions equals the
number of partitions of into distinct parts. In the second
case, an explicit bijection is provided
Rigid commutators and a normalizer chain
The novel notion of rigid commutators is introduced to determine the sequence
of the logarithms of the indices of a certain normalizer chain in the Sylow
2-subgroup of the symmetric group on 2^n letters. The terms of this sequence
are proved to be those of the partial sums of the partitions of an integer into
at least two distinct parts, that relates to a famous Euler's partition
theorem