1,070 research outputs found
Representations of Circular Words
In this article we give two different ways of representations of circular
words. Representations with tuples are intended as a compact notation, while
representations with trees give a way to easily process all conjugates of a
word. The latter form can also be used as a graphical representation of
periodic properties of finite (in some cases, infinite) words. We also define
iterative representations which can be seen as an encoding utilizing the
flexible properties of circular words. Every word over the two letter alphabet
can be constructed starting from ab by applying the fractional power and the
cyclic shift operators one after the other, iteratively.Comment: In Proceedings AFL 2014, arXiv:1405.527
Minimal weight expansions in Pisot bases
For applications to cryptography, it is important to represent numbers with a
small number of non-zero digits (Hamming weight) or with small absolute sum of
digits. The problem of finding representations with minimal weight has been
solved for integer bases, e.g. by the non-adjacent form in base~2. In this
paper, we consider numeration systems with respect to real bases which
are Pisot numbers and prove that the expansions with minimal absolute sum of
digits are recognizable by finite automata. When is the Golden Ratio,
the Tribonacci number or the smallest Pisot number, we determine expansions
with minimal number of digits and give explicitely the finite automata
recognizing all these expansions. The average weight is lower than for the
non-adjacent form
Redundancy of minimal weight expansions in Pisot bases
Motivated by multiplication algorithms based on redundant number
representations, we study representations of an integer as a sum , where the digits are taken from a finite alphabet
and is a linear recurrent sequence of Pisot type with
. The most prominent example of a base sequence is the
sequence of Fibonacci numbers. We prove that the representations of minimal
weight are recognised by a finite automaton and obtain an
asymptotic formula for the average number of representations of minimal weight.
Furthermore, we relate the maximal order of magnitude of the number of
representations of a given integer to the joint spectral radius of a certain
set of matrices
Revisiting LFSMs
Linear Finite State Machines (LFSMs) are particular primitives widely used in
information theory, coding theory and cryptography. Among those linear
automata, a particular case of study is Linear Feedback Shift Registers (LFSRs)
used in many cryptographic applications such as design of stream ciphers or
pseudo-random generation. LFSRs could be seen as particular LFSMs without
inputs.
In this paper, we first recall the description of LFSMs using traditional
matrices representation. Then, we introduce a new matrices representation with
polynomial fractional coefficients. This new representation leads to sparse
representations and implementations. As direct applications, we focus our work
on the Windmill LFSRs case, used for example in the E0 stream cipher and on
other general applications that use this new representation.
In a second part, a new design criterion called diffusion delay for LFSRs is
introduced and well compared with existing related notions. This criterion
represents the diffusion capacity of an LFSR. Thus, using the matrices
representation, we present a new algorithm to randomly pick LFSRs with good
properties (including the new one) and sparse descriptions dedicated to
hardware and software designs. We present some examples of LFSRs generated
using our algorithm to show the relevance of our approach.Comment: Submitted to IEEE-I
Breadth-first serialisation of trees and rational languages
We present here the notion of breadth-first signature and its relationship
with numeration system theory. It is the serialisation into an infinite word of
an ordered infinite tree of finite degree. We study which class of languages
corresponds to which class of words and,more specifically, using a known
construction from numeration system theory, we prove that the signature of
rational languages are substitutive sequences.Comment: 15 page
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