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Graham H. Norton

Publications and source records attributed to Graham H. Norton.

11 recordsLinked to original sources
arXiv2024

On $n$-Dimensional Sequences. I

Let $R$ be a commutative ring and let $n \geq 1.$ We study $Γ(s)$, the generating function and Ann$(s)$, the ideal of characteristic polynomials of $s$, an $n$--dimensional sequence over $R$. We express $f(X_1,\ldots,X_n) \cdot Γ(s)(X_1^{-1},\ldots ,X_n^{-1})$ as a partitioned sum. That is, we give (i) a $2^n$--fold ``border'' partition (ii) an explicit expression for the product as a $2^n$--fold sum; the support of each summand is contained in precisely one member of the partition. A key summand is $β_0(f,s)$, the ``border polynomial'' of $f$ and $s$, which is divisible by $X_1\cdots X_n$. We say that $s$ is {\em eventually rectilinear} if the elimination ideals Ann$(s)\cap R[X_i]$ contain an $f_i(X_i)$ for $1 \leq i \leq n$. In this case, we show that $\mbox{Ann}(s)$ is the ideal quotient $(\sum_{i=1}^n(f_i)\ :\ β_0(f,s)/(X_1\cdots X_n)).$ When $R$ and $R[[X_1,X_2, \ldots ,X_n]]$ are factorial domains (e.g. $R$ a principal ideal domain or ${\Bbb F}[X_1,\ldots,X_n]$), we compute {\em the monic generator} $γ_i$ of $\mbox{Ann}(s) \cap R[X_i]$ from known $f_i \in \mbox{Ann}(s) \cap R[X_i]$ or from a finite number of $1$--dimensional linear recurring sequences over $R$. Over a field ${\Bbb F}$ this gives an $O(\prod_{i=1}^n δγ_i^3)$ algorithm to compute an ${\Bbb F}$--basis for $\mbox{Ann}(s)$.

math.AC↗
arXiv2023

On Rueppel's Linear Complexity Conjecture

Rueppel's conjecture on the linear complexity of the first $n$ terms of the sequence $(1,1,0,1,0^3,1,0^7,1,0^{15},\ldots)$ was first proved by Dai using the Euclidean algorithm. We have previously shown that we can attach a homogeneous (annihilator) ideal of $F[x,z]$ to the first $n$ terms of a sequence over a field $F$ and construct a pair of generating forms for it. This approach gives another proof of Rueppel's conjecture. We also prove additional properties of these forms and deduce the outputs of the LFSR synthesis algorithm applied to the first $n$ terms. Further, dehomogenising the leading generators yields the minimal polynomials of Dai.

cs.SC↗
arXiv2022

A Note on the Games-Chan Algorithm

The Games-Chan algorithm finds the minimal period of a periodic binary sequence of period $2^n$, in $n$ iterations. We generalise this to periodic $q$-ary sequences (where $q$ is a prime power) using generating functions and polynomials and apply this to find the multiplicity of $x-1$ in a $q$-ary polynomial $f$ in $\log_{\,q}°(f)$ iterations.

cs.SC↗
arXiv2018

On the Annihilator Ideal of an Inverse Form

Let $K$ be a field. We simplify and extend work of Althaler \& Dür on finite sequences over $K$ by regarding $K[x^{-1},z^{-1}]$ as a $K[x,z]$ module, and studying forms in $K[x^{-1},z^{-1}]$ from first principles. Then we apply our results to finite sequences. First we define the annihilator ideal $I_F$ of a non-zero form $F\in K[x^{-1},z^{-1}]$, a homogeneous ideal. We inductively construct an ordered pair ($f_1$\,,\,$f_2$) of forms which generate $I_F$\,; our generators are special in that $z$ does not divide the leading grlex monomial of $f_1$ but $z$ divides $f_2$\,, and the sum of their total degrees is always $2-|F|$, where $|F|$ is the total degree of $F$. We show that $f_1,f_2$ is a maximal regular sequence for $I_F$, so that the height of $I_F$ is 2. The corresponding algorithm is $\sim |F|^2/2$. The row vector obtained by accumulating intermediate forms of the construction gives a minimal grlex Gröbner basis for $I_F$ for no extra computational cost other than storage and apply this to determining $\dim_K (K[x,z] /I_F)$\,. We show that either the form vector is reduced or a monomial of $f_1$ can be reduced by $f_2$\,. This enables us to efficiently construct the unique reduced Gröbner basis for $I_F$ from the vector extension of our algorithm. Then we specialise to the inverse form of a finite sequence, obtaining generator forms for its annihilator ideal and a corresponding algorithm which does not use the last 'length change' of Massey. We compute the intersection of two annihilator ideals using syzygies in $K[x,z]^5$. This improves a result of Althaler \& Dür. Finally, dehomogenisation induces a one-to-one correspondence ($f_1$\,,$f_2$) $\mapsto$ (minimal polynomial, auxiliary polynomial), the output of the author's variant of the Berlekamp-Massey algorithm. So we can also solve the LFSR synthesis problem via the corresponding algorithm for sequences.

cs.SC↗
arXiv2018

On the Annihilator Ideal of an Inverse Form. A Simplification

We simplify an earlier paper of the same title by not using syzygy polynomials and by not using a trichotomy of inverse forms. Let $\K$ be a field and $\M=\K[x^{-1},z^{-1}]$ denote Macaulay's $\K[x,z]$ module of inverse polynomials; here $z$ and $z^{-1}$ are homogenising variables. An inverse form $F\in\M$ has a homogeneous annihilator ideal, $\I_F$\,. In an earlier paper we inductively constructed an ordered pair ($f_1$\,,\,$f_2$) of forms in $\K[x,z]$ which generate $\I_F$. We used syzygy polynomials to show that the intermediate forms give a minimal grlex Groebner basis, which can be efficiently reduced. We give a significantly shorter proof that the intermediate forms are a minimal grlex Groebner basis for $\I_F$\,. We also simplify our proof that either $ F$ is already reduced or a monomial of $f_1$ can be reduced by $f_2$\,. The algorithm that computes $f_1\,,f_2$ yields a variant of the Berlekamp-Massey algorithm which does not use the last 'length change' approach of Massey. These new proofs avoid the three separate cases, 'triples' and the technical factorisation of intermediate 'essential' forms. We also show that $f_1,f_2$ is a maximal $\R$ regular sequence for $\I_F$\,, so that $\I_F$ is a complete intersection.

cs.SC↗
arXiv2015

On Sequences, Rational Functions and Decomposition

Our overall goal is to unify and extend some results in the literature related to the approximation of generating functions of finite and infinite sequences over a field by rational functions. In our approach, numerators play a significant role. We revisit a theorem of Niederreiter on (i) linear complexities and (ii) '$n^{th}$ minimal polynomials' of an infinite sequence, proved using partial quotients. We prove (i) and its converse from first principles and generalise (ii) to rational functions where the denominator need not have minimal degree. We prove (ii) in two parts: firstly for geometric sequences and then for sequences with a jump in linear complexity. The basic idea is to decompose the denominator as a sum of polynomial multiples of two polynomials of minimal degree; there is a similar decomposition for the numerators. The decomposition is unique when the denominator has degree at most the length of the sequence. The proof also applies to rational functions related to finite sequences, generalising a result of Massey. We give a number of applications to rational functions associated to sequences.

cs.SC↗
arXiv2011

On Sequences with a Perfect Linear Complexity Profile

We derive Bézout identities for the minimal polynomials of a finite sequence and use them to prove a theorem of Wang and Massey on binary sequences with a perfect linear complexity profile. We give a new proof of Rueppel's conjecture and simplify Dai's original proof. We obtain short proofs of results of Niederreiter relating the linear complexity of a sequence s and K(s), which was defined using continued fractions. We give an upper bound for the sum of the linear complexities of any sequence. This bound is tight for sequences with a perfect linear complexity profile and we apply it to characterise these sequences in two new ways.

cs.IT↗
arXiv2010

Bézout Identities Associated to a Finite Sequence

We consider finite sequences $s\in D^n$ where $D$ is a commutative, unital, integral domain. We prove three sets of identities (possibly with repetitions), each involving $2n$ polynomials associated to $s$. The right-hand side of these identities is a recursively-defined (non-zero) 'product-of-discrepancies'. There are implied iterative algorithms (of quadratic complexity) for the left-hand side coefficients; when the ground domain is factorial, the identities are in effect Bézout identities. We give a number of applications: an algorithm to compute Bézout coefficients over a field; the outputs of the Berlekamp-Massey algorithm; sequences with perfect linear complexity profile; annihilating polynomials which do not vanish at zero and have minimal degree: we simplify and extend an algorithm of Salagean to sequences over $D$. In the Appendix, we give a new proof of a theorem of Imamura and Yoshida on the linear complexity of reverse sequences, initially proved using Hankel matrices over a field and now valid for sequences over a factorial domain.

cs.IT↗
arXiv2010

The Berlekamp-Massey Algorithm via Minimal Polynomials

We present a recursive minimal polynomial theorem for finite sequences over a commutative integral domain $D$. This theorem is relative to any element of $D$. The ingredients are: the arithmetic of Laurent polynomials over $D$, a recursive 'index function' and simple mathematical induction. Taking reciprocals gives a 'Berlekamp-Massey theorem' i.e. a recursive construction of the polynomials arising in the Berlekamp-Massey algorithm, relative to any element of $D$. The recursive theorem readily yields the iterative minimal polynomial algorithm due to the author and a transparent derivation of the iterative Berlekamp-Massey algorithm. We give an upper bound for the sum of the linear complexities of $s$ which is tight if $s$ has a perfect linear complexity profile. This implies that over a field, both iterative algorithms require at most $2\lfloor \frac{n^2}{4}\rfloor$ multiplications.

cs.IT↗
arXiv2010

Shortest Two-way Linear Recurrences

Let $s$ be a finite sequence over a field of length $n$. It is well-known that if $s$ satisfies a linear recurrence of order $d$ with non-zero constant term, then the reverse of $s$ also satisfies a recurrence of order $d$ (with coefficients in reverse order). A recent article of A. Salagean proposed an algorithm to find such a shortest 'two-way' recurrence -- which may be longer than a linear recurrence for $s$ of shortest length $\LC_n$. We give a new and simpler algorithm to compute a shortest two-way linear recurrence. First we show that the pairs of polynomials we use to construct a minimal polynomial iteratively are always relatively prime; we also give the extended multipliers. Then we combine degree lower bounds with a straightforward rewrite of a published algorithm due to the author to obtain our simpler algorithm. The increase in shortest length is $\max\{n+1-2\LC_n,0\}$.

cs.IT↗