Article

# Factoring Modular Polynomials

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## Abstract

This paper gives an algorithm to factor a polynomial f (in one variable) over rings like Z=rZ for r 2 Z or F q [y]=rF q [y] for r 2 F q [y]. The Chinese Remainder Theorem reduces our problem to the case where r is a prime power. Then factorization is not unique, but if r does not divide the discriminant of f , our (probabilistic) algorithm produces a description of all (possibly exponentially many) factorizations into irreducible factors in polynomial time. If r divides the discriminant, we only know how to factor by exhaustive search, in exponential time.

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... So the hard case is when f mod p is power of an irreducible polynomial. The first resolution in this case was achieved by [28] assuming that k is "large". They assumed k to be larger than the maximum power of p dividing the discriminant of the integral f . ...
... Using this observation they could also describe all the factorizations modulo p k , in a compact data structure. The complexity of [28] was improved by [7]. ...
... The study of [27,28] sheds some light on the behaviour of the factoring problem for integral polynomials modulo prime powers. It shows that for "large" k the problem is similar to the factorization over p-adic fields (already solved efficiently by [5]). ...
Conference Paper
Polynomial factoring has famous practical algorithms over fields-- finite, rational and p-adic. However, modulo prime powers, factoring gets harder because there is non-unique factorization and a combinatorial blowup ensues. For example, x^2+p \bmod p^2 is irreducible, but x^2+px \bmod p^2 has exponentially many factors! We present the first randomized poly(\deg f, łog p) time algorithm to factor a given univariate integral f(x) modulo p^k, for a prime p and k łeq 4. Thus, we solve the open question of factoring modulo p^3 posed in (Sircana, ISSAC'17). Our method reduces the general problem of factoring f(x) mod p^k to that of \em root finding in a related polynomial E(y) \bmodłangle p^k, \varphi(x)^\ell \rangle for some irreducible \varphi \bmod p. We can efficiently solve the latter for kłe4, by incrementally transforming E(y). Moreover, we discover an efficient refinement of Hensel lifting to lift factors of f(x) \bmod p to those \bmod\ p^4 (if possible). This was previously unknown, as the case of repeated factors of f(x) \bmod p forbids classical Hensel lifting.
... So the hard case is when f mod p is power of an irreducible polynomial. The first resolution in this case was achieved by [vzGH98] assuming that k is "large". They assumed k to be larger than the maximum power of p dividing the discriminant of the integral f . ...
... Using this observation they could also describe all the factorizations modulo p k , in a compact data structure. The complexity of [vzGH98] was improved by [CL01]. ...
... The study of [vzGH98,vzGH96] sheds some light on the behaviour of the factoring problem for integral polynomials modulo prime powers. It shows that for "large" k the problem is similar to the factorization over p-adic fields (already solved efficiently by [CG00]). ...
Preprint
Polynomial factoring has famous practical algorithms over fields-- finite, rational \& $p$-adic. However, modulo prime powers it gets hard as there is non-unique factorization and a combinatorial blowup ensues. For example, $x^2+p \bmod p^2$ is irreducible, but $x^2+px \bmod p^2$ has exponentially many factors! We present the first randomized poly(deg $f, \log p$) time algorithm to factor a given univariate integral $f(x)$ modulo $p^k$, for a prime $p$ and $k \leq 4$. Thus, we solve the open question of factoring modulo $p^3$ posed in (Sircana, ISSAC'17). Our method reduces the general problem of factoring $f(x) \bmod p^k$ to that of {\em root finding} in a related polynomial $E(y) \bmod\langle p^k, \varphi(x)^\ell \rangle$ for some irreducible $\varphi \bmod p$. We could efficiently solve the latter for $k\le4$, by incrementally transforming $E(y)$. Moreover, we discover an efficient and strong generalization of Hensel lifting to lift factors of $f(x) \bmod p$ to those $\bmod\ p^4$ (if possible). This was previously unknown, as the case of repeated factors of $f(x) \bmod p$ forbids classical Hensel lifting.
... Suppose k, p ∈ N with p prime and f ∈ Z[x] is a univariate polynomial with degree d ≥ 1 and all coefficients having absolute value less than p k . Let N p,k (f ) denote the number of roots of f in Z/ p k (see, e.g., [24,23,2,18,14,28] for further background on prime power rings). Computing N p,k (f ) is a fundamental problem occuring in polynomial factoring [21,9,4,25,15], coding theory [3], and cryptography [19]. ...
... Our use of fast factorization (as in [17]) is why we avail to randomness, but this pays off: Gaining access to individual roots in Z/(p) (as suggested in [8]) enables us to give a more streamlined algorithm. [14] a randomized polynomialtime algorithm to compute all factorizations of certain f ∈ (Z/(p k ))[x]. (Examples like ...
... Unfortunately, their algorithm has the restriction that p k not divide the discriminant of f . Their complexity bound, in our notation, is the sum of d 7 k log(p)(k log(p) + log d) 2 and a term involving the complexity of finding the mod p k reduction of a factorization over Z p [x] (see from [14]). The complexity of just counting the number of possible factorizations (or just the number of possible linear factors) of f from their data structure does not appear to be stated. ...
Preprint
Suppose $k,p\!\in\!\mathbb{N}$ with $p$ prime and $f\!\in\!\mathbb{Z}[x]$ is a univariate polynomial with degree $d$ and all coefficients having absolute value less than $p^k$. We give a Las Vegas randomized algorithm that computes the number of roots of $f$ in $\mathbb{Z}/\!\left(p^k\right)$ within time $d^3(k\log p)^{2+o(1)}$. (We in fact prove a more intricate complexity bound that is slightly better.) The best previous general algorithm had (deterministic) complexity exponential in $k$. We also present some experimental data evincing the potential practicality of our algorithm.
... Even irreducibility testing of a polynomial, with the prime factorization of n given, has no efficient algorithm known. This reduces to prime-power characteristic p k [vzGH98]. Deterministic factoring in such a ring is a much harder question (at least it subsumes deterministic factoring mod p). ...
... Deterministic factoring in such a ring is a much harder question (at least it subsumes deterministic factoring mod p). In fact, even randomized algorithms, or practical solutions, are currently elusive [vzGH96,vzGH98,Kli97,Sȃl05,Sir17,DMS19]. The main obstruction is non-unique factorization. ...
Preprint
Finding an irreducible factor, of a polynomial $f(x)$ modulo a prime $p$, is not known to be in deterministic polynomial time. Though there is such a classical algorithm that {\em counts} the number of irreducible factors of $f\bmod p$. We can ask the same question modulo prime-powers $p^k$. The irreducible factors of $f\bmod p^k$ blow up exponentially in number; making it hard to describe them. Can we count those irreducible factors $\bmod~p^k$ that remain irreducible mod $p$? These are called {\em basic-irreducible}. A simple example is in $f=x^2+px \bmod p^2$; it has $p$ many basic-irreducible factors. Also note that, $x^2+p \bmod p^2$ is irreducible but not basic-irreducible! We give an algorithm to count the number of basic-irreducible factors of $f\bmod p^k$ in deterministic poly(deg$(f),k\log p$)-time. This solves the open questions posed in (Cheng et al, ANTS'18 \& Kopp et al, Math.Comp.'19). In particular, we are counting roots $\bmod\ p^k$; which gives the first deterministic poly-time algorithm to compute Igusa zeta function of $f$. Also, our algorithm efficiently partitions the set of all basic-irreducible factors (possibly exponential) into merely deg$(f)$-many disjoint sets, using a compact tree data structure and {\em split} ideals.
... As all recurrent sequences are periodic, they are in particular linearly recurrent and satisfy the linear recurrence (of not necessarily minimal degree) defined by x n − 1, with n the period of the sequence. An algorithm for determining all factorisations of a polynomial over a ring of the form Z p a (and some other types of rings) was developed in [13]. One factorisation is derived from the factorisation of the polynomial over the p-adic integers (this can be obtained by the algorithms of Chistov, Ford-Zassenhaus, Buchmann-Lenstra, Cantor-Gordon, Pauli, Ford et. ...
... Factoring over the p-adics and then projecting the factorisation to Z p a [x] does not always result in a factorisation into irreducible factors, as irreducible monic polynomials over the p-adic integers may no longer be irreducible when projected (see Example 4.6 for illustration). The advantage of our results compared to [13] is that they hold for all polynomials, regardless of the value of their discriminant. The disadvantage is that they only hold in Galois rings of characteristic p 2 , with no immediate way of extending them to Galois rings of characteristic p a with a > 2. The paper is organised as follows. ...
Article
This article was published in the journal, Finite fields and their applications [© Elsevier] and is also available at: http://www.sciencedirect.com/science/journal/10715797 It is known that univariate polynomials over finite local rings factor uniquely into primary pairwise coprime factors. Primary polynomials are not necessarily irreducible. Here we describe a factorisation into irreducible factors for primary polynomials over Z4 and more generally over Galois rings of characteristic p2. An algorithm is also given. As an application, we factor xn-1 and xn+1 over such rings.
... On obtient alors, s'il y en a, les racines n-ième modulo p de λ. Un algorithme probabiliste qui fonctionne en temps polynômial pour cette factorisation est donné dans [GH98]. ...
Thesis
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Cette thèse porte sur deux éléments actuellement incontournables de la cryptographie à clé publique, qui sont l’arithmétique modulaire avec de grands entiers et la multiplication scalaire sur les courbes elliptiques (ECSM). Pour le premier, nous nous intéressons au système de représentation modulaire adapté (AMNS), qui fut introduit par Bajard et al. en 2004. C’est un système de représentation de restes modulaires dans lequel les éléments sont des polynômes. Nous montrons d’une part que ce système permet d’effectuer l’arithmétique modulaire de façon efficace et d’autre part comment l’utiliser pour la randomisation de cette arithmétique afin de protéger l’implémentation des protocoles cryptographiques contre certaines attaques par canaux auxiliaires. Pour l’ECSM, nous abordons l’utilisation des chaînes d’additions euclidiennes (EAC) pour tirer parti de la formule d’addition de points efficace proposée par Méloni en 2007. L’objectif est d’une part de généraliser au cas d’un point de base quelconque l’utilisation des EAC pour effectuer la multiplication scalaire ; cela, grâce aux courbes munies d’un endomorphisme efficace. D’autre part, nous proposons un algorithme pour effectuer la multiplication scalaire avec les EAC, qui permet la détection de fautes qui seraient commises par un attaquant que nous détaillons.
... There is a long history and very extensive literature dedicated to algorithms on polynomials in finite fields, see, for example [12]. More recently, there was also increasing interest to algorithms for polynomials over residue rings, especially in residue rings modulo prime powers, see [5,9,10,13,17,19,31] and references therein. Here we continue this directions and consider the noisy polynomial interpolation problem modulo prime powers which is analogue to the same problem in finite fields [28,30], which in turn is an extension of the hidden number problem of Boneh and Venkatesan [2,3]. ...
Preprint
Full-text available
We consider the {\it noisy polynomial interpolation problem\/} of recovering an unknown $s$-sparse polynomial $f(X)$ over the ring $\mathbb Z_{p^k}$ of residues modulo $p^k$, where $p$ is a small prime and $k$ is a large integer parameter, from approximate values of the residues of $f(t) \in \mathbb Z_{p^k}$. Similar results are known for residues modulo a large prime $p$, however the case of prime power modulus $p^k$, with small $p$ and large $k$, is new and requires different techniques. We give a deterministic polynomials time algorithm, which for almost given more than a half bits of $f(t)$ for sufficiently many randomly chosen points $t \in \mathbb Z_{p^k}^*$, recovers $f(X)$.
... (2) A similar result as in Proposition 4.3 can be found in [vzGH98]. There the authors show that in case R is a discrete valuation ring, the reduced resultant of f , g ∈ R[x] is equal to the largest elementary divisor of S(f, g) (see Lemma 3.8 of op. ...
Preprint
Full-text available
The resultant of two univariate polynomials is an invariant of great importance in commutative algebra and vastly used in computer algebra systems. Here we present an algorithm to compute it over Artinian principal rings with a modified version of the Euclidean algorithm. Using the same strategy, we show how the reduced resultant and a pair of B\'ezout coefficient can be computed. Particular attention is devoted to the special case of $\mathbf{Z}/n\mathbf{Z}$, where we perform a detailed analysis of the asymptotic cost of the algorithm. Finally, we illustrate how the algorithms can be exploited to improve ideal arithmetic in number fields and polynomial arithmetic over $p$-adic fields.
... Their root will give a nth-root of λ. A probabilistic polynomial time algorithm to achieve this goal is given in [15]. ...
Article
Full-text available
The adapted modular number system (AMNS) is an integer number system which aims to speed up arithmetic operations modulo a prime p. Such a system is defined by a tuple (p,n,γ,ρ,E), where p, n, γ and ρ are integers and E∈Z[X]. In El Mrabet and Gama (in: WAIFI, lecture notes in computer science, Springer, 2012) conditions required to build AMNS with E(X)=Xn+1 are provided. In this paper, we generalise their approach and provide a method to generate multiple AMNS for a given prime p with E(X)=Xn-λ and λ∈Z\{0}. Moreover, we propose a complete set of algorithms without conditional branching to perform arithmetic and conversion operations in the AMNS, using a Montgomery-like method described in Negre and Plantard (in: Information security and privacy, 13th Australasian conference, ACISP 2008, Wollongong, Australia, 2008). We show that our implementation outperforms GNU MP and OpenSSL libraries. Finally, we highlight some properties of the AMNS which state that it could lead to a helpful countermeasure against some side-channel attacks.
... That is, there exist unimodular matrices U and V satisfying U AV = B. Any matrix can be transformed into a diagonal form known as the Smith normal form, which reveals the invariant factors of the matrix [31]. It is also useful for solving systems of linear equations [34,35]. ...
Article
Full-text available
In this thesis we study algorithms for computing normal forms for matrices of Ore polynomials while controlling coe#cient growth. By formulating row reduction as a linear algebra problem, we obtain a fraction-free algorithm for row reduction for matrices of Ore polynomials. The algorithm allows us to compute the rank and a basis of the left nullspace of the input matrix. When the input is restricted to matrices of shift polynomials and ordinary polynomials, we obtain fractionfree algorithms for computing row-reduced forms and weak Popov forms. These algorithms can be used to compute a greatest common right divisor and a least common left multiple of such matrices. Our fraction-free row reduction algorithm can be viewed as a generalization of subresultant algorithms. The linear algebra formulation allows us to obtain bounds on the size of the intermediate results and to analyze the complexity of our algorithms.
... Kaltofen and Lobo use a black box representation for factoring high degree polynomials over finite fields using the Berlekamp algorithm [37]. Algorithms for univariate polynomials over the integers modulo n, which is not a unique factorization domain if n is not prime, are presented by von zur Gathen and Hartlieb [79]. An algorithm for univariate polynomials over Z that relies on factorization over finite fields and uses Hensel lifting to construct integral factors is presented by Zassen- haus [91]. ...
Article
xv Zusammenfassung xvii 1 Introduction 1 1.1 Mathematical preliminaries . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.1 Finite Fields . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.2 Unique Factorization Domains . . . . . . . . . . . . . . . . . 3 1.1.3 Irreducibility . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2 Factorization Algorithms 9 2.1 Square-free Factorization . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1.1 Musser's Algorithm . . . . . . . . . . . . . . . . . . . . . . 12 2.1.2 Yun's Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1.3 The Extended Yun Algorithm . . . . . . . . . . . . . . . . . 14 2.1.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.2 Univariate Factori...
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To whom it may concern: You can freely download all my published papers from my papers page at Texas A&M University: www.math.tamu.edu/~rojas/list2.html
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ac.il Abstract. In this paper we consider the
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Let f be a monic separable polynomial over the rational integers @? and p be a rational prime. We set up an algorithm which yields the extended p-adic values of a given element a in the semi-simple algebra A"f=@?[X]/f. Firstly, we determine the decomposition ...
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New probabilistic algorithms are presented for factoring univariate polynomials over finite fields. The algorithms factor a polynomial of degree n over a finite field of constant cardinality in time O(n 1:815 ). Previous algorithms required time Theta(n 2+o(1) ). The new algorithms rely on fast matrix multiplication techniques. More generally, to factor a polynomial of degree n over the finite field F q with q elements, the algorithms use O(n 1:815 log q) arithmetic operations in F q . The new "baby step/giant step" techniques used in our algorithms also yield new fast practical algorithms at superquadratic asymptotic running time, and subquadratic-time methods for manipulating normal bases of finite fields. 1 INTRODUCTION In this paper, we present a new probabilistic approach for factoring univariate polynomials over finite fields. The resulting algorithms factor a polynomial of degree n over a finite field F q whose cardinality q is constant in time O(n 1:815 ). The best ...
Factorization of polynomials modulo small prime powers
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