Factoring Polynomials over Finite Fields with Linear Galois Groups: An Additive Combinatorics Approach.

Deterministic polynomial factoring over finite fields: A uniform approach via [formula omitted]-schemes

Let D be a division ring with the center F and suppose that D* is the multiplicative group of D. D is called centrally finite if D is a finite dimensional vector space over F and D is locally centrally finite if every finite subset of D generates over F a division subring which is a finite dimensional vector space over F. We say that D is a linear division ring if every finite subset of D generates over Fa centrally finite division subring. It is obvious that every locally centrally finite division ring is linear. In this report we show that the inverse is not true by giving an example of a linear division ring which is not locally centrally finite. Further, we give some properties of subgroups in linear division rings. In particular, we show that every finitely generated subnormal subgroup in a linear ring is central. An interesting corollary is obtained as the following: If D is a linear division ring and D* is finitely generated, then D is a finite field.

Cycle indices of linear, affine, and projective groups

In this chapter we review the main ideas of linear algebra. The one twist is that we allow our scalars to come from any field instead of just the field of real numbers. In particular, the notion of a vector space over the field \(\mathbb{Z}_{2}\) will be essential in our study of coding theory. We will also need to look at other finite fields when we discuss Reed–Solomon codes. One benefit to working with finite dimensional vector spaces over finite fields is that all sets in question are finite, and so computers can be useful in working with them.

We give a quantitative version of a result due to Katz about L-functions of elliptic curves over function fields over finite fields. Roughly speaking, Katz's theorem states that, on average over a suitably chosen algebraic family, the L-function of an elliptic curve over a function field becomes “as irreducible as possible” when seen as a polynomial with rational coefficients, as the cardinality of the field of constants grows. A quantitative refinement is obtained as a corollary of our main result, which gives an estimate for the proportion of elliptic curves studied whose L-functions have “big” Galois group. To do so, we make use of Kowalski's idea to apply large sieve methods in algebro-geometric contexts. Besides large sieve techniques, we use results of Hall on finite orthogonal monodromy and previous work of the author on orthogonal groups over finite fields.

The aim of this paper is to justify the common cryptographic practice of selecting elliptic curves using their order as the primary criterion. We can formalize this issue by asking whether the discrete log problem (DLOG) has the same difficulty for all curves over a given finite field with the same order. We prove that this is essentially true by showing polynomial time random reducibility of DLOG among such curves, assuming the Generalized Riemann Hypothesis (GRH). We do so by constructing certain expander graphs, similar to Ramanujan graphs, with elliptic curves as nodes and low degree isogenies as edges. The result is obtained from the rapid mixing of random walks on this graph. Our proof works only for curves with (nearly) the same endomorphism rings. Without this technical restriction such a DLOG equivalence might be false; however, in practice the restriction may be moot, because all known polynomial time techniques for constructing equal order curves produce only curves with nearly equal endomorphism rings.

An affine algebraic group

We consider the asymptotic complexity of manipulating matrix groups over finite fields. The question is, given a matrix group G by a list of generators, what can we say in polynomial time about the structure of G? While considerable progress has been made recently in identifying the nonabelian composition factors of a matrix group, the fundamental question of recognizing the simplicity of a non-abelian matrix group remains open. For the purposes of polynomial time computation, we reduce this problem to the following “affine case:” Let A be an elementary abelian p-group which is a non-central minimal normal subgroup of a finite group G. Assume further that the quotient G/A is a finite simple group S of Lie type of characteristic p. The algorithmic goal is to distinguish the simple group S from the nonsimple group G. Both S and G are given as “black-box groups of characteristic p.” The reduction to this basic problem involves a large number of recent techniques and results which we review along the way. We address the affine case for the groups S = PSL(2, q) where q = p , p prime. We show that PSL(2, p) can be recognized in Monte Carlo polynomial time among all black-box groups of known characteristic. The situation for f ≥ 2 seems much harder; we exhibit challenging open cases for every f ≥ 2 when q is large. Along with S = PSL(2, p), the positive result holds for all simple groups S of Lie type of characteristic p with the property that in every nontrivial S-module in characteristic p, every element of S has a fixed point. We call these groups “unisingular.” Very recently, Guralnick, Saxl, and Tiep classified these groups. They found that a large number of classes of simple groups are unisingular, including certain classes of linear and unitary groups, all symplectic groups over an odd prime field, all orthogonal groups of odd degree over an odd prime field, many orthogonal groups of even degree, and a number of classes of exceptional groups. 2 Laszlo Babai and Aner Shalev

The number of classes of linearly equivalent functions

We prove that the number of unimodular integral n × n matrices in a norm ball whose characteristic polynomial has Galois group different than the full symmetric group Sn is of strictly lower order of magnitude than the number of all such matrices in the ball, as the radius increases. More generally, we prove a similar result for the Galois groups associated with elements in any connected semisimple linear algebraic group defined and simple over a number field F . Our method is based on the abstract large sieve method developed by Kowalski, and the study of Galois groups via reductions modulo primes developed by Jouve, Kowalski and Zywina. The two key ingredients are a uniform quantitative lattice point counting result, and a non-concentration phenomenon for lattice points in algebraic subvarieties of the group variety, both established previously by the authors. The results answer a question posed by Rivin and by Jouve, Kowalski and Zywina, who have considered Galois groups of random products of elements in algebraic groups.

This thesis presents some contributions in additive combinatorics and arithmetic Ramsey theory. More specifically, it deals with the interaction between combinatorics, number theory and additive combinatorics. This area saw a great improvement with the Szemerédi Regularity Lemma and some of the results that followed. The Regularity Lemma and its consequences have become a widely used tool in graph theory, combinatorics and number theory. Furthermore, its language and point of view has deeply changed the face of additive number theory, a fact universally acknowledged by the Abel award given to Szemerédi in 2012. One of the main reasons for the prize has been Szemerédi's theorem, a result regarding the existence of arbitrarily long arithmetic progressions in dense sets of the integers, the proof of which uses the Regularity Lemma in a key step. One of the earlier consequences of the Regularity Lemma was the Removal Lemma for graphs that was used by Ruzsa and Szemerédi to show Roth theorem, regarding the existence of 3-term arithmetic progressions in dense sets of the integers, in a combinatorial way. The Removal Lemma states that in any graph K with few copies of a subgraph, say a triangle, we can remove few edges from K so that the result contains no copy of the subgraph. This has become a key tool in the applications of the so-called Regularity Method, which has extensive literature in combinatorics, graph theory, number theory and computer science. In 2005 Green introduced a regularity lemma for Abelian groups as well as an algebraic removal lemma. The removal lemma for groups states that, for a given finite Abelian group G, if there are o(|G|^3) solution to x+y+z+t=0 with the variables taking values in S, a subset of G, then we can remove o(|G|) elements from S to make the set S solution-free. The main contributions of this work corresponds to extensions of the removal lemma for groups to either more general contexts, like non-necessary Abelian finite groups, or to linear systems of equations for finite Abelian groups. The main goal is to give a comprehensive and more general framework for many results in additive number theory like Szemerédi Theorem. In particular, we show that the removal lemma for groups by Green can be extended to non-necessary Abelian finite groups. Moreover, we prove a removal lemma for linear systems on finite fields: for every e>0 there exists a d>0 such that if A is a (k x m) linear system of equations with coefficients in a finite field F and the number of solutions to Ax=b, where each variable takes values from a subset Si in F is less than d times |F| raised to m-k, then by removing less than e|F| elements in each Si we can make the resulting sets solution-free, thus solving a conjecture by Green to that respect. Even more, if A is an integer linear system, G is a finite Abelian group, and the determinantal of A and |G| are coprime, then a similar statement holds. Let us mention that the last result allows us to characterize those linear systems where any set S with size proportional to G has a nontrivial solution in S, provided |G| is large enough. This extends the validity of Szmerédi's theorem to finite Abelian groups. These extensions of the removal lemma have been used in arithmetic Ramsey theory to obtain counting results for the number of monochromatic solutions of linear systems. The main result from a work by Frankl, Graham and Rödl in '88 states that the number of monochromatic solutions of regular systems in integer intervals is in fact a positive proportion of the total number of solutions. We give analogous results for solutions in Abelian groups with bounded exponent, for which the main tool in the torsion-free case cannot be applied. Density versions of these counting results are also obtained, in this case with a full characterization.

Let G → GL ( V ) be a representation of a reductive linear algebraic group G on a finite-dimensional vector space V , defined over an algebraically closed field of characteristic zero. The categorical quotient X = V // G carries a natural stratification, due to D. Luna. This paper addresses the following questions: (i) Is the Luna stratification of X intrinsic? That is, does every automorphism of V // G map each stratum to another stratum? (ii) Are the individual Luna strata in X intrinsic? That is, does every automorphism of V // G map each stratum to itself? In general, the Luna stratification is not intrinsic. Nevertheless, we give positive answers to questions (i) and (ii) for interesting families of representations.

Let P(V) be the partial linear transformation semigroup of a vector space V under composition. Given a fixed subspace W of V, define the following subsemigroups of P(V): $$\begin{aligned} PT(V,W)&=\{\alpha \in P(V)\ |\ V\alpha \subseteq W\},\\ T(V,W)&=\{\alpha \in P(V,W)\ |\ \mathrm{dom}\ \alpha =V\},\\ I(V,W)&=\{\alpha \in P(V,W)\ |\ \alpha \text { is injective}\}. \end{aligned}$$ In this paper, we prove certain isomorphism theorems and compute the ranks of these three semigroups for any proper subspace W of V when V is a finite-dimensional vector space over a finite field. Gaussian binomial coefficients play an essential role in these computations.

In this paper we develop new algorithms for factoring polynomials over finite fields by exploring an interesting connection between the algebraic factoring problem and the combinatorial problem of stable coloring of tournaments. We present an algorithm which can be viewed as a recursive refinement scheme through which most cases of polynomials are completely factored in deterministic polynomial time within the first level of refinement, most of the remaining cases are factored completely before the end of the second level refinement, and so on. The algorithm has average polynomial time complexity and (n \(^{\rm log{\it n}}\) logp)O(1) worst case complexity. Under a purely combinatorial conjecture concerning tournaments, the algorithm has worst case complexity (n \(^{\rm loglog{\it n}}\) logp)O(1). Our approach is also useful in reducing the amount of randomness needed to factor a polynomial completely in expected polynomial time. We present a random polynomial time algorithm for factoring polynomials over finite fields which requires only log p random bits. All these results assume the Extended Riemann Hypothesis.KeywordsPolynomial TimeZero DivisorStable ColoringPrimitive IdempotentFactoring PolynomialThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Finite fields play important roles in many application areas such as coding theory, cryptography, Monte Carlo and quasi-Monte Carlo methods, pseudorandom number generation, quantum computing, and wireless communication. In this book we will focus on sequences, character sums, and polynomials over finite fields in view of the above mentioned application areas: Chapters 1 and 2 deal with sequences mainly constructed via characters and analyzed using bounds on character sums. Chapters 3, 5, and 6 deal with polynomials over finite fields. Chapters 4 and 9 consider problems related to coding theory studied via finite geometry and additive combinatorics, respectively. Chapter 7 deals with quasirandom points in view of applications to numerical integration using quasi-Monte Carlo methods and simulation. Chapter 8 studies aspects of iterations of rational functions from which pseudorandom numbers for Monte Carlo methods can be derived. The goal of this book is giving an overview of several recent research directions as well as stimulating research in sequences and polynomials under the unified framework of character theory.