Univariate polynomial factorization over finite fields with large extension degree

van der Hoeven J., Lecerf G.

The efficient evaluation of multivariate polynomials at many points is an important operation for polynomial system solving. Kedlaya and Umans have recently devised a theoretically efficient algorithm for this task when the coefficients are integers or when they lie in a finite field. In this paper, we assume that the set of points where we need to evaluate is fixed and “sufficiently generic”. Under these restrictions, we present a quasi-optimal algorithm for multi-point evaluation over general fields. We also present a quasi-optimal algorithm for the opposite interpolation task.

Neiger V., Rosenkilde J., Solomatov G.

Suppose K is a large enough field and P ⊂ K2 is a fixed, generic set of points which is available for precomputation. We introduce a technique called reshaping which allows us to design quasi-linear algorithms for both: computing the evaluations of an input polynomial f ∈ K [x, y] at all points of P and computing an interpolant f ∈ K[x, y] which takes prescribed values on P and satisfies an input y-degree bound. Our genericity assumption is explicit and we prove that it holds for most point sets over a large enough field. If P violates the assumption, our algorithms still work and the performance degrades smoothly according to a distance from being generic. To show that the reshaping technique may have an impact on other related problems, we apply it to modular composition: suppose generic polynomials M ∈ K[x] and A ∈ K[x] are available for precomputation, then given an input f ∈ K[x, y] we show how to compute f (x, A(x)) rem M(x) in quasi-linear time.

van der Hoeven J., Lecerf G.

In 2008, Kedlaya and Umans designed the first multivariate multi-point evaluation algorithm over finite fields with an asymptotic complexity that can be made arbitrarily close to linear. However, it remains a major challenge to make their algorithm efficient for practical input sizes. In this paper, we revisit and improve their algorithm, while keeping this ultimate goal in mind. In addition we sharpen the known complexity bounds for modular composition of univariate polynomials over finite fields.

van der Hoeven J., Lecerf G.

Nowadays, asymptotically fast algorithms are widely used in computer algebra for computations in towers of algebraic field extensions of small height. Yet it is still unknown how to reach softly linear time for products and inversions in towers of arbitrary height. In this paper we design the first algorithm for general ground fields with a complexity exponent that can be made arbitrarily close to one from the asymptotic point of view. We deduce new faster algorithms for changes of tower representations, including the computation of primitive element representations in subquadratic time.

Harvey D., van der Hoeven J.

We prove that for a fixed prime  p , polynomials in F p [ X ] of degree  n may be multiplied in O ( n log n 4 log ∗ n ) bit operations. Previously, the best known bound was O ( n log n 8 log ∗ n ) .

van der Hoeven J., Lecerf G.

Modular composition is the problem to compute the composition of two univariate polynomials modulo a third one. For polynomials with coefficients in a finite field, Kedlaya and Umans proved in 2008 that the theoretical bit complexity for performing this task could be made arbitrarily close to linear. Unfortunately, beyond its major theoretical impact, this result has not led to practically faster implementations yet. In this article, we explore how polynomial factorization may help modular composition. Most of our algorithms assume that the modulus is fixed, which allows us to precompute factorizations of the modulus over suitable extension fields. This significant restriction with respect to Kedlaya and Umans’ algorithm is acceptable whenever we need to perform many compositions modulo the same modulus. For polynomials over a finite field, we show that compositions modulo a fixed modulus of composite degree are cheaper than general modular compositions. In some very favorable cases of fixed moduli of very smooth degree, we show that the complexity even becomes quasi-linear.

Gall F.L., Urrutia F.

Mullen G.L., Panario D.

von zur Gathen J., Gerhard J.

Computer algebra systems are now ubiquitous in all areas of science and engineering. This highly successful textbook, widely regarded as the 'bible of computer algebra', gives a thorough introduction to the algorithmic basis of the mathematical engine in computer algebra systems. Designed to accompany one- or two-semester courses for advanced undergraduate or graduate students in computer science or mathematics, its comprehensiveness and reliability has also made it an essential reference for professionals in the area. Special features include: detailed study of algorithms including time analysis; implementation reports on several topics; complete proofs of the mathematical underpinnings; and a wide variety of applications (among others, in chemistry, coding theory, cryptography, computational logic, and the design of calendars and musical scales). A great deal of historical information and illustration enlivens the text. In this third edition, errors have been corrected and much of the Fast Euclidean Algorithm chapter has been renovated.

Poteaux A., Schost É.

We generalize Kedlaya and Umans’ modular composition algorithm to the multivariate case. As a main application, we give fast algorithms for many operations involving triangular sets (over a finite field), such as modular multiplication, inversion, or change of order. For the first time, we are able to exhibit running times for these operations that are almost linear, without any overhead exponential in the number of variables. As a further application, we show that, from the complexity viewpoint, Charlap, Coley, and Robbins’ approach to elliptic curve point counting can be competitive with the better known approach due to Elkies.

Kedlaya K.S., Umans C.

We obtain randomized algorithms for factoring degree $n$ univariate polynomials over $\mathbb{F}_q$ requiring $O(n^{1.5 + o(1)}\,{\rm log}^{1+o(1)} q+ n^{1 + o(1)}\,{\rm log}^{2+o(1)} q)$ bit operations. When ${\rm log}\, q < n$, this is asymptotically faster than the best previous algorithms [J. von zur Gathen and V. Shoup, Comput. Complexity, 2 (1992), pp. 187-224; E. Kaltofen and V. Shoup, Math. Comp., 67 (1998), pp. 1179-1197]; for ${\rm log}\, q \ge n$, it matches the asymptotic running time of the best known algorithms. The improvements come from new algorithms for modular composition of degree $n$ univariate polynomials, which is the asymptotic bottleneck in fast algorithms for factoring polynomials over finite fields. The best previous algorithms for modular composition use $O(n^{(\omega + 1)/2})$ field operations, where $\omega$ is the exponent of matrix multiplication [R. P. Brent and H. T. Kung, J. Assoc. Comput. Mach., 25 (1978), pp. 581-595], with a slight improvement in the exponent achieved by employing fast rectangular matrix multiplication [X. Huang and V. Y. Pan, J. Complexity, 14 (1998), pp. 257-299]. We show that modular composition and multipoint evaluation of multivariate polynomials are essentially equivalent, in the sense that an algorithm for one achieving exponent $\alpha$ implies an algorithm for the other with exponent $\alpha + o(1)$, and vice versa. We then give two new algorithms that solve the problem near-optimally: an algebraic algorithm for fields of characteristic at most $n^{o(1)}$, and a nonalgebraic algorithm that works in arbitrary characteristic. The latter algorithm works by lifting to characteristic 0, applying a small number of rounds of multimodular reduction, and finishing with a small number of multidimensional FFTs. The final evaluations are reconstructed using the Chinese remainder theorem. As a bonus, this algorithm produces a very efficient data structure supporting polynomial evaluation queries, which is of independent interest. Our algorithms use techniques that are commonly employed in practice, in contrast to all previous subquadratic algorithms for these problems, which relied on fast matrix multiplication.

Cantor D.G., Zassenhaus H.

We present a new probabilistic algorithm for factoring polynomials over finite fields.

Berlekamp E.R.

This paper reviews some of the known algorithms for factoring polynomials over finite fields and presents a new deterministic procedure for reducing the problem of factoring an arbitrary polynomial over the Galois field GF ( p m ) {\text {GF}}({p^m}) to the problem of finding the roots in GF ( p ) {\text {GF}}(p) of certain other polynomials over GF ( p ) {\text {GF}}(p) . The amount of computation and the storage space required by these algorithms are algebraic in both the degree of the polynomial to be factored and the logarithm of the order of the finite field. Certain observations on the application of these methods to the factorization of polynomials over the rational integers are also included.

Shoup V.

Number theory and algebra play an increasingly significant role in computing and communications, as evidenced by the striking applications of these subjects to such fields as cryptography and coding theory. This introductory book emphasizes algorithms and applications, such as cryptography and error correcting codes, and is accessible to a broad audience. The presentation alternates between theory and applications in order to motivate and illustrate the mathematics. The mathematical coverage includes the basics of number theory, abstract algebra and discrete probability theory. This edition now includes over 150 new exercises, ranging from the routine to the challenging, that flesh out the material presented in the body of the text, and which further develop the theory and present new applications. The material has also been reorganized to improve clarity of exposition and presentation. Ideal as a textbook for introductory courses in number theory and algebra, especially those geared towards computer science students.

Kedlaya K.S., Umans C.

Demin A., van der Hoeven J.

van der Hoeven J., Lecerf G.

Given an algebraic germ of a plane curve at the origin, in terms of a bivariate polynomial, we analyze the complexity of computing an irreducible decomposition up to any given truncation order. With a suitable representation of the irreducible components, and whenever the characteristic of the ground field is zero or larger than the degree of the germ, we design a new algorithm that involves a nearly linear number of arithmetic operations in the ground field plus a small amount of irreducible univariate polynomial factorizations.

Jin C., Xu Y.