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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.

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We give an eecient algorithm for factoring polynomials over nite algebraic extensions of the p-adic numbers. This algorithm uses ideas of Chistov's random polynomial-time algorithm, and is suitable for practical implementation.

In this paper we present a polynomial time algorithm to compute the local zeta function Z(s,f) attached to a polynomial f(x) in Z[x] (in one variable, with splitting field Q) and a prime p. The algorithm reduces in polynomial time the computation of Z(s,f) to the computation of a factorization of f(x) over Q. This reduction is accomplished by constructing a weighted tree from the p-adic expansion of the roots of f(x) modulo a certain power of p, and then associating a generating function to this tree. The generating function constructed in this way coincides with the local zeta function of f(x). We also propose a new class of candidates for one-way functions based on Igusa's zeta functions attached to polynomials in one variable.

In this paper we present a polynomial-time algorithm to solve the following problem: given a non-zero polynomial fe Q(X) in one variable with rational coefficients, find the decomposition of f into irreducible factors in Q(X). It is well known that this is equivalent to factoring primitive polynomials feZ(X) into irreducible factors in Z(X). Here we call f~ Z(X) primitive if the greatest common divisor of its coefficients (the content of f) is 1. Our algorithm performs well in practice, cf. (8). Its running time, measured in bit operations, is O(nl2+n9(log(fD3).

We present a deterministic polynomial time algorithm for computing the zeta function of an arbitrary variety of fixed dimension over a finite field of small characteristic. One consequence of this result is an efficient method for computing the order of the group of rational points on the Jacobian of a smooth geometrically connected projective curve over a finite field of small characteristic.

This paper presents a new framework for computing Grobner bases for ideals and syzygy modules. It is proposed to work in a module that accommodates any given ideal and the corresponding syzygy module (for the given generators of the ideal). A strong Grobner basis for this module contains Grobner bases for both the ideal and the syzygy module. The main result is a simple characterization of strong Grobner bases. This characterization can detect useless S-polynomials without reductions, thus yields an efficient algorithm. It also explains all the rewritten rules used in F5 and the recent papers in the literature. Rigorous proofs are given for the correctness and finite termination of the algorithm. For any term order for an ideal, one may vary signature orders (i.e. the term orders for the syzygy module). It is shown by computer experiments on benchmark examples that signature orders based on weighted terms are much better than other signature orders. This is useful for practical computation. Also, since computing Gobner bases for syzygies is a main computational task for free resolutions in commutative algebra, the algorithm of this paper should be useful for computing free resolutions in practice.

Let f be a polynomial in one variable with integer coefficients, and p a prime. A solution of the congruence f(x)≡0(modp) may branch out into several solutions modulo p 2 , or it may be extended to just one solution, or it may not extend to any solution. Again, a solution modulo p 2 may or may not be extendable to solutions modulo p 3 , etc. In this way one obtains the “solution tree” T=T(f) of congruences modulo p λ for λ=1,2,.... The authors deal with the following questions: What is the structure of such solution trees? How many “isomorphism classes” are there of trees T(f) when f ranges through polynomials of bounded degree and height? They also give bounds for the number of solutions of congruences f(x)≡0(modp λ ) in terms of p,λ and the degree of f.

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.

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.

This article is devoted to algorithms for computing all the roots of a univariate polynomial with coefficients in a complete commutative Noetherian unramified regular local domain, which are given to a fixed common finite precision. We study the cost of our algorithms, discuss their practical performances, and apply our results to the Guruswami and Sudan list decoding algorithm over Galois rings.

We present a polynomial-time algorithm for computing the zeta function
of a smooth projective hypersurface of degree d over a finite field of characteristic
p, under the assumption that p is a suitably small odd prime and does not divide d.
This improves significantly upon an earlier algorithm of the author and Wan which
is only polynomial-time when the dimension is fixed.

Let $f(x)$ be a separable polynomial over a local field. Montes algorithm
computes certain approximations to the different irreducible factors of $f(x)$,
with strong arithmetic properties. In this paper we develop an algorithm to
improve any one of these approximations, till a prescribed precision is
attained. The most natural application of this "single-factor lifting" routine
is to combine it with Montes algorithm to provide a fast polynomial
factorization algorithm. Moreover, the single-factor lifting algorithm may be
applied as well to accelerate the computational resolution of several global
arithmetic problems in which the improvement of an approximation to a single
local irreducible factor of a polynomial is required.

We obtain randomized algorithms for factoring degree n univariate polynomials over F_q requiring O(n^(1.5+o(1)) log^(1+o(1))q + n^(1+o(1)) log^(2+o(1))q) bit operations. When 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 log q ≥ 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^((ω+1)/2)) field operations, where ω 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 α implies an algorithm for the other with exponent α+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.

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.

This paper characterizes all the factorizations of a polynomial with coefficients in the ring Z n where n is a composite number. We give algorithms to compute such factorizations along with algebraic classifications. Contents 1 Introduction 3 1.1 Circuit complexity theory . . . . . . . . . . . . . . . . . . . . . . 3 2 Some Important Tools in Z n [x] 4 2.1 The Z n [x] phenomena . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 The Chinese Remainder Theorem . . . . . . . . . . . . . . . . . . 5 2.3 Irreducibility criteria in Z p k [x] . . . . . . . . . . . . . . . . . . . 7 2.4 Hensel's Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.5 A naive approach to factoring . . . . . . . . . . . . . . . . . . . . 11 3 The Case of Small Discriminants 12 3.1 The p-adic numbers . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 Resultants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.3 The correspondence to factoring over the p-adics . . . . ....

The number of solutions in finite fields of a system of polynomial equations obeys a very strong regularity, reflected for example by the rationality of the zeta function of an algebraic variety defined over a finite field, or the modularity of Hasse-Weil's $L$-function of an elliptic curve over $\Q$. Since two decades, efficient methods have been invented to compute effectively this number of solutions, notably in view of cryptographic applications. This expos\'e presents some of these methods, generally relying on the use of Lefshetz's trace formula in an adequate cohomology theory and discusses their respective advantages. ----- Le nombre de solutions dans les corps finis d'un syst\`eme d'\'equations polynomiales ob\'eit \`a une tr\`es forte r\'egularit\'e, refl\'et\'ee par exemple par la rationalit\'e de la fonction z\^eta d'une vari\'et\'e alg\'ebrique sur un corps fini, ou la modularit\'e de la fonction $L$ de Hasse-Weil d'une courbe elliptique sur $\Q$. Depuis une vingtaine d'ann\'ees des m\'ethodes efficaces ont \'et\'e invent\'ees pour calculer effectivement ce nombre de solutions, notamment en vue d'applications \`a la cryptographie. L'expos\'e en pr\'esentera quelques-unes, g\'en\'eralement fond\'ees l'utilisation de la formule des traces de Lefschetz dans une th\'eorie cohomologique convenable, et expliquera leurs avantages respectifs. Comment: S\'eminaire Bourbaki, 50e ann\'ee, expos\'e 968, Novembre 2006. 48 pages, in french. Final version to appear in Ast\'erisque

We present a p-adic algorithm to compute the zeta function of a nondegenerate curve over a finite field using Monsky-Washnitzer cohomology.
The paper vastly generalizes previous work since in practice all known cases, for example, hyperelliptic, superelliptic, and
Cab curves, can be transformed to fit the nondegenerate case. For curves with a fixed Newton polytope, the property of being
nondegenerate is generic, so that the algorithm works for almost all curves with given Newton polytope. For a genus g curve over Fpn, the expected running time is Õ(n3g6 + n2g6.5), whereas the space complexity amounts to Õ(n3g4), assuming p is fixed.

In August 2002, Agrawal, Kayal and Saxena announced the first deterministic and polynomial-time primality-testing algorithm. For an input n, the Agarwal-Kayal-Saxena (AKS) algorithm runs in time Omicron (log(7.5) n) (heuristic time Omicron (log(6) n)). Verification takes roughly the same amount of time. On the other hand, the Elliptic Curve Primality Proving algorithm (ECPP) runs in random heuristic time 6 (log6 n) (some variant has heuristic time complexity Omicron(log(4) n)) and generates certificates which can be easily verified. However, it is hard to analyze the provable time complexity of ECPP even for a small portion of primes. More recently, Berrizbeitia gave a variant of the AKS algorithm, in which some primes (of density Omicron(1/log(2) n)) cost much less time to prove than a general prime does. Building on these celebrated results, this paper explores the possibility of designing a randomized primality-proving algorithm based on the AKS algorithm. We first generalize Berrizbeitia's algorithm to one which has higher density (Omega (1/log log n)) of primes whose primality can be proved in time complexity 6(log4 n). For a general prime, one round of ECPP is deployed to reduce its primality proof to the proof of a random easily

Journal of Symbolic Computation, special issue in honor of 60th birthday of

- Maurice Rojas
- Korben Rusek

J. Maurice Rojas; and Korben Rusek, "Faster p-adic Feasibility for
Certain Multivariate Sparse Polynomials," Journal of Symbolic Computation, special issue in honor of
60th birthday of Joachim von zur Gathen, vol. 47, no. 4, pp. 454-479 (April 2012).

- Eric Bach
- Jeff Shallit

Eric Bach and Jeff Shallit, Algorithmic Number Theory, Vol. I: Efficient Algorithms, MIT Press, Cambridge, MA, 1996.

Counting Roots for Polynomials Modulo Prime Powers

- Maurice Rojas
- Daqing Wan

J. Maurice Rojas; and Daqing Wan, "Counting Roots for Polynomials Modulo
Prime Powers," Proceedings of ANTS XIII (Algorithmic Number Theory Symposium, July 16-20, 2018,
University of Wisconsin, Madison), to appear.

Efficient Factoring [of ] Polynomials over Local Fields and its Applications

- Alexander L Chistov

Alexander L. Chistov, "Efficient Factoring [of ] Polynomials over Local Fields and its Applications," in
I. Satake, editor, Proc. 1990 International Congress of Mathematicians, pp. 1509-1519, Springer-Verlag,
1991.

- Henri Cohen

Henri Cohen, A course in computational algebraic number theory, Graduate Texts in Mathematics, 138,
Springer-Verlag, Berlin, 1993.

Algorithmic number theory: lattices, number fields, curves and cryptography

- Daqing Wan

Daqing Wan, "Algorithmic theory of zeta functions over finite fields," Algorithmic number theory:
lattices, number fields, curves and cryptography, pp. 551-578, Math. Sci. Res. Inst. Publ., 44, Cambridge
Univ. Press, Cambridge, 2008.