... German cryptographic procedures. I enjoyed interesting discussions with Joachim von zur Gathen, my father. Jamshid Shokrollahi helped with the photographs, ...
Tópico(s): German History and Society
2007 - Taylor & Francis | Cryptologia
... Die For" ( SIGACT News Complexity Theory Column 16), Joachim von zur Gathen, the editor-in-chief of computational complexity, has written me pointing out, quite correctly, that I cheated his journal of a point. His journal, in the issue checked, does include the email address of each author. I apologize for the missing point. Joachim also asked me to mention that, while the ...
Tópico(s): Computability, Logic, AI Algorithms
1997 - Association for Computing Machinery | ACM SIGACT News
Joachim von zur Gathen, Malte Sieveking,
Consider a system of linear equalities and inequalities with integer coefficients. We describe the set of rational solutions by a finite generating set of solution vectors. The entries of these vectors can be bounded by the absolute value of a certain subdeterminant. The smallest integer solution of the system has coefficients not larger than this subdeterminant times the number of indeterminates. Up to the latter factor, the bound is sharp.
Tópico(s): Advanced Graph Theory Research
1978 - American Mathematical Society | Proceedings of the American Mathematical Society
Joachim von zur Gathen, Malte Sieveking,
Consider a system of linear equalities and inequalities with integer coefficients. We describe the set of rational solutions by a finite generating set of solution vectors. The entries of these vectors can be bounded by the absolute value of a certain subdeterminant. The smallest integer solution of the system has coefficients not larger than this subdeterminant times the number of indeterminates. Up to the latter factor, the bound is sharp.
1978 - American Mathematical Society | Proceedings of the American Mathematical Society
Joachim von zur Gathen, Malte Sieveking,
e n wiederum sine v o l l s t ~n d i g e Sprache in NP ergebsn.Im l e t z t e n T a i l w i r d die R e d u z i b i l i t ~t yon ganzen Zahlen und von Polynomen b e h a n d e l t .Die Bezeichnungen sind die g l e i c h e n wie in I I I , wo auch die O-1-Kodierungen
Tópico(s): Coding theory and cryptography
1976 - Springer Science+Business Media | Lecture notes in computer science
Allan Borodin, Joachim von zur Gathen, John E. Hopcroft,
Parallel algorithms to compute the determinant and characteristic polynomial of matrices and the gcd of polynomials are presented. The rank of matrices and solutions of arbitrary systems of linear equations are computed by parallel Las Vegas algorithms. All algorithms work over arbitrary fields. They run in parallel time O(log2 n) (where n is the number of inputs) and use a polynomial number of processors.
Tópico(s): Numerical Methods and Algorithms
1982 - Academic Press | Information and Control
Fast parallel algorithms are presented for the following problems in symbolic manipulation of univariate polynomials: computing all entries of the extended Euclidean scheme of two polynomials over an arbitrary field, gcd and 1cm of many polynomials, factoring polynomials over finite fields, and the squarefree decomposition of polynomials over fields of characteristic zero and over finite fields. For the following estimates, assume that the input polynomials have degree at most n, and the finite ...
Tópico(s): Cryptography and Residue Arithmetic
1984 - Society for Industrial and Applied Mathematics | SIAM Journal on Computing
We give a computational description of Hensel’s method for lifting approximate factorizations of polynomials. The general setting of valuation rings provides the framework for this and the other results of the paper. We describe a Newton method for solving algebraic and differential equations. Finally, we discuss a fast algorithm for factoring polynomials via computing short vectors in modules.
Tópico(s): Complexity and Algorithms in Graphs
1984 - American Mathematical Society | Mathematics of Computation
Tópico(s): Cryptographic Implementations and Security
1985 - Elsevier BV | Journal of Computer and System Sciences
Joachim von zur Gathen, Erich Kaltofen,
We present a probabilistic algorithm that finds the irreducible factors of a bivariate polynomial with coefficients from a finite field in time polynomial in the input size, i.e., in the degree of the polynomial and log (cardinality of field). The algorithm generalizes to multivariate polynomials and has polynomial running time for densely encoded inputs. A deterministic version of the algorithm is also discussed, whose running time is polynomial in the degree of the input polynomial and the size ...
Tópico(s): Cryptography and Residue Arithmetic
1985 - American Mathematical Society | Mathematics of Computation
Joachim von zur Gathen, Erich Kaltofen,
This paper presents a probabilistic reduction for factoring polynomials from multivariate to the bivariate case, over an arbitrary (effectively computable) field. It uses an expected number of field operations (and certain random choices) that is polynomial in the size of sparse representations of input plus output, provided the number of irreducible factors is bounded. We thus obtain probabilistic polynomial-time factoring procedures over algebraic number fields and over finite fields. The reduction ...
Tópico(s): Cryptography and Residue Arithmetic
1985 - Elsevier BV | Journal of Computer and System Sciences
Tópico(s): Commutative Algebra and Its Applications
1990 - Elsevier BV | Journal of Symbolic Computation
Joachim von zur Gathen, Mark Giesbrecht,
Tópico(s): Electromagnetic Scattering and Analysis
1990 - Elsevier BV | Journal of Symbolic Computation
If g and h are polynomials of degrees r and s over a field, their functional composition f = g(h) has degree n = rs. The functional decomposition problem is: given f of degree n = rs, determine whether such g and h exist, and, in the affirmative case, compute them. An apparently difficult case is when the characteristic p of the ground field divides r. This paper presents a polynomial-time partial solution for this "wild" case; it works, e.g., when p2 ⌿ r.
Tópico(s): Algebraic Geometry and Number Theory
1990 - Elsevier BV | Journal of Symbolic Computation
Fast parallel computations are presented for large powers modulo an element that has only small prime factors. They work for integers and polynomials over small finite fields.MSC codes68C2010A3012C05MSC codesparallel processingcircuit depthalgebraic computingsymbolic manipulationpowers of integerspowers of polynomials
Tópico(s): Numerical Methods and Algorithms
1987 - Society for Industrial and Applied Mathematics | SIAM Journal on Computing
The n×n permanent is not a projection of the m×m determinant if m ⩽ √2n− 6√n.
Tópico(s): Random Matrices and Applications
1987 - Elsevier BV | Linear Algebra and its Applications
For those prime numbers p, for which all prime factors of p−1 are small, the two problems of finding a primitive element modulo p and of factoring univariate polynomials over finite fields of characteristic p are (deterministically) polynomial-time equivalent. Assuming the Extended Riemann Hypothesis, they can be solved in polynomial time.
Tópico(s): Cryptography and Residue Arithmetic
1987 - Elsevier BV | Theoretical Computer Science
An account of Valiant's theory of p-computable versus p-definable polynomials, an arithmetic analogue of the Boolean theory of P versus NP, is presented, with detailed proofs of Valiant's central results.
Tópico(s): semigroups and automata theory
1987 - Elsevier BV | Journal of Symbolic Computation
A survey of parallel algorithms for algebraic problems is presented.
Tópico(s): Polynomial and algebraic computation
1986 - Springer Science+Business Media | Lecture notes in computer science
Several methods of computing irreducible polynomials over finite fields are presented. If preprocessing, depending only on p , is allowed for free, then an irreducible polynomial of degree at least n over Z p can be computed deterministically with O(n logp), i.e. O(output size), bit operations. The estimates for the preprocessing time depend on unproven conjectures.
Tópico(s): Cryptography and Residue Arithmetic
1986 - Springer Science+Business Media | Lecture notes in computer science
Representations of univariate rational functions over a given base of polynomials are considered, and a fast parallel algorithm for converting from one base representation to another is given. Special cases of this conversion include the following symbolic manipulation problems: Taylor expansion, partial fraction decomposition, Chinese remainder algorithm, elementary symmetric functions, Padé approximation, and various interpolation problems. If n is the input size, then all algorithms run in parallel ...
Tópico(s): Digital Filter Design and Implementation
1986 - Society for Industrial and Applied Mathematics | SIAM Journal on Computing
Tópico(s): Computability, Logic, AI Algorithms
1988 - Annual Reviews | Annual Review of Computer Science
Joachim von zur Gathen, G. Seroussi,
We compare the two computational models of Boolean circuits and arithmetic circuits in cases where they both apply, namely the computation of polynomials over the rational numbers or over finite fields. Over Q and finite fields, Boolean circuits can simulate arithmetic circuits efficiently with respect to size. Over finite fields of small characteristic, the two models are equally powerful when size is considered, but Boolean circuits are exponentially more powerful than arithmetic circuits with ...
Tópico(s): Machine Learning and Algorithms
1991 - Elsevier BV | Information and Computation
Let q be a prime power, Fq a field with q elements, f ∈ Fq[x] a polynomial of degree n ≥ 1, V(f) = #f(Fq) the number of different values f(α) of f, with α ∈ Fq, and p = q – V(f). It is shown that either ρ = 0 or 4n4 > q or 2pn > q. Hence, if q is "large" and f is not a permutation polynomial, then either n or ρ is "large".
Tópico(s): Cryptography and Data Security
1991 - Cambridge University Press | Bulletin of the Australian Mathematical Society
Tópico(s): Cellular Automata and Applications
1991 - Birkhäuser | Computational Complexity
Joachim von zur Gathen, Victor Shoup,
Tópico(s): Cryptography and Residue Arithmetic
1992 - Birkhäuser | Computational Complexity
... 65111 0541.68019 CrossrefISIGoogle Scholar[2] Allan Borodin, , Joachim von zur Gathen and , John Hopcroft, Fast parallel matrix and GCD ...
Tópico(s): Logic, programming, and type systems
1992 - Society for Industrial and Applied Mathematics | SIAM Journal on Computing
Joachim von zur Gathen, Jürgen Gerhard,
We describe algorithms for polynomial factorization over the binary field ${\mathbb F}_2$, and their implementation. They allow polynomials of degree up to $250 000$ to be factored in about one day of CPU time, distributing the work on two processors.
Tópico(s): Polynomial and algebraic computation
2002 - American Mathematical Society | Mathematics of Computation
Joachim von zur Gathen, Jaime Gutiérrez, Rosario Rubio,
Tópico(s): Coding theory and cryptography
2003 - Springer Science+Business Media | Applicable Algebra in Engineering Communication and Computing
A necessary condition for irreducibility of a trinomial over a finite field, based on classical results of Stickelberger and Swan, is established. It is applied in the special case $\mathbb {F}_{3}$, and some experimental discoveries are reported.
Tópico(s): graph theory and CDMA systems
2003 - American Mathematical Society | Mathematics of Computation