Polynomial GCD Computation Research Articles

Some results on counting roots of polynomials and the Sylvester resultant.

We present two results, the first on the distribution of the roots of a polynomial over the ring of integers modulo n and the second on the distribution of the roots of the Sylvester resultant of two multivariate polynomials. The second result has application to polynomial GCD computation and solving polynomial diophantine equations.

Polynomial Algebra in Form 4

New features of the symbolic algebra package Form 4 are discussed. Most importantly, these features include polynomial factorization and polynomial gcd computation. Examples of their use are shown. One of them is an exact version of Mincer which gives answers in terms of rational polynomials and 5 master integrals.

A New Characterization of Semi-bent and Bent Functions on Finite Fields*

We present a new characterization of semi-bent and bent quadratic functions on finite fields. First, we determine when a GF(2)-linear combination of Gold functions Tr(x2i+1) is semi-bent over GF(2n), n odd, by a polynomial GCD computation. By analyzing this GCD condition, we provide simpler characterizations of semi-bent functions. For example, we deduce that all linear combinations of Gold functions give rise to semi-bent functions over GF(2p) when p belongs to a certain class of primes. Second, we generalize our results to fields GF(pn) where p is an odd prime and n is odd. In that case, we can determine whether a GF(p)-linear combination of Gold functions Tr(xpi+1) is (generalized) semi-bent or bent by a polynomial GCD computation. Similar to the binary case, simple characterizations of these p-ary semi-bent and bent functions are provided.

A New Class of Bent Functions

In this letter, using techniques from linear algebra and coding theory, we characterize the quadratic Boolean functions represented by trace. We show that a linear combination of trace-terms over finite field can be determined to be bent by a polynomial GCD computation. Then we derive some new families of bent functions.

Interval Arithmetic in Cylindrical Algebraic Decomposition

Cylindrical algebraic decomposition requires many very time consuming operations, including resultant computation, polynomial factorization, algebraic polynomial gcd computation and polynomial real root isolation. We show how the time for algebraic polynomial real root isolation can be greatly reduced by using interval arithmetic instead of exact computation. This substantially reduces the overall time for cylindrical algebraic decomposition.

Effective Computation of Rational Approximants and Interpolants

This paper considers the problem of effective algorithms for some problems having structured coefficient matrices. Examples of such problems include rational approximation and rational interpolation. The corresponding coefficient matrices include Hankel, Toeplitz and Vandermonde-like matrices. Effective implies that the algorithms studied are suitable for implementation in either a numeric environment or else a symbolic environment. The paper includes two algorithms for the computation of rational interpolants which are both effective in symbolic environments. The algorithms use arithmetic that is free of fractions but at the same time control the growth of coefficients during intermediate computations. One algorithm is a look-around procedure which computes along a path of closest normal points to an offdiagonal path while the second computes along an arbitrary path using a look-ahead strategy. Along an antidiagonal path the look-ahead recurrence is closely related to the Subresultant PRS algorithm for polynomial GCD computation. Both algorithms are an order of magnitude faster than alternative methods which are effective in symbolic environments.

VLSI array synthesis for polynomial GCD computation and application to finite field division

Many practical algorithms have dynamic (or data-dependent) dependency structure in their computation, which is not desirable for VLSI hardware implementation. Polynomial GCD computation by Euclid's algorithm is a typical example of dynamic dependency. In this paper, we use an algorithmic transformation technique to derive static (or data-independent) dependencies for Euclid's GCD algorithm. The resulting algorithm is mapped to a linear systolic array which is area-efficient and achieves maximum throughput with pipelining. It has m/sub 0/+n/sub 0/+1 processing elements, where m/sub 0/ and n/sub 0/ are degrees of two polynomials. We have applied the technique to the extended GCD algorithm and developed a systolic finite field divider, which can be efficiently used in decoding a variety of error correcting codes. >

GCDHEU: Heuristic polynomial GCD algorithm based on integer GCD computation

A heuristic algorithm, GCDHEU, is described for polynomial GCD computation over the integers. The algorithm is based on evaluation at a single large integer value (for each variable), integer GCD computation, and a single-point interpolation scheme. Timing comparisons show that this algorithm is very efficient for most univariate problems and it is also the algorithm of choice for many problems in up to four variables.