GCD Computation Research Articles

FIRE 6.5: Feynman integral reduction with new simplification library

FIRE is a program which performs integration-by-parts (IBP) reductions of Feynman integrals. Originally, the C++ version of FIRE relies on the computer algebra system Fermat by Robert Lewis to simplify rational functions. We present an upgrade of FIRE which incorporates a new library FUEL initially described in a separate publication, which enables a flexible choice of third-party computer algebra systems as simplifiers, as well as efficient communications with some of the simplifiers as C++ libraries rather than through Unix pipes. We achieve significant speedups for IBP reductions of Feynman integrals involving many kinematic variables, when using an open source backend based on FLINT newly added in this work, or the Symbolica backend developed by Ben Ruijl as a potential successor of FORM. Program summaryProgram title: FIRE, version 6.5 (FIRE 6.5)CPC Library link to program files:https://doi.org/10.17632/cy6k69pb3y.2Developer's repository link:https://gitlab.com/feynmanintegrals/fire.gitLicensing provisions: GPLv2Programming language:Wolfram Mathematica 8.0 or higher, C++17Supplementary material: See linked repository for installation instructions.Journal reference of previous version: Comput. Phys. Commun. 247 (2020) 106877Does the new version supersede the previous version?: Yes.Reasons for the new version: The new version no longer relies on a single computer algebra system, Fermat [1], but instead allows a flexible choice of several systems, some of which offer higher performance, especially when the number of variables is large.Summary of revisions: A new library FUEL [2] is used as a core component of the new version of FIRE to access different computer algebra systems as simplifiers of rational function expressions. Since the first release of FUEL described elsewhere, FUEL version 1.0 here has been enhanced with a new backend based on the open source library FLINT [3] that provides highly performant simplification of rational functions.Nature of problem: Feynman integrals of a given family are reduced to a finite set of master integrals, by solving linear equations arising from integration by parts, using Gaussian elimination. The coefficients of the linear equations are generally rational functions in kinematic variables and the spacetime dimension, and the simplification of such rational functions during Gaussian elimination is a key task that is improved in this upgrade of FIRE.Solution method: Computer algebra systems with state-of-the-art capabilities for polynomial GCD computations are used as simplification backends, or simplifiers in short. Due to the design of FIRE, text strings are used as the exchange format for rational functions before and after simplification. A fast C++ parser is written to parse strings into the internal format of an external simplifer, FLINT [3], with state-of-the-art performance for multivariate polynomial calculations. Similarly, the simplifier Symbolica [4] has high performance in both GCD computations and parsing, and has been integrated into FIRE.

Complete group law for genus 2 Jacobians on Jacobian coordinates

AbstractThis manuscript provides complete, inversion-free, and explicit group law formulas in Jacobian coordinates for the genus 2 hyperelliptic curves of the form $$y^2 = x^5+a_3 x^3+a_2 x^2+a_1 x+a_0$$ y 2 = x 5 + a 3 x 3 + a 2 x 2 + a 1 x + a 0 over a field K with $$\textrm{char}(K) \ne 2$$ char ( K ) ≠ 2 . The formulas do not require the use of polynomial arithmetic operations such as resultant, mod, or gcd computations but only operations in K.

An efficient algorithm deciding chaos for linear cellular automata over [formula omitted] with applications to data encryption

We provide an efficient algorithm deciding chaos for linear cellular automata (LCA) over (Z/mZ)n, a large and important class of cellular automata (CA) which may exhibit many of the complex features typical of general CA and are used in many applications. The efficiency of our algorithm is mainly due to fact that it avoids the computation of the prime factor decomposition of m which is a well-known difficult task. Instead of factoring m we make use of a new and efficient generalized technique for computing the greatest common divisor (gcd) of polynomials with coefficients not belonging to a field, which in itself is an interesting result. We wish also to emphasize that the gcd computations required by our algorithm always involve polynomials of degree at most n.We also illustrate the impact of our algorithm in real-world applications regarding the growing domain of cryptosystems, the latter being often based on LCA over (Z/mZ)n with n>1. As a matter of facts, since cryptosystems have to satisfy the so-called confusion and diffusion properties (which are ensured if the involved LCA is chaotic) our algorithm turns out to be an important tool for building chaotic LCA over (Z/mZ)n and, hence, for improving the existing methods based on them.

Profiling Attack against RSA Key Generation Based on a Euclidean Algorithm

A profiling attack is a powerful variant among the noninvasive side channel attacks. In this work, we target RSA key generation relying on the binary version of the extended Euclidean algorithm for modular inverse and GCD computations. To date, this algorithm has only been exploited by simple power analysis; therefore, the countermeasures described in the literature are focused on mitigating only this kind of attack. We demonstrate that one of those countermeasures is not effective in preventing profiling attacks. The feasibility of our approach relies on the extraction of several leakage vectors from a single power trace. Moreover, because there are known relationships between the secrets and the public modulo in RSA, the uncertainty in some of the guessed secrets can be reduced by simple tests. This increases the effectiveness of the proposed attack.

Programming of an Approximating K-Ary GCD Algorithm

In our paper, we consider the procedure for effective programming of an approximating k-ary GCD algorithm of natural numbers in the C++ programming language in the Visual Studio programming environment with the installed MPIR library of long numbers. There are also some theoretical calculations that justify the application of certain solutions. The task of GCD computation has an important efficient in different applications of Number Theory and Cybersecurity.

Affine Equivalences of Trigonometric Curves

We provide an efficient algorithm to detect whether two given trigonometric curves, i.e. two parametrized curves whose components are truncated Fourier series, in any dimension, are affinely equivalent, i.e. whether there exists an affine mapping transforming one of the curves onto the other. If the coefficients of the parametrizations are known exactly (the exact case), the algorithm boils down to univariate gcd computation, so it is efficient and fast. If the coefficients of the parametrizations are known with finite precision, e.g. floating point numbers (the approximate case), the univariate gcd computation is replaced by the computation of approximate gcds. Our experiments show that the method works well, even for high degrees.

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.

ReFORM: designing a new symbolic manipulation toolkit

Since modern-day multi-loop Feynman diagram computations often require manipulating billions of terms, taking up terabytes of memory, a powerful symbolic manipulation toolkit (SMT) is essential. The de facto solution is Form, but it has several shortcomings. In this work we present reForm, a new SMT in early development, that will handle the same workload as Form but does not have its shortcomings. We showcase some features of reForm, including Python and C APIs. Finally, we provide benchmarks for polynomial GCD computations, which show that reForm often outperforms its competitors. A link to the source code of the technical preview version is provided.

Cache-Timing Attacks on RSA Key Generation

During the last decade, constant-time cryptographic software has quickly transitioned from an academic construct to a concrete security requirement for real-world libraries. Most of OpenSSL’s constant-time code paths are driven by cryptosystem implementations enabling a dedicated flag at runtime. This process is perilous, with several examples emerging in the past few years of the flag either not being set or software defects directly mishandling the flag. In this work, we propose a methodology to analyze security-critical software for side-channel insecure code path traversal. Applying our methodology to OpenSSL, we identify three new code paths during RSA key generation that potentially leak critical algorithm state. Exploiting one of these leaks, we design, implement, and mount a single trace cache-timing attack on the GCD computation step. We overcome several hurdles in the process, including but not limited to: (1) granularity issues due to word-size operands to the GCD function; (2) bulk processing of desynchronized trace data; (3) non-trivial error rate during information extraction; and (4) limited high-confidence information on the modulus factors. Formulating lattice problem instances after obtaining and processing this limited information, our attack achieves roughly a 27% success rate for key recovery using the empirical data from 10K trials.

Numerical Proper Reparametrization of Space Curves and Surfaces

Simplifying rational parametrizations of surfaces is a basic problem in CAD (computer-aided design). Reducing their tracing index, called proper reparametrization, is an important simplification way. Most existing proper reparametrization work is symbolic. Yet in practical environments the surfaces are usually given with perturbed coefficients hence need a numerical technique of reparametrization considering the intrinsic properness of the perturbed surfaces. We present algorithms for reparametrizing a numerically rational space curve or surface. First, we provide an efficient way to find a parametric support transformation and compute a reparametrization with proper parametric support. Second, we develop a numerical algorithm to further reduce the tracing index, where numerical techniques such as sparse interpolation and approximated GCD computations are involved. We finally provide the error bound between the given rational curve/surface and our reparametrization result.

Enhancing the extended hensel construction by using Gröbner basis

Contrary to that the general Hensel construction (GHC: [3]) uses univariate initial Hensel factors, the extended Hensel construction (EHC: [8]) uses multivariate initial Hensel factors determined by the Newton polygon (see below) of the given multivariate polynomial F (x, u ) ∈ K[ x , u ], where ( u ) = ( u 1 ,..., u ℓ ), with ℓ ≥ 2, and K is a number field. The F ( x, u ) may be such that its leading coefficient may vanish at ( u ) = ( 0 ) = (0,...,0), and even may be F ( x , 0 ) = 0. The EHC was used so far for computing series expansion of multivariate algebraic function determined by F ( x, u ) = 0, at critical points [8, 5] and for factorization [4, 1] and GCD computation [7] of F ( x , u ), without shifting the origin of u . It allows us to construct efficient algorithms for sparse multivariate polynomials [1, 7]. The EHC is another and promising approach than Zippel's sparse Hensel lifting [9, 10].

Multiplicity-preserving triangular set decomposition of two polynomials

In this paper, a multiplicity-preserving triangular set decomposition algorithm is proposed for a system of two polynomials, which involves only computing the primitive polynomial remainder sequence of two polynomials once and certain GCD computations. The algorithm decomposes the unmixed variety defined by two polynomials into square free and disjoint (for non-vertical components, see Definition 4) algebraic cycles represented by triangular sets, which may have negative multiplicities. Thus, the authors can count the multiplicities of the non-vertical components. In the bivariate case, the authors give a complete algorithm to decompose the system into zeros represented by triangular sets with multiplicities. The authors also analyze the complexity of the algorithm in the bivariate case. The authors implement the algorithm and show the effectiveness of the method with extensive

Exact symbolic–numeric computation of planar algebraic curves

We present a certified and complete algorithm to compute arrangements of real planar algebraic curves. It computes the decomposition of the plane induced by a finite number of algebraic curves in terms of a cylindrical algebraic decomposition. From a high-level perspective, the overall method splits into two main subroutines, namely an algorithm denoted Bisolve to isolate the real solutions of a zero-dimensional bivariate system, and an algorithm denoted GeoTop to compute the topology of a single algebraic curve.Compared to existing approaches based on elimination techniques, we considerably improve the corresponding lifting steps in both subroutines. As a result, generic position of the input system is never assumed, and thus our algorithm never demands for any change of coordinates. In addition, we significantly limit the types of symbolic operations involved, that is, we only use resultant and gcd computations as purely symbolic operations. The latter results are achieved by combining techniques from different fields such as (modular) symbolic computation, numerical analysis and algebraic geometry.We have implemented our algorithms as prototypical contributions to the C++-project Cgal. We exploit graphics hardware to expedite the remaining symbolic computations. We have also compared our implementation with the current reference implementations, that is, Lgp and Maple’s Isolate for polynomial system solving, and Cgal’s bivariate algebraic kernel for analyses and arrangement computations of algebraic curves. For various series of challenging instances, our exhaustive experiments show that the new implementations outperform the existing ones.

Approximate Gröbner Bases, Overdetermined Polynomial Systems, and Approximate GCDs

We discuss computation of Gröbner bases using approximate arithmetic for coefficients. We show how certain considerations of tolerance, corresponding roughly to absolute and relative error from numeric computation, allow us to obtain good approximate solutions to problems that are overdetermined. We provide examples of solving overdetermined systems of polynomial equations. As a secondary feature we show handling of approximate polynomial GCD computations, using benchmarks from the literature.

Variable projection methods for approximate GCD computations

This paper presents optimization methods and software for the approximate GCD problem of multiple univariate polynomials in the weighted 2-norm. Backward error minimization and Sylvester low-rank approximation formulations of the problem are solved by the variable projection method. Optimization methods are implemented in publicly available C++ software package with an interface to MATLAB. Results on computational complexity are presented.

Small algorithms for small systems

s of Recent Doctoral Dissertations in Computer Algebra Communicated by Jeremy Johnson Each month we are pleased to present abstracts of recent doctoral dissertations in Computer Algebra and Symbolic Computation. We encourage all recent Ph.D. graduates (and their supervisors), who have defended in the past two years, to submit their abstracts for publication in CCA. Please send abstracts to the CCA editors for consideration. Author: Seyed Mohammad Mahdi Javadi Title: Efficient Algorithms for Computation with Sparse Polynomials Institution: Simon Fraser University Thesis Advisor: Michael Monagan Committee Members: Arvind Gupta, Marni Mishna External examiner: Mark Giesbrecht Defended: January 2011. The problem of interpolating a sparse polynomial has always been one of the central objects of research in the area of computer algebra. It is the key part of many algorithms such as polyno- mial GCD computation. We present a probabilistic algorithm to interpolate a sparse multivariate polynomial over a Finite Field, represented with a black box. Our algorithm modifies the Ben- Or/Tiwari algorithm from 1988 for interpolating polynomials over rings with characteristic zero to positive characteristics by doing additional probes. To interpolate a polynomial in n variables with t non-zero terms, Zippel’s algorithm interpolates one variable at a time using O(ndt) probes to the black box where d bounds the degree of the polynomial. Our new algorithm does O(nt) probes. We provide benchmarks comparing our algorithm to Zippel’s algorithm and the racing algorithm of Kaltofen/Lee. The benchmarks demonstrate that for sparse polynomials our algorithm often makes fewer probes. A key advantage in our new algorithm is, unlike the other two algorithms, it can be parallelized efficiently. Our main application for an efficient sparse interpolation algorithm is computing GCDs of polynomials. We are especially interested in polynomials over algebraic function Fields. The best GCD algorithm available is SparseModGcd, presented by Javadi and Monagan in 2006. We further improve this algorithm in three ways. First we prove that we can eliminate the trial divisions in positive characteristic. Trial divisions are the bottleneck of the algorithm for denser polynomials. Second, we give a new (and correct) solution to the normalization problem. Finally we will present a new in-place library of functions for computing GCDs of univariate polynomials over algebraic number Fields. Furthermore we present an efficient algorithm for factoring multivariate polynomials oover al- gebraic fields with multiple field extensions and parameters. Our algorithm uses Hensel lifting and extends the EEZ algorithm of Wang which was designed for factorization over the rationals. We

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.

Multi-exponentiation algorithm based on binary GCD computation and its application to side-channel countermeasure

A series of algorithms for evaluation of multi-exponentiation are proposed based on the binary greatest common divisor algorithm. The proposed algorithms are inversion free and have the capability to evaluate double or multi-exponentiation with non-fixed base numbers and exponents. They can also be employed in developing side-channel countermeasures. For n-bit double and triple exponentiation, they achieve the average complexity of 1.53n and 1.75n multiplications (including squarings), respectively. The proposed algorithms can be very useful for the implementation of many public-key cryptosystems on small devices with limited memory space, e.g., smart cards.

Nearly optimal algorithms for the decomposition of multivariate rational functions and the extended Lüroth Theorem

The extended Lüroth Theorem says that if the transcendence degree of K ( f 1 , … , f m ) / K is 1 then there exists f ∈ K ( X ¯ ) such that K ( f 1 , … , f m ) is equal to K ( f ) . In this paper we show how to compute f with a probabilistic algorithm. We also describe a probabilistic and a deterministic algorithm for the decomposition of multivariate rational functions. The probabilistic algorithms proposed in this paper are softly optimal when n is fixed and d tends to infinity. We also give an indecomposability test based on gcd computations and Newton’s polytope. In the last section, we show that we get a polynomial time algorithm, with a minor modification in the exponential time decomposition algorithm proposed by Gutierez–Rubio–Sevilla in 2001.

Lower bounds for decision problems in imaginary, norm-Euclidean quadratic integer rings

We prove lower bounds for the complexity of deciding several relations in imaginary, norm-Euclidean quadratic integer rings, where computations are assumed to be relative to a basis of piecewise-linear operations. In particular, we establish lower bounds for deciding coprimality in these rings, which yield lower bounds for gcd computations. In each imaginary, norm-Euclidean quadratic integer ring, a known binary-like gcd algorithm has complexity that is quadratic in our lower bound.