Related papers: Qin's Algorithm, Continued Fractions and 2-dimensi…
Modulo inverse is an important arithmetic operation. Many famous algorithms in public key cryptography require to compute modulo inverse. It is argued that the method of DaYan deriving one of Jiushao Qin provides the most concise and…
Using an adaptation of Qin Jiushao's method from the 13th century, it is possible to prove that a system of linear modular equations a(i,1) x(i) + ... + a(i,n) x(n) = b(i) mod m(i), i=1, ..., n has integer solutions if m(i)>1 are pairwise…
This paper explores the ability of the Chinese Remainder Theorem formalism to model Montgomery-type algorithms. A derivation of CRT based on Qin's Identity gives Montgomery reduction algorithm immediately. This establishes a unified…
Our aim is to find a complex continued fraction algorithm finding all the best Diophantine approximations to a complex number. Using the sequence of minimal vectors in a two dimensional lattice over Gaussian integers, we obtain an algorithm…
This paper introduces two forms of modular inverses and proves their reciprocity formulas respectively. These formulas are then applied to formulate new and generalized algorithm for computing these modular inverses. The same algorithm is…
In this paper, some issues concerning the Chinese remaindering representation are discussed. Some new converting methods, including an efficient probabilistic algorithm based on a recent result of von zur Gathen and Shparlinski…
Since the invention of the famous LLL algorithm, lattice reduction has been an extremely useful tool in computational number theory. By construction, the LLL algorithm deals with lattices living in a vector space endowed with a positive…
We propose a generic design for Chinese remainder algorithms. A Chinese remainder computation consists in reconstructing an integer value from its residues modulo non coprime integers. We also propose an efficient linear data structure, a…
We propose a generic design for Chinese remainder algorithms. A Chinese remainder computation consists in reconstructing an integer value from its residues modulo non coprime integers. We also propose an efficient linear data structure, a…
Inspired by the integrable structures appearing in weakly coupled planar N=4 super Yang-Mills theory, we study Q-operators and Yangian invariants of rational integrable spin chains. We review the quantum inverse scattering method along with…
We present several continued fraction algorithms, each of which gives an eventually periodic expansion for every quadratic element of ${\mathbb Q}_p$ over ${\mathbb Q}$ and gives a finite expansion for every rational number. We also give,…
Cartan matrices are of fundamental importance in representation theory. For algebras defined by quivers (i.e. directed graphs) with relations the computation of the entries of the Cartan matrix amounts to counting nonzero paths in the…
We first survey the current state of the art concerning the dynamical properties of multidimensional continued fraction algorithms defined dynamically as piecewise fractional maps and compare them with algorithms based on lattice reduction.…
A basic theory on the first order right and left linear quaternion differential systems (LQDS) is given systematic in this paper. To proceed the theory of LQDS we adopt the theory of column-row determinants recently introduced by the…
We introduce the notion of matrices graph, defining continued fraction algorithms where the past and the future are almost independent. We provide an algorithm to convert more general algorithms into matrices graphs. We present an algorithm…
In this paper we propose several strategies for the exact computation of the determinant of a rational matrix. First, we use the Chinese Remaindering Theorem and the rational reconstruction to recover the rational determinant from its…
The Chinese remainder theorem (CRT) provides an efficient way to reconstruct an integer from its remainders modulo several integer moduli, and has been widely applied in signal processing and information theory. Its multidimensional…
Residue arithmetic is an elegant and convenient way of computing with integers that exceed the natural word size of a computer. The algorithms are highly parallel and hence naturally adapted to quantum computation. The process differs from…
In this paper, we give a detailed account of the algorithm outlined in [1] for Feynman integral reduction and $\varepsilon$-factorised differential equations. The algorithm consists of two steps. In the first step, we use a new geometric…
We show that the additive-slow-Farey version of the traditional continued fractions algorithm has a natural interpretation as a method for producing integer partitions of a positive number $n$ into two smaller numbers, with multiplicity. We…