Related papers: Square Root Computation In Finite Fields
We present an algorithm for computing the integral closure of a reduced ring that is finitely generated over a finite field.
In 2015, Abatzoglou, Silverberg, Sutherland, and Wong presented a framework for primality proving algorithms for special sequences of integers using an elliptic curve with complex multiplication. They applied their framework to obtain…
We design and analyze an algorithm for computing solutions with coefficients in a finite field $\mathbb{F}_q$ of underdetermined systems defined over $\mathbb{F}_q$. The algorithm is based on reductions to zero-dimensional searches. The…
We consider the problem of optimizing a multivariate quadratic function where each decision variable is constrained to be a complex $m$'th root of unity. Such problems have applications in signal processing, MIMO detection, and the…
We design and analyze an algorithm for computing rational points of hypersurfaces defined over a finite field based on searches on "vertical strips", namely searches on parallel lines in a given direction. Our results show that, on average,…
Some geometry on non-singular cubic curves, mainly over finite fields, is surveyed. Such a curve has 9,3,1 or 0 points of inflexion, and cubic curves are classified accordingly. The group structure and the possible numbers of rational…
A finite semifield is a division algebra over a finite field where multiplication is not necessarily associative. We consider here the complexity of the multiplication in small semifields and finite field extensions. For this operation, the…
In this paper we provide, first, a general symbolic algorithm for computing the symmetries of a given rational surface, based on the classical differential invariants of surfaces, i.e. Gauss curvature and mean curvature. In practice, the…
In this paper we describe an algorithm for implicitizing rational hypersurfaces in case there exists at most a finite number of base points. It is based on a technique exposed in math.AG/0210096, where implicit equations are obtained as…
In this paper, we study the arithmetics of skew polynomial rings over finite fields, mostly from an algorithmic point of view. We give various algorithms for fast multiplication, division and extended Euclidean division. We give a precise…
We describe some "unrestricted" algorithms which are useful for the computation of elementary and special functions when the precision required is not known in advance. Several general classes of algorithms are identified and illustrated by…
We introduce new rounding methods to improve the accuracy of finite precision quantum arithmetic. These quantum rounding methods are applicable when multiple samples are being taken from a quantum program. We show how to use multiple…
This paper addresses the numerical solution of the matrix square root problem. Two fixed point iterations are proposed by rearranging the nonlinear matrix equation $A - X^2 = 0$ and incorporating a positive scaling parameter. The proposals…
How can complexity theory and algorithms benefit from practical advances in computing? We give a short overview of some prior work using practical computing to attack problems in computational complexity and algorithms, informally describe…
In recent decades, the field of quantum computing has experienced remarkable progress. This progress is marked by the superior performance of many quantum algorithms compared to their classical counterparts, with Shor's algorithm serving as…
The square root velocity transformation provides a convenient and numerically efficient approach to functional and shape data analysis of curves. We study fundamental geometric properties of curves under this transformation. Moreover,…
This paper shows that, for matrix multiplications and convolutions, it is possible to asymptotically replace each real multiplication with a single squaring operation. Similarly, a single complex multiplication can be replaced with 3…
We introduce an arbitrary order, computationally efficient method to smooth corners on curves in the plane, as well as edges and vertices on surfaces in $\mathbb R^3$. The method is local, only modifying the original surface in a…
We propose an improved algorithm for finding roots of polynomials over finite fields. This makes possible significant speedup of the decoding process of Bose-Chaudhuri-Hocquenghem, Reed-Solomon, and some other error-correcting codes.
Non-linear polynomial systems over finite fields are used to model functional behavior of cryptosystems, with applications in system security, computer cryptography, and post-quantum cryptography. Solving polynomial systems is also one of…