An Analysis of the Generalized Binary GCD Algorithm
نویسنده
چکیده
In this paper we analyze a slight modification of Jebelean’s version of the k-ary GCD algorithm. Jebelean had shown that on n-bit inputs, the algorithm runs in O(n) time. In this paper, we show that the average running time of our modified algorithm is O(n/ logn). This analysis involves exploring the behavior of spurious factors introduced during the main loop of the algorithm. We also introduce a Jebeleanstyle left-shift k-ary GCD algorithm with a similar complexity that performs well in practice.
منابع مشابه
Computational Number Theory and Applications to Cryptography
• Greatest common divisor (GCD) algorithms. We begin with Euclid’s algorithm, and the extended Euclidean algorithm [2, 12]. We will then discuss variations and improvements such as Lehmer’s algorithm [14], the binary algorithms [12], generalized binary algorithms [20], and FFT-based methods. We will also discuss how to adapt GCD algorithms to compute modular inverses and to compute the Jacobi a...
متن کاملComparing several GCD algorithms
0 binary, I-binary: The binary GCD algorithm ([lS]) and its improvement for multidigit integers (Gosper, see [12]). We compare the executron times of several algoixtliiiis for computing the G‘C‘U of arbitrary precasion iirlegers. These algorithms are the known ones (Euclidean, brnary, plus-mrnus), and the improved variants of these for multidigit compzltation (Lehmer and similar), as well as ne...
متن کاملLocally GCD domains and the ring $D+XD_S[X]$
An integral domain $D$ is called a emph{locally GCD domain} if $D_{M}$ is aGCD domain for every maximal ideal $M$ of $D$. We study somering-theoretic properties of locally GCD domains. E.g., we show that $%D$ is a locally GCD domain if and only if $aDcap bD$ is locally principalfor all $0neq a,bin D$, and flat overrings of a locally GCD domain arelocally GCD. We also show that the t-class group...
متن کاملBinary GCD Like Algorithms for Some Complex Quadratic Rings
On the lines of the binary gcd algorithm for rational integers, algorithms for computing the gcd are presented for the ring of integers in Q( √ d) where d ∈ {−2,−7,−11,−19}. Thus a binary gcd like algorithm is presented for a unique factorization domain which is not Euclidean (case d = −19). Together with the earlier known binary gcd like algorithms for the ring of integers in Q( √−1) and Q(√−3...
متن کاملFPGA Implementation of an Extended Binary GCD Algorithm for Systolic Reduction of Rational Numbers
We present the FPGA implementation of an extension of the binary plus–minus systolic algorithm which computes the GCD (greatest common divisor) and also the normal form of a rational number, without using division. A sample array for 8 bit operands consumes 83.4% of an Atmel 40K10 chip and operates at 25 MHz.
متن کامل