Results 1  10
of
26
Tisserand, Towards Correctly Rounded Transcendentals
 Proc. 13th IEEE Symp. Comput. Arithmetic
, 1997
"... ..."
(Show Context)
Multidigit Multiplication For Mathematicians
, 2001
"... This paper surveys techniques for multiplying elements of various commutative rings. It covers Karatsuba multiplication, dual Karatsuba multiplication, Toom multiplication, dual Toom multiplication, the FFT trick, the twisted FFT trick, the splitradix FFT trick, Good's trick, the SchönhageStr ..."
Abstract

Cited by 35 (8 self)
 Add to MetaCart
This paper surveys techniques for multiplying elements of various commutative rings. It covers Karatsuba multiplication, dual Karatsuba multiplication, Toom multiplication, dual Toom multiplication, the FFT trick, the twisted FFT trick, the splitradix FFT trick, Good's trick, the SchönhageStrassen trick, Schönhage's trick, Nussbaumer's trick, the cyclic SchönhageStrassen trick, and the CantorKaltofen theorem. It emphasizes the underlying ring homomorphisms.
Small MultiplierBased Multiplication and Division Operators for VirtexII Devices
 In FieldProgrammable Logic and Applications, volume 2438 of LNCS
, 2002
"... This paper presents integer mu l iplk( ion and division operators dedicated to VirtexII FPGAs from Xil# x. Those operators are basedonsmal l 1818 mu l ipl8 b cks avail# l e in the VirtexII device fam il . Various tradeo#s are expl ored (computation decomposition, radix, digit sets ... ) using ..."
Abstract

Cited by 14 (1 self)
 Add to MetaCart
This paper presents integer mu l iplk( ion and division operators dedicated to VirtexII FPGAs from Xil# x. Those operators are basedonsmal l 1818 mu l ipl8 b cks avail# l e in the VirtexII device fam il . Various tradeo#s are expl ored (computation decomposition, radix, digit sets ... ) using specific VHDL generators. The obtained operatorsl ead to speed improvements up to 18% for mul tiplp#.k( and 40% for division compared to standardsol tions onl y based on CLBs. 1
Asymmetric squaring formulae
, 2006
"... We present efficient squaring formulae based on the ToomCook multiplication algorithm. The latter always requires at least one nontrivial constant division in the interpolation step. We show such nontrivial divisions are not needed in the case two operands are equal for three, four and fiveway s ..."
Abstract

Cited by 13 (0 self)
 Add to MetaCart
(Show Context)
We present efficient squaring formulae based on the ToomCook multiplication algorithm. The latter always requires at least one nontrivial constant division in the interpolation step. We show such nontrivial divisions are not needed in the case two operands are equal for three, four and fiveway squarings. Our analysis shows that our 3way squaring algorithms have much less overhead than the best known 3way ToomCook algorithm. Our experimental results show that one of our new 3way squaring methods performs faster than mpz_mul() in GNU multiple precision library (GMP) for squaring integers of approximately 2400–6700 bits on Pentium IV Prescott 3.2GHz. For squaring in Z[x], our 3way squaring algorithms are much superior to other known squaring algorithms for small input size. In addition, we present 4way and 5way squaring formulae which do not require any constant divisions by integers other than a power of 2. Under some reasonable assumptions, our 5way squaring formula is faster than the recently proposed Montgomery’s 5way Karatsubalike formulae. Keywords: Squaring, Karatsuba algorithm, Toom
Multiplication by an Integer Constant
, 2001
"... We present and compare various algorithms, including a new one, allowing to perform multiplications by integer constants using elementary operations. Such algorithms are useful, as they occur in several problems, such as the ToomCooklike algorithms to multiply large multipleprecision integers, th ..."
Abstract

Cited by 10 (0 self)
 Add to MetaCart
We present and compare various algorithms, including a new one, allowing to perform multiplications by integer constants using elementary operations. Such algorithms are useful, as they occur in several problems, such as the ToomCooklike algorithms to multiply large multipleprecision integers, the approximate computation of consecutive values of a polynomial, and the generation of integer multiplications by compilers.
Robust multiplexed codes for compression of heterogeneous data
 n o 4, Apr. 2005, p. 13931403. TEMICS 27
"... Abstract — Compression systems of real signals (images, video, audio) generate sources of information with different levels of priority which are then encoded with variable length codes (VLC). This paper addresses the issue of robust transmission of such VLC encoded heterogeneous sources over error ..."
Abstract

Cited by 6 (1 self)
 Add to MetaCart
Abstract — Compression systems of real signals (images, video, audio) generate sources of information with different levels of priority which are then encoded with variable length codes (VLC). This paper addresses the issue of robust transmission of such VLC encoded heterogeneous sources over errorprone channels. VLCs are very sensitive to channel noise: when some bits are altered, synchronization losses can occur at the receiver. This paper describes a new family of codes, called multiplexed codes, that confine the desynchronization phenomenon to low priority data while asymptotically reaching the entropy bound for both (low and high priority) sources. The idea consists in creating fixed length codes for high priority information and in using the inherent redundancy to describe low priority data, hence the name multiplexed codes. Theoretical and simulation results reveal a very high error resilience at almost no cost in compression efficiency. Index Terms — source coding, variable length codes, data compression, data communication, entropy codes I.
The Table Maker's Dilemma
, 1998
"... The Table Maker's Dilemma is the problem of always getting correctly rounded results when computing the elementary functions. After a brief presentation of this problem, we present new developments that have helped us to solve this problem for the doubleprecision exponential function in a s ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
The Table Maker's Dilemma is the problem of always getting correctly rounded results when computing the elementary functions. After a brief presentation of this problem, we present new developments that have helped us to solve this problem for the doubleprecision exponential function in a small domain. These new results show that this problem can be solved, at least for the doubleprecision format, for the most usual functions. Keywords: Table Maker's Dilemma, Elementary Functions, Correct Rounding, FloatingPoint Arithmetic. R#sum# Le dilemme du concepteur de tables est le probl#me de toujours fournir des r#sultats arrondis correctement lors du calcul de fonctions #l#mentaires. Apr#s une br#ve pr#sentation du probl#me, nous pr#sentons de nouveaux r#sultats qui permettent de r#soudre ce probl#me pour l'exponentielle en double pr#cision dans un petit domaine. Ces r#sultats montrent que le probl#me peut #tre r#solu, au moins pour le format double pr#cision, pour la plupart d...
Retiming DAGs
, 1998
"... This paper is devoted to a lowcomplexity algorithm for retiming circuits without cycles, i.e. whose network graph is a Direct Acyclic Graph (DAG). On one hand DAGs have a great practical importance, as shown by the online arithmetic circuits used as a target application in this paper. On the o ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
This paper is devoted to a lowcomplexity algorithm for retiming circuits without cycles, i.e. whose network graph is a Direct Acyclic Graph (DAG). On one hand DAGs have a great practical importance, as shown by the online arithmetic circuits used as a target application in this paper. On the other hand retiming is a costly design optimization technique, in particular when applied to large circuits. Hence the need to design a specialized retiming algorithm to handle DAGs more eciently than generalpurpose retiming algorithms. Our algorithm dramatically improves on current solutions in the literature: we gain an order of magnitude in the worstcase complexity, and we show convincing experimental results at the end of the paper.
VLSI Architectures and Arithmetic Operations with Application to the Fermat Number Transform
 Linköping Studies in Science and Technology 425 (dissertation
, 1996
"... and to our children ..."
(Show Context)
Fast Modular Reduction With Precomputation
 In Proceedings of KoreaJapan Joint Workshop on Information Security and Cryptology, Lecture
"... Multiplication and modular reduction of long integers are two primitive operations for the implementation of most public key crypto algorithms. Multiplication can be best performed using Karatsuba's divideandconquer technique. However, the modular reduction process is more complicated and tim ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
Multiplication and modular reduction of long integers are two primitive operations for the implementation of most public key crypto algorithms. Multiplication can be best performed using Karatsuba's divideandconquer technique. However, the modular reduction process is more complicated and timeconsuming. Thus an efficient implementation of modular reduction operation is one of main factors affecting the performance of public key cryptosystems. In this paper, we investigate a method for speeding up modular reduction using more or less precomputation based on the modulus, and present implementation results of various algorithms including our proposed methods. 1 Introduction There are two approaches to reducing the computation time for modular exponentiation; reducing the number of modular multiplications required and reducing the computation time for modular multiplication. Since modular exponentiation requires hundreds of modular multiplications, a small improvement by the latter app...