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On a generalization of addition chains: Addition–multiplication chains
2008
Discrete Mathematics
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We show that some notations and facts on addition chains can be generalized to addition-multiplication chains. ...
In other words, we show that addition-multiplication chains resemble addition chains in many aspects. ...
The aim of this paper is to show that some facts on addition chains can be generalized to addition-multiplication chains. The paper is organized as follows. ...
doi:10.1016/j.disc.2007.04.015
fatcat:zh7mguaisvdd3eiozfpmnq4onm
Addition Chains Meet Postage Stamps: Reducing the Number of Multiplications
[article]
2014
arXiv
pre-print
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We introduce stamp chains. A stamp chain is a finite integer sequence that is both an addition chain and an additive 2-basis, i.e., a solution to the postage stamp problem. ...
We provide a simple method for converting known postage stamp solutions of length k into stamp chains of length k+1. Using stamp chains, we construct an algorithm that computes u(x^i) for i=1,... ...
An extremal stamp chain (of length k) is one that attains this maximum. Example 1. A 5 = {1, 2, 4, 8, 16} is an addition chain, and in fact a minimallength addition chain ending at 16. ...
arXiv:1310.7090v2
fatcat:zhtznp32uvd5tdmdzkfintrame
Multiplicative ergodicity of Laplace transforms for additive functional of Markov chains
[article]
2015
arXiv
pre-print
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We develop a general approach based on the use of the operator perturbation method. We apply our general results to two examples of Markov chains, including a linear autoregressive model. ...
We study properties of the Laplace transforms of non-negative additive functionals of Markov chains. ...
We namely state general conditions ensuring the multiplicative ergodicity of an additive functional of a geometrically ergodic Markov chain. ...
arXiv:1506.07659v3
fatcat:gjqmfytrsbh6xkxleu62iwqsxa
EFFICIENT GENERATION OF SHORTEST ADDITION-MULTIPLICATION CHAINS
2018
Journal of the Egyptian Mathematical Society
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The aim of this paper is to generalize some results on addition chains to addition-multiplication chains. The paper concerns with generating shortest addition-multiplication chains. ...
It first presents two methods for generating short addition-multiplication chains. Second, it presents an algorithm for generating a shortest addition-multiplication chain. ...
For examples:
Generating Shortest Addition-Multiplication Chains This section contains two subsections. ...
doi:10.21608/joems.2018.2691.1052
fatcat:wccvvumq4nconeoptebvqujz5a
Multiple constant multiplication by time-multiplexed mapping of addition chains
2004
Proceedings of the 41st annual conference on Design automation - DAC '04
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We propose a novel implementation of this circuit based on combining the addition chains of the constituent constants. ...
We compare the area and latency efficiency of this addition chain based approach against a straightforward approach based on a constant table and a full multiplier. ...
Tummeltshammer was supported through the University of Technology, Vienna, Austria by a fellowship for short-term scientific research abroad. ...
doi:10.1145/996566.996787
dblp:conf/dac/TummeltshammerHP04
fatcat:y46trq35afccvfsjfqqqtdxhhy
On Geometric Ergodicity of Additive and Multiplicative Transformation Based Markov Chain Monte Carlo in High Dimensions
[article]
2017
arXiv
pre-print
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transformations of a one-dimensional random variable drawn from any arbitrary distribution defined on a relevant support. ...
Two significant transformations associated with TMCMC are additive and multiplicative transformations. Combinations of additive and multiplicative transformations are also of much interest. ...
Acknowledgment We are sincerely grateful to three anonymous reviewers whose comments led to a much improved version of our manuscript. ...
arXiv:1312.0915v4
fatcat:a5zxel3wajanvdtsj5digqkyau
Error detection in addition chain based ECC Point Multiplication
2009
2009 15th IEEE International On-Line Testing Symposium
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The paper takes into account the algorithm for elliptic Curve Point Multiplication based on a sequence of additions called "addition chain" and shows how suitable modifications of the algorithms used for ...
One of the mode used to break the security of cryptosystem is the injection of a fault in the hardware realizing the cryptographic algorithm. ...
POINT MULTIPLICATION BY MEANS OF EUCLIDEAN ADDITION CHAIN One of the methods proposed to compute the multiplication presented in equation (3) is based on the Euclidean Addition Chain [7] . ...
doi:10.1109/iolts.2009.5196010
dblp:conf/iolts/PontarelliCRS09
fatcat:ih77iwx7tjd6zo4zcka3llqyq4
Generalized Additive Model Multiple Imputation by Chained Equations With Package ImputeRobust
2018
The R Journal
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We present a package for robust multiple imputation (the ImputeRobust package) that allows the use of generalized additive models for location, scale, and shape in the context of chained equations. ...
The paper describes the basics of the imputation technique which builds on a semi-parametric regression model (GAMLSS) and the algorithms and functions provided with the corresponding package. ...
Robust imputation with gamlss and mice The imputation method realized by the package ImputeRobust is based on a generalized additive model for location, scale and shape of the variable to be imputed. ...
doi:10.32614/rj-2018-014
fatcat:3zriu5o5s5b7bok5rp6qn6k4ai
Euclidean addition chains scalar multiplication on curves with efficient endomorphism
2018
Journal of Cryptographic Engineering
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Euclidean addition chains scalar multiplication on curves with efficient endomorphism. ...
In order to do so we generalize results from [21] on the relation of random Euclidean chains generation and elliptic curve point distribution obtained from those chains. ...
To obtain a proven security of n bits, that is to say to be able to guarantee that one can generate 2 n different chains computing 2 n different points, one can work with chains of length n but has to ...
doi:10.1007/s13389-018-0190-0
fatcat:g4eobbjb7rd3dnnxzz6dyn27rm
Effect of improved multiplication efficiency on exponentiation algorithms derived from addition chains
1986
Mathematics of Computation
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The interaction between the efficiency of the basic multiplication algorithm and the addition chain used to compute x" is studied. ...
We conclude that either repeated multiplication by x or repeated squaring should be used and the provenance of each technique is established. ...
Next we look at some exponentiation algorithms based on addition chains, (a) Repeated Multiplication by x. (c) Factor Chain. ...
doi:10.1090/s0025-5718-1986-0829630-0
fatcat:ppjp34bhtjb2dpmjkthvnkxppa
Effect of Improved Multiplication Efficiency on Exponentiation Algorithms Derived from Addition Chains
1986
Mathematics of Computation
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The interaction between the efficiency of the basic multiplication algorithm and the addition chain used to compute x" is studied. ...
We conclude that either repeated multiplication by x or repeated squaring should be used and the provenance of each technique is established. ...
Next we look at some exponentiation algorithms based on addition chains, (a) Repeated Multiplication by x. (c) Factor Chain. ...
doi:10.2307/2007998
fatcat:hamml6gsrfgofej6jvvougx7oq
Random Euclidean Addition Chain Generation and Its Application to Point Multiplication
[chapter]
2010
Lecture Notes in Computer Science
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In this paper, we use a similar approach. We propose to modify the key generation process using a small Euclidean addition chain c instead of a scalar k. ...
We propose two different ways to generate short Euclidean addition chains and give a first theoretical analysis of the size and distribution of the obtained keys. ...
This section is dedicated to a specific scalar multiplication algorithm based on Euclidean addition chains.
Euclidean Addition Chains Definition 1. ...
doi:10.1007/978-3-642-17401-8_18
fatcat:6bj6dwcl5zhnpmgq3eb7lf47u4
Multiplicative ergodicity of laplace transforms for additive functional of markov chains with application to age-dependent branching process
[article]
2017
arXiv
pre-print
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Our approach is related to the behaviour of the Laplace transform of nonnegative additive functional of Markov chains and require weak moment assumption (no exponential moment is needed). ...
We suppose here that the lifetimes of the cells are dependent random variables, that the numbers of new cells are random and dependent. ...
For this, we study this multiplicative ergodicity property in the context of additive functional of Markov chains with Markov kernel P and initial law µ. ...
arXiv:1701.00718v1
fatcat:o76o3ajvnbaoxcacqro5eag4wu
A New Method of Generating Optimal Addition Chain Based on Graph
2017
International Journal of Mathematical Sciences and Computing
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the move from one number to another number in the addition chain. ...
The problem of determining correct sequence of multiplications requires in performing modular exponentiation can be elegantly formulated using the concept of addition chains. ...
The method generates an algorithm which provides the multiplication logic using a multiplication circuit. Daniel J. ...
doi:10.5815/ijmsc.2017.02.04
fatcat:3xji5gyls5hl7irrsyqmpulw2m
Multiplicative ergodicity of Laplace transforms for additive functional of Markov chains
2019
E S A I M: Probability & Statistics
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In this respect, a key property is the multiplicative ergodicity, which deals with the asymptotic behaviour of some Laplace-type transform of nonnegative additive functional of a Markov chain. ...
We establish a spectral version of this multiplicative ergodicity property in a general framework. Our approach is based on the use of the operator perturbation method. ...
In this model, at each step, either we follow a Markov chain Z = (Z n ) n (with probability (1 − α)) or we generate an independent random variable with distribution the invariant probability measure of ...
doi:10.1051/ps/2019003
fatcat:sr2hlax3tfexxcu2tlh7324omi
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