Minimal weight digit set conversions

Copyright © 2004 IEEEWe consider the problem of recoding a number to minimize the number of nonzero digits in its representation, that is, to minimize the weight of the representation. A general sliding window scheme is described that extends minimal binary sliding window conversion to arbitrary radix and to encompass signed digit sets. This new conversion expresses a number of known recoding techniques as special cases. Proof that this scheme achieves minimal weight for a given digit set is provided and results concerning the theoretical average and worst-case weight are derived.Braden Phillips and Neil Burges

Optimal Average Joint Hamming Weight and Minimal Weight Conversion of d Integers

In this paper, we propose the minimal joint Hamming weight conversion for any binary expansions of $d$ integers. With redundant representations, we may represent a number by many expansions, and the minimal joint Hamming weight conversion is the algorithm to select the expansion that has the least joint Hamming weight. As the computation time of the cryptosystem strongly depends on the joint Hamming weight, the conversion can make the cryptosystem faster. Most of existing conversions are limited to some specific representations, and are difficult to apply to other representations. On the other hand, our conversion is applicable to any binary expansions. The proposed can explore the minimal average weights in a class of representation that have not been found. One of the most interesting results is that, for the expansion of integer pairs when the digit set is $\{0, \pm 1, \pm 3\}$, we show that the minimal average joint Hamming weight is $0.3575$. This improves the upper bound value, $0.3616$, proposed by Dahmen, Okeya, and Takagi

Optimality of the Width-ww Non-adjacent Form: General Characterisation and the Case of Imaginary Quadratic Bases

Efficient scalar multiplication in Abelian groups (which is an important operation in public key cryptography) can be performed using digital expansions. Apart from rational integer bases (double-and-add algorithm), imaginary quadratic integer bases are of interest for elliptic curve cryptography, because the Frobenius endomorphism fulfils a quadratic equation. One strategy for improving the efficiency is to increase the digit set (at the prize of additional precomputations). A common choice is the width\nbd-$w$ non-adjacent form (\wNAF): each block of $w$ consecutive digits contains at most one non-zero digit. Heuristically, this ensures a low weight, i.e.\ number of non-zero digits, which translates in few costly curve operations. This paper investigates the following question: Is the \wNAF{}-expansion optimal, where optimality means minimising the weight over all possible expansions with the same digit set? The main characterisation of optimality of \wNAF{}s can be formulated in the following more general setting: We consider an Abelian group together with an endomorphism (e.g., multiplication by a base element in a ring) and a finite digit set. We show that each group element has an optimal \wNAF{}-expansion if and only if this is the case for each sum of two expansions of weight 1. This leads both to an algorithmic criterion and to generic answers for various cases. Imaginary quadratic integers of trace at least 3 (in absolute value) have optimal \wNAF{}s for $w\ge 4$. The same holds for the special case of base $(\pm 3\pm\sqrt{-3})/2$ and $w\ge 2$, which corresponds to Koblitz curves in characteristic three. In the case of $\tau=\pm1\pm i$, optimality depends on the parity of $w$. Computational results for small trace are given

Framework for the Analysis of the Adaptability, Extensibility, and Scalability of Semantic Information Integration and the Context Mediation Approach

Technological advances such as Service Oriented Architecture (SOA) have increased the feasibility and importance of effectively integrating information from an ever widening number of systems within and across enterprises. A key difficulty of achieving this goal comes from the pervasive heterogeneity in all levels of information systems. A robust solution to this problem needs to be adaptable, extensible, and scalable. In this paper, we identify the deficiencies of traditional semantic integration approaches. The COntext INterchange (COIN) approach overcomes these deficiencies by declaratively representing data semantics and using a mediator to create the necessary conversion programs from a small number of conversion rules. The capabilities of COIN is demonstrated using an example with 150 data sources, where COIN can automatically generate the over 22,000 conversion programs needed to enable semantic interoperability using only six parametizable conversion rules. This paper presents a framework for evaluating adaptability, extensibility, and scalability of semantic integration approaches. The application of the framework is demonstrated with a systematic evaluation of COIN and other commonly practiced approaches.This work has been supported, in part, by MITRE Corp., the MIT-MUST project, the Singapore-MIT Alliance, and Suruga Bank

Semantic Information Integration in the Large: Adaptability, Extensibility, and Scalability of the Context Mediation Approach

There is pressing need for effectively integrating information from an ever increasing number of available sources both on the web and in other existing systems. A key difficulty of achieving this goal comes from the pervasive heterogeneities in all levels of information systems. Existing and emerging technologies, such as the Web, ODBC, XML, and Web Services, provide essential capabilities in resolving heterogeneities in the hardware and software platforms, but they do not address the semantic heterogeneity of the data itself. A robust solution to this problem needs to be adaptable, extensible, and scalable. In this paper, we identify the deficiencies of traditional approaches that address this problem using hand-coded programs or require complete data standardization. The COntext INterchange (COIN) approach overcomes these deficiencies by declaratively representing data semantics and using a mediator to create the necessary conversion programs using a small number of conversion rules. The capabilities of COIN is demonstrated using an intelligence information integration example consisting of 150 data sources, where COIN can automatically generate the over 22,000 conversion programs needed to enable semantic integration using only six parametizable conversion rules. This paper makes a unique contribution by providing a systematic evaluation of COIN and other commonly practiced approaches

Semantic Information Integration in the Large: Adaptability, Extensibility, and Scalability of the Context Mediation Approach

There is pressing need for effectively integrating information from an ever increasing number of available sources both on the web and in other existing systems. A key difficulty of achieving this goal comes from the pervasive heterogeneities in all levels of information systems. Existing and emerging technologies, such as the Web, ODBC, XML, and Web Services, provide essential capabilities in resolving heterogeneities in the hardware and software platforms, but they do not address the semantic heterogeneity of the data itself. A robust solution to this problem needs to be adaptable, extensible, and scalable. In this paper, we identify the deficiencies of traditional approaches that address this problem using hand-coded programs or require complete data standardization. The COntext INterchange (COIN) approach overcomes these deficiencies by declaratively representing data semantics and using a mediator to create the necessary conversion programs using a small number of conversion rules. The capabilities of COIN is demonstrated using an intelligence information integration example consisting of 150 data sources, where COIN can automatically generate the over 22,000 conversion programs needed to enable semantic integration using only six parametizable conversion rules. This paper makes a unique contribution by providing a systematic evaluation of COIN and other commonly practiced approaches