Energy-Efficient Software Implementation of Long Integer Modular Arithmetic

Großschädl, Johann; Avanzi, Roberto; Savaş, Erkay; Tillich, Stefan

doi:10.1007/11545262_6

Cited by 41 publications

(62 citation statements)

References 15 publications

(26 reference statements)

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“…A simple and easy-to-implement technique for multi-precision multiplication is the operand scanning method [10], also known as schoolbook method [7]. The operand scanning method, as specified in [7,Algorithm 1], has a characteristic nested-loop structure with an outer loop iterates through the s words b i of the operand B, starting with the least-significant word b 0 .…”

Section: Operand Scanning Methodsmentioning

confidence: 99%

“…The operand scanning method, as specified in [7,Algorithm 1], has a characteristic nested-loop structure with an outer loop iterates through the s words b i of the operand B, starting with the least-significant word b 0 . In the inner loop, b i is multiplied with a word a j of operand A and the 2w-bit product is added to the intermediate result obtained so far.…”

Section: Operand Scanning Methodsmentioning

confidence: 99%

“…add-with-carry) instructions. The operand scanning method is fairly easy to program in a high-level language like C or Java [7], but is generally less efficient than the product scanning method (described below) if both are written in Assembly language. In summary, when multiplying two s-word operands, the operand scanning method has to execute s 2 mul, 4s 2 add (resp.…”

Section: Operand Scanning Methodsmentioning

confidence: 99%

“…The product scanning method (specified in [7, Algorithm 2]) comprises two nested loops; one computes the lower half of the result and the second contributes the upper half. As the name suggests, the two outer loops of the product scanning method move through the product itself, starting at the least significant word [6,7]. More precisely, the product is obtained one word at a time, whereby the i-th word contains all partial products a j · b k with j + k = i.…”

Section: Product Scanning Methodsmentioning

confidence: 99%

“…Besides reducing the delay of public-key primitives and protocols, fast multi-precision arithmetic is crucial for another reason, namely energy efficiency. In general, the energy consumption of cryptographic software executed on a microprocessor increases proportionally with the execution time [7]. This, together with the fact that a large portion of embedded systems are battery powered, makes a good case for minimizing execution time, even if the target application does not impose stringent delay constraints on cryptographic primitives.…”

Section: Requirementsmentioning

confidence: 99%

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Reverse Product-Scanning Multiplication and Squaring on 8-Bit AVR Processors

Liu

Seo

Groβschädl

et al. 2015

Information and Communications Security

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Abstract. High performance, small code size, and good scalability are important requirements for software implementations of multi-precision arithmetic algorithms to fit resource-limited embedded systems. In this paper, we describe optimization techniques to speed up multi-precision multiplication and squaring on the AVR ATmega series of 8-bit microcontrollers. First, we present a new approach to perform multi-precision multiplication, called Reverse Product Scanning (RPS), that resembles the hybrid technique of Gura et al., but calculates the byte-products in the inner loop in reverse order. The RPS method processes four bytes of the two operands in each iteration of the inner loop and employs two carry-catcher registers to minimize the number of add instructions. We also describe an optimized algorithm for multi-precision squaring based on the RPS technique that is, depending on the operand length, up to 44.3% faster than multiplication. Our AVR Assembly implementations of RPS multiplication and RPS squaring occupy less than 1 kB of code space each and are written in a parameterized fashion so that they can support operands of varying length without recompilation. Despite this high level of flexibility, our RPS multiplication outperforms the looped variant of Hutter et al.'s operand-caching technique and saves between 40 and 51% of code size. We also combine our RPS multiplication and squaring routines with Karatsuba's method to further reduce execution time. When executed on an ATmega128 processor, the "karatsubarized RPS method" needs only 85 k clock cycles for a 1024-bit multiplication (or 48 k cycles for a squaring). These results show that it is possible to achieve high performance without sacrificing code size or scalability.

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Section: Operand Scanning Methodsmentioning

confidence: 99%

Section: Operand Scanning Methodsmentioning

confidence: 99%

Section: Operand Scanning Methodsmentioning

confidence: 99%

Section: Product Scanning Methodsmentioning

confidence: 99%

Section: Requirementsmentioning

confidence: 99%

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Reverse Product-Scanning Multiplication and Squaring on 8-Bit AVR Processors

Liu

Seo

Groβschädl

et al. 2015

Information and Communications Security

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Green Cloud Computing and Environmental Sustainability

Kumar

Buyya

2012

Harnessing Green It

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Cloud computing is a highly scalable and cost-effective infrastructure for running HPC, enterprise and Web applications. However, the growing demand of Cloud infrastructure has drastically increased the energy consumption of data centers, which has become a critical issue. High energy consumption not only translates to high operational cost, which reduces the profit margin of Cloud providers, but also leads to high carbon emissions which is not environmentally friendly. Hence, energy-efficient solutions are required to minimize the impact of Cloud computing on the environment. In order to design such solutions, deep analysis of Cloud is required with respect to their power efficiency. Thus, in this chapter, we discuss various elements of Clouds which contribute to the total energy consumption and how it is addressed in the literature. We also discuss the implication of these solutions for future research directions to enable green Cloud computing. The chapter also explains the role of Cloud users in achieving this goal. and generally include specific component for developing applications, advanced services for application monitoring, management, and reporting.

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Green Software

and

Agrawal

2012

Harnessing Green It

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In a career spanning over three decades in academia and industry, Dr Murugesan has led several innovative IT projects, provided leadership in teaching and research, and consulted to business, industry and educational institutions.His work has focussed on the development, application and management of IT with expertise and interests spanning a range of areas, including green computing, cloud computing, Web 2.0 and 3.0, mobile computing applications, Web engineering, e-business and IT for emerging markets. He has over 150 publications which include journal and conference papers, executive reports, edited books, IEEE CS EssentialSets and e-mail advisories. He has developed and delivered professional certificate training programs on green IT and cloud computing. He serves as Associate Editor-in-Chief of IEEE's IT Professional magazine and on the editorial boards of other international journals. He also edits and contributes to the IT in Emerging Markets Department of IT Professional .H e is a Fellow of the Australian Computer Society, a Fellow of IETE, a Senior Member of IEEE and a distinguished visitor and tutorial speaker of the IEEE Computer Society. You can follow him on Twitter @santweets and at LinkedIn and contact him at

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Energy-Efficient Software Implementation of Long Integer Modular Arithmetic

Cited by 41 publications

References 15 publications

Reverse Product-Scanning Multiplication and Squaring on 8-Bit AVR Processors

Reverse Product-Scanning Multiplication and Squaring on 8-Bit AVR Processors

Green Cloud Computing and Environmental Sustainability

Green Software

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